title: "[Branch-Bound] Circle Permutation" date: 2022-04-17 11:03:00 tags:
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<h1>Circle Permutation</h1>
<div contenteditable="true" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n2" cid="n2" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.968ex" height="2.828ex" role="img" focusable="false" viewBox="0 -972 1312 1250" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.629ex;" class="in-text-selection"><defs><path id="MJX-1-TEX-N-40" d="M56 347Q56 429 86 498T164 612T270 680T386 705Q522 705 622 603T722 349Q722 126 608 126Q541 126 513 176Q512 177 512 179T510 182L509 183Q508 183 503 177T487 163T464 146T429 132T385 126Q311 126 251 186T190 347Q190 448 251 508T385 568Q426 568 460 548T509 511T531 479H555Q580 479 582 478Q586 477 587 468Q588 454 588 338V260Q588 200 593 182T619 163Q641 163 655 178T674 223T680 273T682 325V330Q682 426 647 500Q611 569 544 618T388 668Q271 668 184 577T96 347Q96 216 180 121T396 26Q421 26 446 28T493 34T535 43T573 52T605 63T629 72T647 80T657 84H716Q722 78 722 74Q722 65 675 45T547 7T392 -11Q255 -11 156 90T56 347ZM274 347Q274 266 308 214T390 162Q420 162 449 182T498 235L504 245V449L498 459Q453 532 387 532Q347 532 311 483T274 347Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="menclose"><g transform="translate(267, 0)"><g data-mml-node="mstyle"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="40" xlink:href="#MJX-1-TEX-N-40"></use></g></g></g></g></g><rect x="33.5" y="-244.5" width="1245" height="1183" fill="none" stroke-width="67"></rect></g></g></g></svg></mjx-container></div></div> <h2>Description</h2> <h2>Input</h2> <p>对于给定的 n 个圆，设计一个优先队列式分支限界法，计算 n 个圆的最佳排列方案，使 其长度达到最小。</p> <h2>Output</h2> <p>由文件 input.txt 给出输入数据。第一行有 1 个正整数 n (1≤n≤20)。接下来的 1 行有 n 个数，表示 n 个圆的半径。</p> <h2>Sample</h2> <p>将计算出的最小圆排列的长度输出到文件 output.txt。</p> <p><strong>输入文件示例</strong></p> <p>input.txt</p> <p>3 1 1 2</p> <p><strong>输出文件示例</strong></p> <p>output.txt</p> <p>7.65685</p> <h2>Analysis</h2> <h3>Brute-Force Method (DFS/BFS)</h3> <p>对于输入，题目给定 <code>n个圆</code>，要求给出 <code>长度最小的圆的排列</code>。</p> <p>则不妨考虑一个 <code>Brute-Force Method</code>思路：</p> <ul> <li>生成 <code>n个圆的排列</code>方案</li> <li>计算 <code>所有排列方案</code>中的 <code>长度</code>，得到 <code>最小长度</code></li> </ul> <hr> <blockquote><p>推导： <code>两圆相切时圆心的水平距公式</code></p> </blockquote> <p>对于 <code>生成n个圆的所有排列</code>来说，这是容易的。</p> <p>现在我们考虑，如果确实给定了 <code>某个特定的圆排列方案</code>，我们应如何 <code>计算出该方案的长度</code>。</p> <p>我们设 <code>两个圆</code>的 <code>半径</code> 分别为 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.008ex" height="1.339ex" role="img" focusable="false" viewBox="0 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data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-47-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-47-TEX-N-31"></use></g></g></g></g></svg></mjx-container><script type="math/tex">r_1</script> 和 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.008ex" height="1.339ex" role="img" focusable="false" viewBox="0 -442 887.6 592" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-48-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-48-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-48-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-48-TEX-N-32"></use></g></g></g></g></svg></mjx-container><script type="math/tex">r_2</script>，设 <code>两个圆的圆心水平距离</code>为 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.176ex" height="1.593ex" role="img" focusable="false" viewBox="0 -694 520 704" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-49-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-49-TEX-I-1D451"></use></g></g></g></svg></mjx-container><script type="math/tex">d</script></p> <ul> <li><p>考虑 <code>只有2个圆</code>的情况</p> <ul> <li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.033ex" height="1.658ex" role="img" focusable="false" viewBox="0 -583 3108.7 733" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-50-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-50-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-50-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-50-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-50-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-50-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1165.3,0)"><use data-c="3D" xlink:href="#MJX-50-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(2221.1,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-50-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-50-TEX-N-32"></use></g></g></g></g></svg></mjx-container><script type="math/tex">r_1 = r_2</script></p> <p>该情况下，<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.975ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 4851.1 844" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-51-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-51-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-51-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-51-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-51-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-51-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-51-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(797.8,0)"><use data-c="3D" xlink:href="#MJX-51-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(1853.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-51-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-51-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(2963.3,0)"><use data-c="2B" xlink:href="#MJX-51-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(3963.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-51-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-51-TEX-N-32"></use></g></g></g></g></svg></mjx-container><script type="math/tex">d = r_1 + r_2</script></p> </li> <li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.033ex" height="2.106ex" role="img" focusable="false" viewBox="0 -716 3108.7 931" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.486ex;"><defs><path id="MJX-52-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-52-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-52-TEX-N-2260" d="M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z"></path><path id="MJX-52-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-52-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-52-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1165.3,0)"><use data-c="2260" xlink:href="#MJX-52-TEX-N-2260"></use></g><g data-mml-node="msub" transform="translate(2221.1,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-52-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-52-TEX-N-32"></use></g></g></g></g></svg></mjx-container><script type="math/tex">r_1 \ne r_2</script></p> <ul> <li>可以 <code>部分遮盖</code> 另一个圆</li> </ul> <ul> <li>可以 <code>完全遮盖</code> 另一个圆</li> </ul> <p> </p> <p>则我们得出 <code>公式</code></p> <div contenteditable="true" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n52" cid="n52" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div 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xlink:href="#MJX-3-TEX-N-2B"></use></g><g data-mml-node="mo" transform="translate(7055.6,0)"><use data-c="22EF" xlink:href="#MJX-3-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(8449.8,0)"><use data-c="2B" xlink:href="#MJX-3-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(9450,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-3-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-3-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(10408.3,0)"><use data-c="29" xlink:href="#MJX-3-TEX-N-29"></use></g></g></g></g></g></g></svg></mjx-container></div></div> <p>所以，我们需要一些额外信息：<code>各个圆的圆心水平坐标</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.719ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6064 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-53-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-53-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-53-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-53-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path><path id="MJX-53-TEX-N-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z"></path><path id="MJX-53-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-53-TEX-N-5B" d="M118 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="63" xlink:href="#MJX-53-TEX-N-63"></use><use data-c="65" xlink:href="#MJX-53-TEX-N-65" transform="translate(444,0)"></use><use data-c="6E" xlink:href="#MJX-53-TEX-N-6E" transform="translate(888,0)"></use><use data-c="74" xlink:href="#MJX-53-TEX-N-74" transform="translate(1444,0)"></use><use data-c="65" xlink:href="#MJX-53-TEX-N-65" transform="translate(1833,0)"></use><use data-c="72" xlink:href="#MJX-53-TEX-N-72" transform="translate(2277,0)"></use><use data-c="73" xlink:href="#MJX-53-TEX-N-73" transform="translate(2669,0)"></use><use data-c="5B" xlink:href="#MJX-53-TEX-N-5B" transform="translate(3063,0)"></use><use data-c="30" xlink:href="#MJX-53-TEX-N-30" transform="translate(3341,0)"></use><use data-c="2E" xlink:href="#MJX-53-TEX-N-2E" transform="translate(3841,0)"></use><use data-c="2E" xlink:href="#MJX-53-TEX-N-2E" transform="translate(4119,0)"></use><use data-c="6E" xlink:href="#MJX-53-TEX-N-6E" transform="translate(4397,0)"></use><use data-c="2D" xlink:href="#MJX-53-TEX-N-2D" transform="translate(4953,0)"></use><use data-c="31" xlink:href="#MJX-53-TEX-N-31" transform="translate(5286,0)"></use><use data-c="5D" xlink:href="#MJX-53-TEX-N-5D" transform="translate(5786,0)"></use></g></g></g></svg></mjx-container><script type="math/tex">\t{centers[0..n-1]}</script></p> <p>对于计算 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.188ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3619 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-54-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-54-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-54-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-54-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path><path id="MJX-54-TEX-N-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z"></path><path id="MJX-54-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-54-TEX-N-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path id="MJX-54-TEX-N-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="63" xlink:href="#MJX-54-TEX-N-63"></use><use data-c="65" xlink:href="#MJX-54-TEX-N-65" transform="translate(444,0)"></use><use data-c="6E" xlink:href="#MJX-54-TEX-N-6E" transform="translate(888,0)"></use><use data-c="74" xlink:href="#MJX-54-TEX-N-74" transform="translate(1444,0)"></use><use data-c="65" xlink:href="#MJX-54-TEX-N-65" transform="translate(1833,0)"></use><use data-c="72" xlink:href="#MJX-54-TEX-N-72" transform="translate(2277,0)"></use><use data-c="73" xlink:href="#MJX-54-TEX-N-73" transform="translate(2669,0)"></use><use data-c="5B" xlink:href="#MJX-54-TEX-N-5B" transform="translate(3063,0)"></use><use data-c="5D" xlink:href="#MJX-54-TEX-N-5D" transform="translate(3341,0)"></use></g></g></g></svg></mjx-container><script type="math/tex">\t{centers[]}</script> 数组，我们知道 <code>某个圆的圆心水平坐标</code>肯定与 <code>该圆左侧的圆的圆心水平坐标</code> <code>存在关系</code></p> <blockquote><p>n.b. 我们这里说的是 <code>该圆左侧的圆</code>，而不是 <code>该圆左边的那1个圆</code>。</p> <p>实际上，我们如果要计算 <code>某个圆的圆心水平坐标</code>，需要把 <code>该圆左侧的所有圆</code>都做考虑。</p> </blockquote> <p>设我们有 <code>圆的排列</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="11.634ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 5142.4 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-69-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-69-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-69-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-69-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-69-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-69-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-69-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-69-TEX-N-30"></use></g></g><g data-mml-node="msub" transform="translate(887.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-69-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-69-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1941.8,0)"><use data-c="22EF" xlink:href="#MJX-69-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(3280.4,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-69-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-69-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-69-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-69-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_0r_1\cdots r_{n-1}</script></p> <p>对 <code>圆的数量</code> 运用 <code>归纳法</code>：</p> <ul> <li><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.506ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2433.6 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-56-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-56-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-56-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-56-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(877.8,0)"><use data-c="3D" xlink:href="#MJX-56-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1933.6,0)"><use data-c="31" xlink:href="#MJX-56-TEX-N-31"></use></g></g></g></svg></mjx-container><script type="math/tex">n = 1</script>，我们假设 <code>第一个圆</code>位于 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.029ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2222.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-57-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-57-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-57-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-57-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-57-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="30" xlink:href="#MJX-57-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="2C" xlink:href="#MJX-57-TEX-N-2C"></use></g><g data-mml-node="mn" transform="translate(1333.7,0)"><use data-c="30" xlink:href="#MJX-57-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(1833.7,0)"><use data-c="29" xlink:href="#MJX-57-TEX-N-29"></use></g></g></g></svg></mjx-container><script type="math/tex">(0,0)</script> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="18.934ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8368.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-58-TEX-N-27F9" d="M1218 514Q1218 525 1234 525Q1239 525 1242 525T1247 525T1251 524T1253 523T1255 520T1257 517T1260 512Q1297 438 1358 381T1469 300T1565 263Q1582 258 1582 250T1573 239T1536 228T1478 204Q1334 134 1260 -12Q1256 -21 1253 -22T1238 -24Q1218 -24 1218 -17Q1218 -13 1223 0Q1258 69 1309 123L1319 133H70Q56 140 56 153Q56 168 72 173H1363L1373 181Q1412 211 1490 250Q1489 251 1472 259T1427 283T1373 319L1363 327H710L707 328L390 327H72Q56 332 56 347Q56 360 70 367H1319L1309 377Q1276 412 1247 458T1218 514Z"></path><path id="MJX-58-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-58-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-58-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-58-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path><path id="MJX-58-TEX-N-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z"></path><path id="MJX-58-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-58-TEX-N-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path id="MJX-58-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-58-TEX-N-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path id="MJX-58-TEX-N-20" d=""></path><path id="MJX-58-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mstyle"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(278,0)"><use data-c="27F9" xlink:href="#MJX-58-TEX-N-27F9"></use></g><g data-mml-node="mstyle" transform="translate(1916,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mtext" transform="translate(2471.8,0)"><use data-c="63" xlink:href="#MJX-58-TEX-N-63"></use><use data-c="65" xlink:href="#MJX-58-TEX-N-65" transform="translate(444,0)"></use><use data-c="6E" xlink:href="#MJX-58-TEX-N-6E" transform="translate(888,0)"></use><use data-c="74" xlink:href="#MJX-58-TEX-N-74" transform="translate(1444,0)"></use><use data-c="65" xlink:href="#MJX-58-TEX-N-65" transform="translate(1833,0)"></use><use data-c="72" xlink:href="#MJX-58-TEX-N-72" transform="translate(2277,0)"></use><use data-c="73" xlink:href="#MJX-58-TEX-N-73" transform="translate(2669,0)"></use><use data-c="5B" xlink:href="#MJX-58-TEX-N-5B" transform="translate(3063,0)"></use><use data-c="30" xlink:href="#MJX-58-TEX-N-30" transform="translate(3341,0)"></use><use data-c="5D" xlink:href="#MJX-58-TEX-N-5D" transform="translate(3841,0)"></use><use data-c="20" xlink:href="#MJX-58-TEX-N-20" transform="translate(4119,0)"></use><use data-c="3D" xlink:href="#MJX-58-TEX-N-3D" transform="translate(4369,0)"></use><use data-c="20" xlink:href="#MJX-58-TEX-N-20" transform="translate(5147,0)"></use><use data-c="30" xlink:href="#MJX-58-TEX-N-30" transform="translate(5397,0)"></use></g></g></g></svg></mjx-container><script type="math/tex">\implies \t{centers[0] = 0}</script></li> <li><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.506ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2433.6 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-59-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-59-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-59-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-59-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(877.8,0)"><use data-c="3D" xlink:href="#MJX-59-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1933.6,0)"><use data-c="32" xlink:href="#MJX-59-TEX-N-32"></use></g></g></g></svg></mjx-container><script type="math/tex">n=2</script>，我们可以使用 <code>两圆相切时的圆心的水平距公式</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="37.53ex" height="2.398ex" role="img" focusable="false" viewBox="0 -765.7 16588.3 1060" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-60-TEX-N-27F9" d="M1218 514Q1218 525 1234 525Q1239 525 1242 525T1247 525T1251 524T1253 523T1255 520T1257 517T1260 512Q1297 438 1358 381T1469 300T1565 263Q1582 258 1582 250T1573 239T1536 228T1478 204Q1334 134 1260 -12Q1256 -21 1253 -22T1238 -24Q1218 -24 1218 -17Q1218 -13 1223 0Q1258 69 1309 123L1319 133H70Q56 140 56 153Q56 168 72 173H1363L1373 181Q1412 211 1490 250Q1489 251 1472 259T1427 283T1373 319L1363 327H710L707 328L390 327H72Q56 332 56 347Q56 360 70 367H1319L1309 377Q1276 412 1247 458T1218 514Z"></path><path id="MJX-60-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-60-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path 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data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-61-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-61-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,-86.5)"><use data-c="221A" xlink:href="#MJX-61-TEX-N-221A"></use></g><rect width="1775.1" height="60" x="853" y="653.5"></rect></g></g></g></g></svg></mjx-container><script type="math/tex">\r{n =3，\t{centers[2]} \overset{?}{=} \t{centers[1] + } 2\sqrt{r_1r_2}}</script></li> </ul> <p>当我们考虑 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.506ex" height="1.69ex" role="img" focusable="false" viewBox="0 -665 2433.6 747" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-62-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-62-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-62-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-62-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(877.8,0)"><use data-c="3D" xlink:href="#MJX-62-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1933.6,0)"><use data-c="33" xlink:href="#MJX-62-TEX-N-33"></use></g></g></g></svg></mjx-container><script type="math/tex">n=3</script>时，存在一些问题。</p> <p>不妨假设，<code>第2个圆</code>是 <code>无穷小</code>的，以至于 <code>我们可以忽略掉</code> <code>第2个圆</code>，在这个假设下，我们可以得到 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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xlink:href="#MJX-63-TEX-N-6E" transform="translate(888,0)"></use><use data-c="74" xlink:href="#MJX-63-TEX-N-74" transform="translate(1444,0)"></use><use data-c="65" xlink:href="#MJX-63-TEX-N-65" transform="translate(1833,0)"></use><use data-c="72" xlink:href="#MJX-63-TEX-N-72" transform="translate(2277,0)"></use><use data-c="73" xlink:href="#MJX-63-TEX-N-73" transform="translate(2669,0)"></use><use data-c="5B" xlink:href="#MJX-63-TEX-N-5B" transform="translate(3063,0)"></use><use data-c="30" xlink:href="#MJX-63-TEX-N-30" transform="translate(3341,0)"></use><use data-c="5D" xlink:href="#MJX-63-TEX-N-5D" transform="translate(3841,0)"></use></g></g></g><g data-mml-node="mo" transform="translate(9738.2,0)"><use data-c="2B" xlink:href="#MJX-63-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(10738.4,0)"><use data-c="32" xlink:href="#MJX-63-TEX-N-32"></use></g><g data-mml-node="msqrt" transform="translate(11238.4,0)"><g transform="translate(853,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-63-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-63-TEX-N-30"></use></g></g><g data-mml-node="msub" transform="translate(887.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-63-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-63-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,-94.3)"><use data-c="221A" xlink:href="#MJX-63-TEX-N-221A"></use></g><rect width="1775.1" height="60" x="853" y="645.7"></rect></g></g></g></svg></mjx-container><script type="math/tex">\t{centers[2] = \b{centers[0]}} + 2\sqrt{r_0r_2}</script></p> <p>同理，我们可以假设 <code>某个圆左侧的所有圆 (除第一个圆外)</code>是 <code>无穷小</code>的，这样我们就可以 <code>忽略掉这些圆</code>。</p> <p>进而得到 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" 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data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-64-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-64-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-64-TEX-N-31"></use></g></g></g><g data-mml-node="msub" transform="translate(1806.1,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-64-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-64-TEX-I-1D458"></use></g></g></g></g><g data-mml-node="mo" transform="translate(0,-115.5)"><use data-c="221A" xlink:href="#MJX-64-TEX-N-221A"></use></g><rect width="2708.5" height="60" x="853" y="624.5"></rect></g></g></g></svg></mjx-container><script type="math/tex">\t{centers[k] = \b{centers[0]}} + 2\sqrt{r_{k-1} r_{k}}</script></p> <blockquote><p>这里并不是说 <code>某圆</code> 必须要和 <code>第一个圆</code> 发生 <code>相切</code>，只是拿 <code>第一个圆</code>进行举例。</p> <p>实际上，我们可以假设 <code>任何圆</code>是 <code>无穷小的</code>，从而使得 <code>我们想要的两个圆</code> 发生 <code>相切</code></p> </blockquote> <p>换句话说：<code>某个圆</code>有可能与 <code>该圆左侧的所有圆</code>发生 <code>相切</code>。</p> <blockquote><p>而不仅仅是与 <code>该圆左边的那一个圆</code>发生 <code>相切</code></p> </blockquote> <p>则可得</p> <div contenteditable="true" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n85" cid="n85" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="58.226ex" height="2.415ex" role="img" focusable="false" viewBox="0 -783.7 25736 1067.4" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.642ex;" class="in-text-selection"><defs><path 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xlink:href="#MJX-4-TEX-N-22EF"></use></g><g data-mml-node="mi" transform="translate(23168.6,0)"><use data-c="1D456" xlink:href="#MJX-4-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(23735.8,0)"><use data-c="2212" xlink:href="#MJX-4-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(24736,0)"><use data-c="31" xlink:href="#MJX-4-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(25236,0)"><use data-c="7D" xlink:href="#MJX-4-TEX-N-7D"></use></g></g></g></g></g></g></svg></mjx-container></div></div> <blockquote><p>n.b. 由于我们的公式是 <code>假设两圆之间相切</code>而计算出的 <code>两圆的圆心距</code>。</p> <p>实际上，我们得到的 <code>centers[k]</code> 是 <code>两个圆恰好发生相切</code>时的 <code>阈值</code>。</p> <p>如果 <code>圆A</code>和 <code>圆B</code> 之间存在 <code>某个足够大的圆C</code>，使得： 当 <code>圆B</code>和 <code>圆A</code> 发生 <code>相切时</code>， <code>圆B</code>和 <code>圆C</code>必定早已 <code>重叠</code> （而题目不允许圆发生重叠）</p> <p>我们通过 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-85-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(878,0)"><use data-c="1D44E" xlink:href="#MJX-85-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(1407,0)"><use data-c="1D465" xlink:href="#MJX-85-TEX-I-1D465"></use></g></g></g></svg></mjx-container><script type="math/tex">max</script> 函数可以非常巧妙地避开这个情形。</p> <p>若 <code>在圆A和圆B之间</code>确实存在 <code>足够大的圆C</code>，则有：</p> <div contenteditable="true" spellcheck="false" 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transform="translate(7640.4,0)"><g transform="translate(853,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-5-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(378,-150) scale(0.707)"><use data-c="1D436" xlink:href="#MJX-5-TEX-I-1D436"></use></g></g></g></g><g data-mml-node="msub" transform="translate(1216.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-5-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(378,-150) scale(0.707)"><use data-c="1D435" xlink:href="#MJX-5-TEX-I-1D435"></use></g></g></g></g></g><g data-mml-node="mo" transform="translate(0,-97.3)"><use data-c="221A" xlink:href="#MJX-5-TEX-N-221A"></use></g><rect width="2432.8" height="60" x="853" y="642.7"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0,-667.3)"><g data-mml-node="mtd" transform="translate(4583.7,0)"></g><g data-mml-node="mtd" transform="translate(4583.7,0)"><g data-mml-node="mstyle" fill="red" stroke="red"><g data-mml-node="mo"><use data-c="2265" xlink:href="#MJX-5-TEX-N-2265"></use></g><g data-mml-node="mtext" transform="translate(1055.8,0)"><use data-c="63" xlink:href="#MJX-5-TEX-N-63"></use><use data-c="65" xlink:href="#MJX-5-TEX-N-65" transform="translate(444,0)"></use><use data-c="6E" xlink:href="#MJX-5-TEX-N-6E" transform="translate(888,0)"></use><use data-c="74" xlink:href="#MJX-5-TEX-N-74" transform="translate(1444,0)"></use><use data-c="65" xlink:href="#MJX-5-TEX-N-65" transform="translate(1833,0)"></use><use data-c="72" xlink:href="#MJX-5-TEX-N-72" transform="translate(2277,0)"></use><use data-c="73" xlink:href="#MJX-5-TEX-N-73" transform="translate(2669,0)"></use></g><g data-mml-node="mo" transform="translate(4118.8,0)"><use data-c="5B" xlink:href="#MJX-5-TEX-N-5B"></use></g><g data-mml-node="msub" transform="translate(4396.8,0)"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(378,-152.7) scale(0.707)"><use data-c="1D434" xlink:href="#MJX-5-TEX-I-1D434"></use></g></g><g data-mml-node="mo" transform="translate(5355.1,0)"><use data-c="5D" xlink:href="#MJX-5-TEX-N-5D"></use></g><g data-mml-node="mo" transform="translate(5855.3,0)"><use data-c="2B" xlink:href="#MJX-5-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(6855.6,0)"><use data-c="32" xlink:href="#MJX-5-TEX-N-32"></use></g><g data-mml-node="msqrt" transform="translate(7355.6,0)"><g transform="translate(853,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-5-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(378,-152.7) scale(0.707)"><use data-c="1D434" xlink:href="#MJX-5-TEX-I-1D434"></use></g></g></g></g><g data-mml-node="msub" transform="translate(1211.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-5-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(378,-150) scale(0.707)"><use data-c="1D435" xlink:href="#MJX-5-TEX-I-1D435"></use></g></g></g></g></g><g data-mml-node="mo" transform="translate(0,-92.7)"><use data-c="221A" xlink:href="#MJX-5-TEX-N-221A"></use></g><rect width="2427.8" height="60" x="853" y="647.3"></rect></g></g></g></g></g></g></g></g></g></g></svg></mjx-container></div></div> <p>此时，<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.477ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 1979 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-85-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-85-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-85-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(878,0)"><use data-c="1D44E" xlink:href="#MJX-85-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(1407,0)"><use data-c="1D465" xlink:href="#MJX-85-TEX-I-1D465"></use></g></g></g></svg></mjx-container><script type="math/tex">max</script>函数将会 <code>选择让 圆B和圆C发生相切</code>，而 <code>放弃让 圆B和圆A相切</code></p> <blockquote><p>由于我们的 <code>centers[k]</code>计算的是 <code>两圆相切时的阈值</code>：所以如果 <code>圆B</code> 与 <code>圆A</code> 和 <code>圆C</code> 能 <code>同时相切</code>，</p> <p>则说明 <code>圆A</code>和 <code>圆B</code>的 <code>大小相同的圆</code></p> </blockquote> </blockquote> <hr> <blockquote><p>计算 <code>圆的排列长度</code></p> </blockquote> <p>在已知 <code>各个圆的半径</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.626ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 4254.8 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-67-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-67-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-67-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-67-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-67-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-67-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-67-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-67-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1054.2,0)"><use data-c="22EF" xlink:href="#MJX-67-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(2392.9,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-67-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-67-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-67-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-67-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_0\cdots r_{n-1}</script> 和 <code>各个圆的圆心的水平坐标</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.545ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 4218.8 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-68-TEX-I-1D450" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path><path id="MJX-68-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-68-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-68-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-68-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-68-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D450" xlink:href="#MJX-68-TEX-I-1D450"></use></g><g data-mml-node="mn" transform="translate(466,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-68-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1036.2,0)"><use data-c="22EF" xlink:href="#MJX-68-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(2374.9,0)"><g data-mml-node="mi"><use data-c="1D450" xlink:href="#MJX-68-TEX-I-1D450"></use></g><g data-mml-node="TeXAtom" transform="translate(466,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-68-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-68-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-68-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">c_0 \cdots c_{n-1}</script> 后</p> <p>易得</p> <div contenteditable="true" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n104" cid="n104" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container 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transform="translate(31575.1,0)"><use data-c="2212" xlink:href="#MJX-6-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(32575.4,0)"><use data-c="31" xlink:href="#MJX-6-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(33075.4,0)"><use data-c="29" xlink:href="#MJX-6-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(33464.4,0)"><use data-c="7D" xlink:href="#MJX-6-TEX-N-7D"></use></g></g></g></g></g></g></svg></mjx-container></div></div> <p>该方法通过求 <code>最右边的区间端点</code>和 <code>最左边的区间端点</code>来求得 <code>圆的排列长度</code></p> <blockquote><p>n.b. <code>最右边的区间端点</code>并不一定是 <code>最后一个圆</code>所给出的。</p> <p>原因很简单，我们可以简单地假设 <code>最后一个圆</code>是 <code>无穷小</code>的，</p> <p>那么此时 <code>最右边的区间端点</code>就与 <code>最后一个圆</code>没有关系了。</p> </blockquote> <h3>BFS with Priority (Branch-Bound)</h3> <h4>Introduction to branch-bound</h4> <p>我们回顾一下，前面所讲的 <code>DFS</code> 和 <code>BFS</code> 本质上均属于 <code>Brute-Force Method</code>。</p> <p><code>DFS</code> 按照 <code>深度优先</code> 方式在 <code>一条路径</code>上进行 <code>不断深入进行搜索</code>。</p> <p><code>BFS</code> 按照 <code>广度优先</code> 方式在 <code>当前节点</code> 上进行 <code>拓展</code> 生成所有的 <code>子节点</code>，然后 <code>按顺序地逐个</code>搜索 <code>子节点</code></p> <p>这两种 <code>搜索方式</code> 是 <code>盲目的 (Blind)</code> 进行 <code>搜索 (Search)</code>，对他们而言，并不考虑 <code>某个节点</code> 所具有的 <code>性质</code>，而是将 <code>所有的节点</code> 都认为是 <code>对等的 (Symmetric)</code>。</p> <p>比如说，<code>DFS</code>会首先沿着 <code>整颗搜索空间树</code> 的 <code>最左侧路径 (The Leftmost Path)</code>进行 <code>搜索</code>。但是，有可能在 <code>这条路径</code> 上根本就不存在 <code>任何的合法解</code>。</p> <blockquote><p>当然，如果 <code>非常幸运的话</code>，<code>DFS</code> 可以很快地在 <code>最左侧路径</code>上找到一个 <code>最优解</code>，然后利用 <code>剪枝 (Prune)</code> 来 <code>大量地裁剪</code>掉 <code>整颗搜索空间树</code> 的 <code>其余部分</code>。</p> <p>此时，<code>DFS</code> 将表现地非常良好。</p> <blockquote><p>如果可以为 <code>DFS</code> <code>尽可能早地</code> 找到一个 <code>较优的合法解</code>，则可以利用 <code>剪枝</code>来 <code>大幅度地</code> 加快搜索。</p> <p>但是，在很多情况下，我们可能 <code>甚至</code> 连一个 <code>合法解</code> 都无法找到，这时 <code>DFS</code>只能 <code>陷入更加盲目的搜索当中</code> 。</p> </blockquote> </blockquote> <p>同样地，<code>BFS</code>也有类似的 <code>困境</code>。</p> <hr> <p>现在，我们基于 <code>BFS</code> 的 <code>特性</code>：每次从 <code>当前节点</code>生成 <code>下一个子节点</code>进行搜索。</p> <p>如果我们使用 <code>优先队列 (Priority Queue)</code> 将 <code>BFS</code> 在 <code>搜索过程</code>中的 <code>节点 (Node)</code> 按照 <code>某种估价策略 (Cost Function)</code> 进行 <code>优先级排序</code>，且每次只取出 <code>当前的优先队列</code>中 <code>价值最小的节点 (Minimum-Cost Node)</code> 作为 <code>下一个进行搜索的节点</code>。</p> <blockquote><p>根据 <code>问题</code>的 <code>优化目标</code>，如果问题是 <code>最大化问题</code>，则取的是 <code>价值最大的节点</code> 。</p> </blockquote> <p>则，我们实际上已经实现了一种 <code>基于节点的价值的BFS搜索方式</code>：</p> <p>与普通的 <code>BFS</code>的区别在于，我们通过 <code>价值函数 (Cost Function)</code> 为 <code>每个节点 (Node)</code> 进行 <code>估价</code>，给该节点设置 <code>价值 (Cost)</code>，然后将该节点加入到 <code>优先队列</code>，且每次只从 <code>优先队列</code> 中取出 <code>当前优先队列中的价值最小的节点</code>。</p> <blockquote><p>根据 <code>Branch-Bound</code>的概念，<code>节点类型</code>被分为下面3种：</p> <ul> <li><code>活节点 (Live-Node)</code>：已经被 <code>产生</code>但仍未被 <code>访问</code>的节点。</li> <li><code>正被探索节点 (E-Node)</code>：当前 <code>正在被访问的节点</code>，也就是 <code>正在被拓展的节点</code>。</li> <li><code>死节点 (Dead-Node)</code>：<code>已经被生成但将来不可能访问或拓展的节点</code> </li> </ul> </blockquote> <hr> <h4>How to define the cost-function</h4> <p>由于题目的目的是 <code>最小化目标值</code>，所以我们定义 <code>价值函数</code>为 <code>目标值的下界</code>。</p> <blockquote><p><code>下界</code>并不一定是 <code>严格下界</code>，可以是 <code>比较宽松的下界</code>。</p> <p>但我们期望的是：<code>界限</code> 尽可能地 <code>逼近最优解的目标值</code></p> </blockquote> <p>假定对于 <code>n个圆</code>的 <code>排列方案</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="11.634ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 5142.4 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-69-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-69-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-69-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-69-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-69-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-69-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-69-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-69-TEX-N-30"></use></g></g><g data-mml-node="msub" transform="translate(887.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-69-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-69-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1941.8,0)"><use data-c="22EF" xlink:href="#MJX-69-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(3280.4,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-69-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-69-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-69-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-69-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_0r_1\cdots r_{n-1}</script>，</p> <p>已经确定了 <code>前k个圆</code>的 <code>排列</code>：<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="21.168ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 9356.2 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-70-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-70-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-70-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-70-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-70-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-70-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-70-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g 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data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-70-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-70-TEX-N-31"></use></g></g></g></g><g data-mml-node="mstyle" fill="red" stroke="red" transform="translate(5086.5,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-70-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-70-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(1069.1,0)"><use data-c="22EF" xlink:href="#MJX-70-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(2407.7,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-70-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-70-TEX-I-1D45B"></use></g><g 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transform="translate(2685.7,0)"><use data-c="2C" xlink:href="#MJX-73-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3130.4,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-73-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-73-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-73-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-73-TEX-N-31"></use></g></g></g></g><g data-mml-node="mo" transform="translate(11943.7,0)"><use data-c="7D" xlink:href="#MJX-73-TEX-N-7D"></use></g><g data-mml-node="mo" transform="translate(12443.7,0)"><use data-c="29" xlink:href="#MJX-73-TEX-N-29"></use></g></g></g></svg></mjx-container><script type="math/tex">r = \min{\b{r_{k-1}}, \r{r_k, \cdots, r_{n-1}}})</script></p> <p>则可以给出一个 <code>下界</code>：<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="30.187ex" height="2.369ex" role="img" focusable="false" viewBox="0 -750 13342.8 1047.1" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.672ex;"><defs><path id="MJX-74-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path id="MJX-74-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 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131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path><path id="MJX-74-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D43F" xlink:href="#MJX-74-TEX-I-1D43F"></use></g><g data-mml-node="TeXAtom" transform="translate(714,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-74-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-74-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(902.4,0)"><use data-c="22EF" xlink:href="#MJX-74-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(2074.4,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-74-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-74-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-74-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-74-TEX-N-31"></use></g></g></g></g></g><g data-mml-node="mo" transform="translate(3825.2,0)"><use data-c="2265" xlink:href="#MJX-74-TEX-N-2265"></use></g><g data-mml-node="mo" transform="translate(4881,0)"><use data-c="28" xlink:href="#MJX-74-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5270,0)"><use data-c="1D45B" xlink:href="#MJX-74-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(6092.2,0)"><use data-c="2212" xlink:href="#MJX-74-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(7092.4,0)"><use data-c="31" xlink:href="#MJX-74-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(7814.6,0)"><use data-c="2212" xlink:href="#MJX-74-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(8814.9,0)"><use data-c="1D458" xlink:href="#MJX-74-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(9558.1,0)"><use data-c="2B" xlink:href="#MJX-74-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(10558.3,0)"><use data-c="31" xlink:href="#MJX-74-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(11058.3,0)"><use data-c="29" xlink:href="#MJX-74-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(11669.5,0)"><use data-c="2217" xlink:href="#MJX-74-TEX-N-2217"></use></g><g data-mml-node="mn" transform="translate(12391.8,0)"><use data-c="32" xlink:href="#MJX-74-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(12891.8,0)"><use data-c="1D45F" xlink:href="#MJX-74-TEX-I-1D45F"></use></g></g></g></svg></mjx-container><script type="math/tex">L_{r_k \cdots r_{n-1}} \ge (n-1-k+1) * 2r</script></p> <p>因此，<code>整合</code>两个式子得到：</p> <div contenteditable="true" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n153" cid="n153" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container 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transform="translate(3869,0)"></use><use data-c="31" xlink:href="#MJX-7-TEX-N-31" transform="translate(4202,0)"></use><use data-c="5D" xlink:href="#MJX-7-TEX-N-5D" transform="translate(4702,0)"></use></g><g data-mml-node="mo" transform="translate(12918.7,0)"><use data-c="2B" xlink:href="#MJX-7-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(13918.9,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-7-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-7-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-7-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-7-TEX-N-31"></use></g></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(15725,0)"><g data-mml-node="mo" transform="translate(0 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transform="translate(22353.3,0)"><use data-c="1D458" xlink:href="#MJX-7-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(23096.6,0)"><use data-c="2B" xlink:href="#MJX-7-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(24096.8,0)"><use data-c="31" xlink:href="#MJX-7-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(24596.8,0)"><use data-c="29" xlink:href="#MJX-7-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(25208,0)"><use data-c="2217" xlink:href="#MJX-7-TEX-N-2217"></use></g><g data-mml-node="mn" transform="translate(25930.2,0)"><use data-c="32" xlink:href="#MJX-7-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(26430.2,0)"><use data-c="1D45F" xlink:href="#MJX-7-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(26881.2,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="29" xlink:href="#MJX-7-TEX-S3-29"></use></g></g><g data-mml-node="mo" transform="translate(27839.4,0)"><use data-c="2212" xlink:href="#MJX-7-TEX-N-2212"></use></g><g data-mml-node="mstyle" fill="red" stroke="red" transform="translate(28839.7,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-7-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-7-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-7-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-7-TEX-N-31"></use></g></g></g></g></g></g></g></g></g></g></g></g></svg></mjx-container></div></div><blockquote><p>n.b. <code>不等式</code> 需要扣除 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.086ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 1806.1 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-82-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-82-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-82-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-82-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-82-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-82-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-82-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-82-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_{k-1}</script></p>
<p>因为对于 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.66ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 4269.7 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-76-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-76-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-76-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-76-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-76-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-76-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-76-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-76-TEX-I-1D458"></use></g></g></g><g data-mml-node="mo" transform="translate(1069.1,0)"><use data-c="22EF" xlink:href="#MJX-76-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(2407.7,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-76-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-76-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-76-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-76-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_{k}\cdots r_{n-1}</script>来说，<code>每个圆的长度 = 该圆半径的两倍</code></p>
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data-c="3D" xlink:href="#MJX-77-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2647.7,0)"><use data-c="32" xlink:href="#MJX-77-TEX-N-32"></use></g><g data-mml-node="msub" transform="translate(3147.7,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-77-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-77-TEX-I-1D456"></use></g></g><g data-mml-node="mstyle" transform="translate(3925.6,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(4925.6,0)"><use data-c="28" xlink:href="#MJX-77-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5314.6,0)"><use data-c="1D456" xlink:href="#MJX-77-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(5937.4,0)"><use data-c="3D" xlink:href="#MJX-77-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(6993.2,0)"><use data-c="1D458" xlink:href="#MJX-77-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(7680.8,0)"><use data-c="22EF" xlink:href="#MJX-77-TEX-N-22EF"></use></g><g data-mml-node="mi" transform="translate(9019.5,0)"><use data-c="1D45B" xlink:href="#MJX-77-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(9841.7,0)"><use data-c="2212" xlink:href="#MJX-77-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(10841.9,0)"><use data-c="31" xlink:href="#MJX-77-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(11341.9,0)"><use data-c="29" xlink:href="#MJX-77-TEX-N-29"></use></g></g></g></svg></mjx-container><script type="math/tex">L_{r_i} = 2r_i \quad(i = k\cdots n-1)</script></p>
<p>但对于处于 <code>接壤处</code> 的 <code>圆</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.086ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 1806.1 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-82-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-82-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-82-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-82-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-82-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-82-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-82-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-82-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_{k-1}</script> 和 <code>圆</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.042ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 902.4 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-81-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-81-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-81-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-81-TEX-I-1D458"></use></g></g></g></g></svg></mjx-container><script type="math/tex">r_k</script>，</p>
<p>我们 <code>实际上</code> 拥有的是<code>左半个圆</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.086ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 1806.1 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-82-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-82-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-82-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-82-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-82-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-82-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-82-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-82-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_{k-1}</script> 和 <code>右半个圆</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.042ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 902.4 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-81-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-81-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-81-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-81-TEX-I-1D458"></use></g></g></g></g></svg></mjx-container><script type="math/tex">r_k</script></p>
<blockquote><p>也就是说，我们并没有拥有 <code>整个圆</code> <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.086ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 1806.1 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-82-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-82-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-82-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-82-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-82-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-82-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-82-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-82-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container><script type="math/tex">r_{k-1}</script></p> </blockquote>
<p>即 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="40.315ex" height="1.977ex" role="img" focusable="false" viewBox="0 -666 17819.3 874" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-83-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-83-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 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data-c="27F9" xlink:href="#MJX-83-TEX-N-27F9"></use></g><g data-mml-node="mstyle" transform="translate(6235.8,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="msub" transform="translate(6791.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-83-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-83-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-83-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-83-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(8819.9,0)"><use data-c="2B" xlink:href="#MJX-83-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(9820.1,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-83-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-83-TEX-I-1D458"></use></g></g></g><g data-mml-node="mo" transform="translate(11000.3,0)"><use data-c="2264" xlink:href="#MJX-83-TEX-N-2264"></use></g><g data-mml-node="msub" transform="translate(12056.1,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-83-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-83-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(13180.7,0)"><use data-c="2B" xlink:href="#MJX-83-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(14180.9,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-83-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-83-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(15361.1,0)"><use data-c="3D" xlink:href="#MJX-83-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(16416.9,0)"><use data-c="32" xlink:href="#MJX-83-TEX-N-32"></use></g><g data-mml-node="msub" transform="translate(16916.9,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-83-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-83-TEX-I-1D458"></use></g></g></g></g></svg></mjx-container><script type="math/tex">r_{k-1} \le r_k \implies r_{k-1} + r_{k} \le r_k + r_k = 2r_k</script></p>
<blockquote><p>n.b. 准确地说，我们实际上是用 <code>图中的红色半圆</code> 来 <code>替代 (Substitute)</code> <code>图中的绿色半圆</code></p> </blockquote> </blockquote>
<p>综上，我们得出了 <code>评估任何给定节点的下界</code>的 <code>价值函数</code></p>
<blockquote><p>通常说，可以将 <code>当前节点的情况</code>进行 <code>极端化假设</code> 来获得 <code>界限</code>：比如说使用 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.124ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 1823 672" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-84-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-84-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-84-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-84-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(878,0)"><use data-c="1D456" xlink:href="#MJX-84-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(1223,0)"><use data-c="1D45B" xlink:href="#MJX-84-TEX-I-1D45B"></use></g></g></g></svg></mjx-container><script type="math/tex">min</script> 和 <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.477ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 1979 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-85-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-85-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-85-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(878,0)"><use data-c="1D44E" xlink:href="#MJX-85-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(1407,0)"><use data-c="1D465" xlink:href="#MJX-85-TEX-I-1D465"></use></g></g></g></svg></mjx-container><script type="math/tex">max</script> 函数来使得 <code>某些值</code>属于 <code>最好的情况</code> 或者 <code>最坏的情况</code></p> </blockquote>
<blockquote><p>另外，还有一种策略是通过为 <code>当前节点</code> 运行一个 <code>贪心算法 (Greedy Method)</code> 来获取 <code>界限</code>。</p>
<p>比如，通过为 <code>0/1 Knapsack</code> 的 <code>节点</code> 运行一个 <code>Fraction Knapsack</code>的 <code>Greedy Method</code> 可以获得一个 <code>非常接近当前节点的最优解的合法解</code>。</p>
<p>使用 <code>Greedy Method</code>来获得 <code>Bound</code> 需要满足：</p>
<ul>
<li><code>贪心选择策略</code>足够简单，使得 <code>贪心算法</code>不至于过慢</li> <li>所获得的 <code>贪心解</code>大部分情况下 <code>比较接近</code> <code>当前节点的最优解</code></li></ul>
<p>但对于 <code>圆排列问题</code>，我们不容易找到一种 <code>贪心选择策略足够简单，且贪心解比较接近当前节点的最优解</code> 的 <code>贪心选择算法</code>。</p>
<blockquote><p>比如说，下面2种 <code>贪心选择策略</code>：</p>
<ul>
<li>按照 <code>降序/升序</code> 排列 <code>剩下的圆</code></li> <li>按照 <code>大圆</code> 和 <code>小圆</code> 交替的方式排列 <code>剩下的圆</code></li></ul>
<p>这是我们直观想得到的策略，但它们给出的 <code>贪心解</code> 可能会 <code>非常偏离</code> <code>当前节点的最优解</code> 。</p>
<blockquote><p>第一个策略：我们其实已经知道，它将包含很多非 <code>最优的子结构</code>。实际上，这是一个 <code>非常糟糕</code> 的 <code>贪心选择策略</code></p>
<p>第二个策略：这个 <code>贪心选择策略</code> 可能会稍微好一些，但仍然有可能 <code>非常偏离</code> <code>当前节点的最优解</code>。</p>
<p>而且，我们可能会想对该策略进行 <code>改进</code>，使之变得更加 <code>贪婪 (More Greedy)</code>，比如说 <code>尽可能多地包含更多小圆</code>。</p>
<p>但是，如果 <code>引入更复杂的贪心选择策略</code> 同时也增加了 <code>价值函数</code> 的 <code>复杂度</code>，在整个 <code>BFS搜索过程</code> 上有可能 <code>得不偿失</code></p> </blockquote> </blockquote> </blockquote>
<hr>
<h4>Accelerate the search by pruning</h4>
<p>对于某些 <code>我们已知不可能从中获得最优解的节点</code>，可以直接将 <code>该节点</code> 进行 <code>丢弃</code>。</p>
<h5>Not better than current solution</h5>
<pre><code class="language-pseudocode" lang="pseudocode">// define the ans as the infinity for min-imization problem ans = infinity </code></pre>
<pre><code class="language-pseudocode" lang="pseudocode">// when we reach a leaf-node, try to update the ans ans = min(ans, node.cost) </code></pre>
<pre><code class="language-pseudocode" lang="pseudocode">// when we are in a non-leaf node, check the cost before expanding it if (node.cost < ans) { expand the node } </code></pre>
<h5>Out of bound</h5>
<pre><code class="language-pseudocode" lang="pseudocode">// before we expand a node, check the bound (cost-function) if (bound(node) < ans) { expand the node } </code></pre>
<h5>Contains substructures that are not optimal </h5>
<p>根据 <code>题目性质</code>，通过 <code>观察</code>可以发现，<code>最优解</code>不应当包含的 <code>子结构</code>。 如对于本题：</p>
<ol> <li><code>最优解</code>必定不包含 <code>连续的3个半径升序的圆</code></li> <li><code>最优解</code>必定不包含 <code>连续的3个半径降序的圆</code></li> </ol> <pre><code class="language-java" lang="java"> // the radius of circles are sorted in ascending order: Arrays.sort(circles); // prune: we ensure that the best solution won't contain 3 successive ascending circles or 3 successive descending circles. if (node.level >= 2) { if (node.plan[node.level] > node.plan[node.level - 1] && node.plan[node.level - 1] > node.plan[node.level - 2]) { return Double.MAX_VALUE; } if (node.plan[node.level] < node.plan[node.level - 1] && node.plan[node.level - 1] < node.plan[node.level - 2]) { return Double.MAX_VALUE; } } </code></pre> <blockquote><p>通过 <code>剪枝</code> 掉 <code>拥有非最优子结构的节点</code>，我们实际上可以非常好地利用 <code>解的局部特征</code>，并且更有可能 <code>尽早地进行剪枝</code>。</p> </blockquote> <h4>Accelerate the search by priority</h4> <p>相比于 <code>BFS</code> 和 <code>DFS</code> 的 <code>盲目的搜索</code> 而言，<code>分支界限法</code>使用了 <code>价值函数</code>来计算出 <code>某个节点的界限</code>，而且 <code>搜索性能</code> 极大程度地取决于 <code>价值函数</code> 是否 <code>足够聪明 (Intellectual)</code>。</p> <p>如果 <code>价值函数</code> 是 <code>绝对聪明的 (Absolutely Intellectual)</code>，那么它就好像可以 <code>预知未来</code>一样，直接从 <code>成千上万条可能的路径</code>之中，挑选出 <code>1条最优解的路径</code>。</p> <p>而如果 <code>价值函数</code> 不够聪明，那么它可能会被 <code>某些节点</code> 所 <code>误导</code>，为 <code>这些节点</code>给出 <code>错误的估价</code>，从而导致 <code>额外的没有必要的搜索</code>，但 <code>后续仍然有机会重回正轨 (Bring us closer to the optimal solution)</code> （如果 <code>最优解</code>所在的路径上的节点没有被 <code>错误地剪枝</code>掉的话）。</p> <p>但对于 <code>DFS</code>而言，它从 <code>最左侧路径</code>开始 <code>深入</code> 进行搜索 ，如果 <code>最优解</code>处于 <code>最右侧路径</code>上的话，则会做 <code>大量无用的搜索</code>。</p> <p>下面例子给出了当 <code>指数爆炸</code> 时，<code>毫无目标地进行盲目搜索的DFS/BFS</code> 和 <code>拥有价值函数的分支界限</code> 的不同 <code>性能表现</code>。</p> <pre><code class="language-java" lang="java">Branch and Bound ----------------------------------------------------- Current Case: CIRCLE15.in & CIRCLE15.out Expected Input: [12, 48, 31, 18, 25, 33, 35, 73, 75, 65, 78, 94, 48] Expected Output: [1121.467] Your Output: [1121.467] Time Cost: 22052.981400 ms (22052981400 ns) Accepted. ----------------------------------------------------- </code></pre> <pre><code class="language-java" lang="java">DFS ----------------------------------------------------- Current Case: CIRCLE15.in & CIRCLE15.out Expected Input: [12, 48, 31, 18, 25, 33, 35, 73, 75, 65, 78, 94, 48] Expected Output: [1121.467] Your Output: [1121.467] Time Cost: 66771.506100 ms (66771506100 ns) Accepted. ----------------------------------------------------- </code></pre> <pre><code class="language-java" lang="java">BFS ----------------------------------------------------- Current Case: CIRCLE15.in & CIRCLE15.out Expected Input: [12, 48, 31, 18, 25, 33, 35, 73, 75, 65, 78, 94, 48] Expected Output: [1121.467] Your Output: [1121.467] Time Cost: 179924.051700 ms (179924051700 ns) Accepted. ----------------------------------------------------- </code></pre> <h2>Solution</h2> <h3>DFS</h3> <h4>Diagram</h4> <pre><code class="language-mermaid" lang="mermaid">graph TD; root((root)) --#0,1--> 0_((2.000)); 0_ --#1,2--> 0_1_((5.828)); 0_1_ --#2,9--> 0_1_2_((21.314)); 0_1_2_ --#3,5--> 0_1_2_3_((30.730)); 0_1_ --#4,5--> 0_1_3_((15.153)); 0_1_3_ --#5,9--> 0_1_3_2_((32.569)); 0_1_3_ --#6,9--> 0_1_3_2_((32.569)); style 0_1_3_2_ fill: lightgray 0_ --#7,9--> 0_2_((18.000)); 0_2_ --#8,2--> 0_2_1_((19.485)); 0_2_1_ --#9,5--> 0_2_1_3_((28.810)); 0_2_ --#10,5--> 0_2_3_((27.416)); 0_2_3_ --#11,2--> 0_2_3_1_((30.741)); 0_ --#12,5--> 0_3_((10.472)); 0_3_ --#13,2--> 0_3_1_((13.797)); 0_3_1_ --#14,9--> 0_3_1_2_((29.282)); 0_3_1_ --#15,9--> 0_3_1_2_((29.282)); style 0_3_1_2_ fill: lightgray 0_3_ --#16,9--> 0_3_2_((27.889)); 0_3_2_ --#17,2--> 0_3_2_1_((29.374)); 0_3_2_ --#18,2--> 0_3_2_1_((29.374)); style 0_3_2_1_ fill: lightgray root((root)) --#19,2--> 1_((4.000)); 1_ --#20,1--> 1_0_((5.828)); 1_0_ --#21,9--> 1_0_2_((19.828)); 1_0_2_ --#22,5--> 1_0_2_3_((29.245)); 1_0_2_ --#23,5--> 1_0_2_3_((29.245)); style 1_0_2_3_ fill: lightgray 1_0_ --#24,5--> 1_0_3_((14.301)); 1_0_3_ --#25,9--> 1_0_3_2_((31.717)); 1_0_3_ --#26,9--> 1_0_3_2_((31.717)); style 1_0_3_2_ fill: lightgray 1_ --#27,9--> 1_2_((19.485)); 1_2_ --#28,1--> 1_2_0_((19.485)); 1_2_0_ --#29,5--> 1_2_0_3_((28.902)); 1_2_0_ --#30,5--> 1_2_0_3_((28.902)); style 1_2_0_3_ fill: lightgray 1_2_ --#31,5--> 1_2_3_((28.902)); 1_2_ --#32,5--> 1_2_3_((28.902)); style 1_2_3_ fill: lightgray 1_ --#33,5--> 1_3_((13.325)); 1_3_ --#34,1--> 1_3_0_((13.797)); 1_3_0_ --#35,9--> 1_3_0_2_((30.741)); 1_3_0_ --#36,9--> 1_3_0_2_((30.741)); style 1_3_0_2_ fill: lightgray 1_3_ --#37,9--> 1_3_2_((30.741)); 1_3_ --#38,9--> 1_3_2_((30.741)); style 1_3_2_ fill: lightgray root((root)) --#39,9--> 2_((18.000)); 2_ --#40,1--> 2_0_((18.000)); 2_0_ --#41,2--> 2_0_1_((19.828)); 2_0_1_ --#42,5--> 2_0_1_3_((29.153)); 2_0_1_ --#43,5--> 2_0_1_3_((29.153)); style 2_0_1_3_ fill: lightgray 2_0_ --#44,5--> 2_0_3_((27.416)); 2_0_3_ --#45,2--> 2_0_3_1_((30.741)); 2_0_3_ --#46,2--> 2_0_3_1_((30.741)); style 2_0_3_1_ fill: lightgray 2_ --#47,2--> 2_1_((19.485)); 2_1_ --#48,1--> 2_1_0_((21.314)); 2_1_0_ --#49,5--> 2_1_0_3_((29.786)); 2_1_0_ --#50,5--> 2_1_0_3_((29.786)); style 2_1_0_3_ fill: lightgray 2_1_ --#51,5--> 2_1_3_((28.810)); 2_1_ --#52,5--> 2_1_3_((28.810)); style 2_1_3_ fill: lightgray 2_ --#53,5--> 2_3_((27.416)); 2_3_ --#54,1--> 2_3_0_((27.889)); 2_3_0_ --#55,2--> 2_3_0_1_((31.717)); 2_3_0_ --#56,2--> 2_3_0_1_((31.717)); style 2_3_0_1_ fill: lightgray 2_3_ --#57,2--> 2_3_1_((30.741)); 2_3_ --#58,2--> 2_3_1_((30.741)); style 2_3_1_ fill: lightgray root((root)) --#59,5--> 3_((10.000)); 3_ --#60,1--> 3_0_((10.472)); 3_0_ --#61,2--> 3_0_1_((14.301)); 3_0_1_ --#62,9--> 3_0_1_2_((29.786)); 3_0_1_ --#63,9--> 3_0_1_2_((29.786)); style 3_0_1_2_ fill: lightgray 3_0_ --#64,9--> 3_0_2_((27.416)); 3_0_2_ --#65,2--> 3_0_2_1_((28.902)); 3_0_2_ --#66,2--> 3_0_2_1_((28.902)); style 3_0_2_1_ fill: lightgray 3_ --#67,2--> 3_1_((13.325)); 3_1_ --#68,1--> 3_1_0_((15.153)); 3_1_0_ --#69,9--> 3_1_0_2_((29.153)); 3_1_0_ --#70,9--> 3_1_0_2_((29.153)); style 3_1_0_2_ fill: lightgray 3_1_ --#71,9--> 3_1_2_((28.810)); 3_1_ --#72,9--> 3_1_2_((28.810)); style 3_1_2_ fill: lightgray 3_ --#73,9--> 3_2_((27.416)); 3_2_ --#74,1--> 3_2_0_((27.416)); 3_2_0_ --#75,2--> 3_2_0_1_((29.245)); 3_2_0_ --#76,2--> 3_2_0_1_((29.245)); style 3_2_0_1_ fill: lightgray 3_2_ --#77,2--> 3_2_1_((28.902)); 3_2_ --#78,2--> 3_2_1_((28.902)); style 3_2_1_ fill: lightgray </code></pre> <h4>Source</h4> <pre><code class="language-java" lang="java">public class CirclePermutationSolver_DFS { static Judger judger = new Judger("/Cases/Lab7/CIRCLE PERMUTATION").redirectErrorToErrorFile().disablePrettyFormat(); /* all the circles should TOUCH THE GROUND, therefore we can't put one circle over another if we have an infinite big circle, we can put ALL the other circles under the big circle ! for total n circles: the FIRST circle and the LAST circle is special. THE OTHER circles should considerate both of them. we can divide a circle into 2 PARTs we assume that: the first circle is in (0, 0) */ public static double calc_center(int[] circles, int[] plan, int plan_index) { // if we only have 1 circle, we assume that the first circle is in (0, 0) double center = 0; /* compare circles[plan[plan_index]] with ALL the previous circles */ for (int i = 0; i < plan_index; i++) { double r1 = circles[plan[i]]; double r2 = circles[plan[plan_index]]; center = Math.max(center, centers[i] + 2 * Math.sqrt(r1 * r2)); } return center; } public static double calc_length(int[] circles, int[] plan) { /* all circles */ // the first circle double interval_left = 0; double interval_right = 0; // it's ok if we only have 1 circle. calcCenter() will return 0 if we only have 1 circle for (int i = 0; i < plan.length; i++) { double r = circles[plan[i]]; // current circle & left circle double c = calc_center(circles, plan, i); interval_left = Math.min(interval_left, c - r); interval_right = Math.max(interval_right, c + r); } return interval_right - interval_left; } /* Global Variables */ static int n; static int[] circles; static boolean[] used; static double[] centers; static int[] plan; static double ans; public static void search(int depth) { /* Base Case */ if (depth >= n) { ans = Math.min(ans, calc_length(circles, plan)); return; } /* Recursion Case */ for (int i = 0; i < circles.length; i++) { /* lock */ if (used[i]) continue; else used[i] = true; /* update */ plan[depth] = i; centers[depth] = calc_center(circles, plan, depth); // pruning increases speed by about 2 to 10 times ! // n.b. the LHS is not the new length ! it's JUST a condition for pruning // if you want to get the new length, you ought to call calcLength() if (centers[depth] + circles[plan[0]] + circles[plan[depth]] < ans) { search(depth + 1); } /* unlock */ used[i] = false; } } public static void solve(Scanner scanner) { /* Initialize */ n = scanner.nextInt(); circles = new int[n]; centers = new double[n]; for (int i = 0; i < n; i++) { circles[i] = scanner.nextInt(); } used = new boolean[n]; plan = new int[n]; ans = Double.MAX_VALUE; /* Algo */ search(0); System.out.printf("%.3f", ans); } public static void main(String[] args) { for (Scanner scanner : judger) { solve(scanner); } } } </code></pre> <h4>Benchmark</h4> <pre><code class="language-yaml" lang="yaml">----------------------------------------------------- Current Case: CIRCLE0.in & CIRCLE0.out Expected Input: [3, 1 1 2] Expected Output: [7.657] Your Output: [7.657] Time Cost: 2.953300 ms (2953300 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE1.in & CIRCLE1.out Expected Input: [5, 59 23 41 70 47 ] Expected Output: [454.388] Your Output: [454.388] Time Cost: 1.378200 ms (1378200 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE10.in & CIRCLE10.out Expected Input: [10, 9 2 117 45 9 3 142 14 9 98 ] Expected Output: [755.928] Your Output: [755.928] Time Cost: 866.089000 ms (866089000 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE11.in & CIRCLE11.out Expected Input: [2, 10000 1] Expected Output: [20000.000] Your Output: [20000.000] Time Cost: 0.695200 ms (695200 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE12.in & CIRCLE12.out Expected Input: [2, 10000 10000] Expected Output: [40000.000] Your Output: [40000.000] Time Cost: 0.688300 ms (688300 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE13.in & CIRCLE13.out Expected Input: [1, 10000] Expected Output: [20000.000] Your Output: [20000.000] Time Cost: 0.700300 ms (700300 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE2.in & CIRCLE2.out Expected Input: [7, 94 35 20 88 55 28 57 ] Expected Output: [666.874] Your Output: [666.874] Time Cost: 1.431800 ms (1431800 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE3.in & CIRCLE3.out Expected Input: [7, 25 1 5 74 47 77 8 ] Expected Output: [415.089] Your Output: [415.089] Time Cost: 1.630000 ms (1630000 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE4.in & CIRCLE4.out Expected Input: [9, 17 49 77 84 86 64 75 88 65 ] Expected Output: [1159.668] Your Output: [1159.668] Time Cost: 44.285300 ms (44285300 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE5.in & CIRCLE5.out Expected Input: [10, 99 22 17 45 91 73 42 14 9 98 ] Expected Output: [858.474] Your Output: [858.474] Time Cost: 447.405000 ms (447405000 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE6.in & CIRCLE6.out Expected Input: [10, 51 100 66 37 30 83 87 98 31 43 ] Expected Output: [1140.471] Your Output: [1140.471] Time Cost: 409.175500 ms (409175500 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE7.in & CIRCLE7.out Expected Input: [10, 51 100 66 37 30 1 87 98 31 3 ] Expected Output: [902.696] Your Output: [902.696] Time Cost: 550.624100 ms (550624100 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE8.in & CIRCLE8.out Expected Input: [9, 1 49 77 8 86 6 75 88 3 ] Expected Output: [738.394] Your Output: [738.394] Time Cost: 83.854200 ms (83854200 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE9.in & CIRCLE9.out Expected Input: [10, 9 22 17 45 9 3 42 14 9 98 ] Expected Output: [400.389] Your Output: [400.389] Time Cost: 343.135900 ms (343135900 ns) Accepted. ----------------------------------------------------- Result Statistics: √ √ √ √ √ √ √ √ √ √ √ √ √ √ </code></pre> <h3>BFS</h3> <h4>Diagram</h4> <pre><code class="language-mermaid" lang="mermaid">graph TD; root((root)) --#0,1--> 0_((2.000)); root((root)) --#1,1--> 1_((4.000)); root((root)) --#2,1--> 2_((18.000)); root((root)) --#3,1--> 3_((10.000)); 3_ --#4,2--> 3_0_((10.472)); 3_ --#5,2--> 3_1_((13.325)); 3_ --#6,2--> 3_2_((27.416)); 3_2_ --#7,9--> 3_2_0_((27.416)); 3_2_ --#8,9--> 3_2_1_((28.902)); 3_2_1_ --#9,5--> 3_2_1_0_((30.730)); 3_2_0_ --#10,5--> 3_2_0_1_((29.245)); 3_1_ --#11,9--> 3_1_0_((15.153)); 3_1_ --#12,9--> 3_1_2_((28.810)); 3_1_2_ --#13,5--> 3_1_2_0_((28.810)); 3_1_0_ --#14,5--> 3_1_0_2_((29.153)); style 3_1_0_2_ fill: lightgray 3_0_ --#15,9--> 3_0_1_((14.301)); 3_0_ --#16,9--> 3_0_2_((27.416)); 3_0_2_ --#17,5--> 3_0_2_1_((28.902)); style 3_0_2_1_ fill: lightgray 3_0_1_ --#18,5--> 3_0_1_2_((29.786)); style 3_0_1_2_ fill: lightgray 2_ --#19,2--> 2_0_((18.000)); 2_ --#20,2--> 2_1_((19.485)); 2_ --#21,2--> 2_3_((27.416)); 2_3_ --#22,9--> 2_3_0_((27.889)); 2_3_ --#23,9--> 2_3_1_((30.741)); style 2_3_1_ fill: lightgray 2_3_0_ --#24,5--> 2_3_0_1_((31.717)); style 2_3_0_1_ fill: lightgray 2_1_ --#25,9--> 2_1_0_((21.314)); 2_1_ --#26,9--> 2_1_3_((28.810)); style 2_1_3_ fill: lightgray 2_1_0_ --#27,5--> 2_1_0_3_((29.786)); style 2_1_0_3_ fill: lightgray 2_0_ --#28,9--> 2_0_1_((19.828)); 2_0_ --#29,9--> 2_0_3_((27.416)); 2_0_3_ --#30,5--> 2_0_3_1_((30.741)); style 2_0_3_1_ fill: lightgray 2_0_1_ --#31,5--> 2_0_1_3_((29.153)); style 2_0_1_3_ fill: lightgray 1_ --#32,2--> 1_0_((5.828)); 1_ --#33,2--> 1_2_((19.485)); 1_ --#34,2--> 1_3_((13.325)); 1_3_ --#35,9--> 1_3_0_((13.797)); 1_3_ --#36,9--> 1_3_2_((30.741)); style 1_3_2_ fill: lightgray 1_3_0_ --#37,5--> 1_3_0_2_((30.741)); style 1_3_0_2_ fill: lightgray 1_2_ --#38,9--> 1_2_0_((19.485)); 1_2_ --#39,9--> 1_2_3_((28.902)); style 1_2_3_ fill: lightgray 1_2_0_ --#40,5--> 1_2_0_3_((28.902)); style 1_2_0_3_ fill: lightgray 1_0_ --#41,9--> 1_0_2_((19.828)); 1_0_ --#42,9--> 1_0_3_((14.301)); 1_0_3_ --#43,5--> 1_0_3_2_((31.717)); style 1_0_3_2_ fill: lightgray 1_0_2_ --#44,5--> 1_0_2_3_((29.245)); style 1_0_2_3_ fill: lightgray 0_ --#45,2--> 0_1_((5.828)); 0_ --#46,2--> 0_2_((18.000)); 0_ --#47,2--> 0_3_((10.472)); 0_3_ --#48,9--> 0_3_1_((13.797)); 0_3_ --#49,9--> 0_3_2_((27.889)); 0_3_2_ --#50,5--> 0_3_2_1_((29.374)); style 0_3_2_1_ fill: lightgray 0_3_1_ --#51,5--> 0_3_1_2_((29.282)); style 0_3_1_2_ fill: lightgray 0_2_ --#52,9--> 0_2_1_((19.485)); 0_2_ --#53,9--> 0_2_3_((27.416)); 0_2_3_ --#54,5--> 0_2_3_1_((30.741)); style 0_2_3_1_ fill: lightgray 0_2_1_ --#55,5--> 0_2_1_3_((28.810)); style 0_2_1_3_ fill: lightgray 0_1_ --#56,9--> 0_1_2_((21.314)); 0_1_ --#57,9--> 0_1_3_((15.153)); 0_1_3_ --#58,5--> 0_1_3_2_((32.569)); style 0_1_3_2_ fill: lightgray 0_1_2_ --#59,5--> 0_1_2_3_((30.730)); style 0_1_2_3_ fill: lightgray </code></pre> <h4>Source</h4> <pre><code class="language-java" lang="java">public class CirclePermutationSolver_BFS { static Judger judger = new Judger("/Cases/Lab7/CIRCLE PERMUTATION").redirectErrorToErrorFile().disablePrettyFormat(); /* all the circles should TOUCH THE GROUND, therefore we can't put one circle over another if we have an infinite big circle, we can put ALL the other circles under the big circle ! for total n circles: the FIRST circle and the LAST circle is special. THE OTHER circles should considerate both of them. we can divide a circle into 2 PARTs --- take 2 circles into considerations: Case1: luckily, we have a big enough circle to cover another circle Case2: we can't cover the other circle, but we can cover the big circle */ static class Node { /* State */ public int level; public int[] plan; public double[] centers; // n.b. if we use swap() to generate circle permutation, we don't need the used field ! public boolean[] used; public Node(int level, int[] plan, double[] centers, boolean[] used) { this.level = level; this.plan = plan; this.centers = centers; this.used = used; } } public static double calc_center(int[] circles, int[] plan, double[] centers, int level) { // if we only have 1 circle, we assume that the first circle is in (0, 0) double center = 0; /* compare circles[plan[plan_index]] with ALL the previous circles */ for (int i = 0; i < level; i++) { double r1 = circles[plan[i]]; double r2 = circles[plan[level]]; center = Math.max(center, centers[i] + 2 * Math.sqrt(r1 * r2)); } return center; } public static double calc_length(int[] circles, int[] plan, double[] centers, int level) { /* all circles */ // the first circle double interval_left = 0; double interval_right = 0; // it's ok if we only have 1 circle. calcCenter() will return 0 if we only have 1 circle for (int i = 0; i <= level; i++) { double r = circles[plan[i]]; // current circle & left circle double c = centers[i]; interval_left = Math.min(interval_left, c - r); interval_right = Math.max(interval_right, c + r); } return interval_right - interval_left; } public static double ans; public static double solve(int n, int[] circles) { /* Initialize */ ans = Double.MAX_VALUE; /* Construct the queue and init */ LinkedList<Node> nodes = new LinkedList<>(); nodes.add(new Node(-1, new int[n], new double[n], new boolean[n])); // BFS while (!nodes.isEmpty()) { // Get one node Node currentNode = nodes.poll(); int $level = currentNode.level + 1; /* Generate all the next nodes */ for (int k = 0; k < n; k++) { // Skip the used circles if (currentNode.used[k]) continue; // Generate the next node int[] $plan = currentNode.plan.clone(); $plan[$level] = k; double[] $centers = currentNode.centers.clone(); $centers[$level] = calc_center(circles, $plan, $centers, $level); boolean[] $used = currentNode.used.clone(); $used[k] = true; /* Update the next node */ double length = calc_length(circles, $plan, $centers, $level); // ADD of DROP the next node ? // prune: if the next-node is NOT BETTER THAN current solution if (length < ans) { /* Try to update ans */ Node node = new Node($level, $plan, $centers, $used); /* Finish permutation ! */ if ($level == n - 1) { ans = Math.min(ans, length); } else nodes.push(node); } } } return ans; } public static void main(String[] args) { for (Scanner scanner : judger) { int n = scanner.nextInt(); int[] a = new int[n]; for (int i = 0; i < n; i++) { a[i] = scanner.nextInt(); } System.out.printf("%.3f", solve(n, a)); } } } </code></pre> <h4>Benchmark</h4> <pre><code class="language-yaml" lang="yaml">----------------------------------------------------- Current Case: CIRCLE0.in & CIRCLE0.out Expected Input: [3, 1 1 2] Expected Output: [7.657] Your Output: [7.657] Time Cost: 3.026100 ms (3026100 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE1.in & CIRCLE1.out Expected Input: [5, 59 23 41 70 47 ] Expected Output: [454.388] Your Output: [454.388] Time Cost: 1.600400 ms (1600400 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE10.in & CIRCLE10.out Expected Input: [10, 9 2 117 45 9 3 142 14 9 98 ] Expected Output: [755.928] Your Output: [755.928] Time Cost: 2300.654100 ms (2300654100 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE11.in & CIRCLE11.out Expected Input: [2, 10000 1] Expected Output: [20000.000] Your Output: [20000.000] Time Cost: 0.799000 ms (799000 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE12.in & CIRCLE12.out Expected Input: [2, 10000 10000] Expected Output: [40000.000] Your Output: [40000.000] Time Cost: 0.685500 ms (685500 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE13.in & CIRCLE13.out Expected Input: [1, 10000] Expected Output: [20000.000] Your Output: [20000.000] Time Cost: 0.686900 ms (686900 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE2.in & CIRCLE2.out Expected Input: [7, 94 35 20 88 55 28 57 ] Expected Output: [666.874] Your Output: [666.874] Time Cost: 6.327000 ms (6327000 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE3.in & CIRCLE3.out Expected Input: [7, 25 1 5 74 47 77 8 ] Expected Output: [415.089] Your Output: [415.089] Time Cost: 5.280500 ms (5280500 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE4.in & CIRCLE4.out Expected Input: [9, 17 49 77 84 86 64 75 88 65 ] Expected Output: [1159.668] Your Output: [1159.668] Time Cost: 316.421300 ms (316421300 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE5.in & CIRCLE5.out Expected Input: [10, 99 22 17 45 91 73 42 14 9 98 ] Expected Output: [858.474] Your Output: [858.474] Time Cost: 3063.122800 ms (3063122800 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE6.in & CIRCLE6.out Expected Input: [10, 51 100 66 37 30 83 87 98 31 43 ] Expected Output: [1140.471] Your Output: [1140.471] Time Cost: 3877.059500 ms (3877059500 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE7.in & CIRCLE7.out Expected Input: [10, 51 100 66 37 30 1 87 98 31 3 ] Expected Output: [902.696] Your Output: [902.696] Time Cost: 3195.105900 ms (3195105900 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE8.in & CIRCLE8.out Expected Input: [9, 1 49 77 8 86 6 75 88 3 ] Expected Output: [738.394] Your Output: [738.394] Time Cost: 216.487600 ms (216487600 ns) Accepted. ----------------------------------------------------- Current Case: CIRCLE9.in & CIRCLE9.out Expected Input: [10, 9 22 17 45 9 3 42 14 9 98 ] Expected Output: [400.389] Your Output: [400.389] Time Cost: 2362.946400 ms (2362946400 ns) Accepted. ----------------------------------------------------- Result Statistics: √ √ √ √ √ √ √ √ √ √ √ √ √ √ </code></pre> <p> </p> <h3>Branch-Bound</h3> <h4>Diagram</h4> <pre><code class="language-mermaid" lang="mermaid">graph TD; root((root)) --#0,1--> 0_((2.000</br>8.000)); root((root)) --#1,2--> 1_((4.000</br>9.000)); root((root)) --#2,5--> 2_((10.000</br>12.000)); root((root)) --#3,9--> 3_((18.000</br>16.000)); 0_ --#4,2--> 0_1_((5.828</br>13.828)); 0_ --#5,5--> 0_2_((10.472</br>15.472)); 0_ --#6,9--> 0_3_((18.000</br>17.000)); 1_ --#7,1--> 1_0_((5.828</br>9.828)); 1_ --#8,5--> 1_2_((13.325</br>13.325)); 1_ --#9,9--> 1_3_((19.485</br>15.485)); 1_0_ --#10,5--> 1_0_2_((14.301</br>24.301)); 1_0_ --#11,9--> 1_0_3_((19.828</br>25.828)); 0_1_ --#12,5--> 0_1_2_((15.153</br>INF)); style 0_1_2_ fill: lightgray 0_1_ --#13,9--> 0_1_3_((21.314</br>INF)); style 0_1_3_ fill: lightgray 2_ --#14,2--> 2_1_((13.325</br>16.325)); 2_ --#15,1--> 2_0_((10.472</br>14.472)); 2_ --#16,9--> 2_3_((27.416</br>23.416)); 1_2_ --#17,1--> 1_2_0_((13.797</br>15.797)); 1_2_ --#18,9--> 1_2_3_((30.741</br>INF)); style 1_2_3_ fill: lightgray 2_0_ --#19,2--> 2_0_1_((14.301</br>18.301)); 2_0_ --#20,9--> 2_0_3_((27.416</br>24.416)); 0_2_ --#21,2--> 0_2_1_((13.797</br>17.797)); 0_2_ --#22,9--> 0_2_3_((27.889</br>INF)); style 0_2_3_ fill: lightgray 1_2_0_ --#23,9--> 1_2_0_3_((30.741</br>30.741)); 0_2_1_ --#24,9--> 0_2_1_3_((29.282</br>29.282)); 2_0_1_ --#25,9--> 2_0_1_3_((29.786</br>INF)); style 2_0_1_3_ fill: lightgray 2_1_ --#26,1--> 2_1_0_((15.153</br>INF)); style 2_1_0_ fill: lightgray 2_1_ --#27,9--> 2_1_3_((28.810</br>22.810)); 1_3_ --#28,5--> 1_3_2_((28.902</br>26.902)); 1_3_ --#29,1--> 1_3_0_((19.485</br>19.485)); 0_3_ --#30,5--> 0_3_2_((27.416</br>26.416)); 0_3_ --#31,2--> 0_3_1_((19.485</br>21.485)); 3_ --#32,2--> 3_1_((19.485</br>22.485)); 3_ --#33,5--> 3_2_((27.416</br>27.416)); 3_ --#34,1--> 3_0_((18.000</br>20.000)); 1_3_0_ --#35,5--> 1_3_0_2_((28.902</br>28.902)); 0_3_1_ --#36,5--> 0_3_1_2_((28.810</br>26.810)); 3_0_ --#37,5--> 3_0_2_((27.416</br>28.416)); 3_0_ --#38,2--> 3_0_1_((19.828</br>23.828)); 3_1_ --#39,5--> 3_1_2_((28.810</br>26.810)); style 3_1_2_ fill: lightgray 3_1_ --#40,1--> 3_1_0_((21.314</br>INF)); style 3_1_0_ fill: lightgray 2_1_3_ --#41,1--> 2_1_3_0_((28.810</br>26.810)); 3_0_1_ --#42,5--> 3_0_1_2_((29.153</br>INF)); style 3_0_1_2_ fill: lightgray 1_0_2_ --#43,9--> 1_0_2_3_((31.717</br>INF)); style 1_0_2_3_ fill: lightgray 2_3_ --#44,1--> 2_3_0_((27.416</br>27.416)); 2_3_ --#45,2--> 2_3_1_((28.902</br>29.902)); style 2_3_1_ fill: lightgray 2_0_3_ --#46,2--> 2_0_3_1_((28.902</br>28.902)); style 2_0_3_1_ fill: lightgray 1_0_3_ --#47,5--> 1_0_3_2_((29.245</br>29.245)); style 1_0_3_2_ fill: lightgray 1_3_2_ --#48,1--> 1_3_2_0_((29.374</br>INF)); style 1_3_2_0_ fill: lightgray 0_3_2_ --#49,2--> 0_3_2_1_((30.741</br>INF)); style 0_3_2_1_ fill: lightgray 2_3_0_ --#50,2--> 2_3_0_1_((29.245</br>29.245)); style 2_3_0_1_ fill: lightgray 3_0_2_ --#51,2--> 3_0_2_1_((30.741</br>30.741)); style 3_0_2_1_ fill: lightgray 3_2_ --#52,2--> 3_2_1_((30.741</br>INF)); style 3_2_1_ fill: lightgray 3_2_ --#53,1--> 3_2_0_((27.889</br>INF)); style 3_2_0_ fill: lightgray </code></pre> <h4>Source</h4> <pre><code class="language-java" lang="java">public class CirclePermutationSolver_BranchBoundMethod { static Judger judger = new Judger("/Cases/Lab7/CIRCLE PERMUTATION").redirectErrorToErrorFile().disablePrettyFormat(); static class Node { /* State */ public int level; // n.b. we use swap() to generate permutation, so used[] is useless. public int[] plan; public double[] centers; /* Model */ public double cost = 0; public static int n; public static int[] circles; public static double ans; public Node(int level, int[] plan, double[] centers) { this.level = level; this.plan = plan; this.centers = centers; } public String radius_string() { int[] radius = new int[this.level + 1]; for (int i = 0; i <= this.level; i++) { radius[i] = Node.circles[this.plan[i]]; } return Arrays.toString(radius); } @Override public String toString() { return "Node{" + "level=" + level + ", radius=" + radius_string() + ", centers=" + Arrays.toString(centers) + ", cost=" + cost + '}'; } } /* n.b. This function use centers[0..level - 1] to calculate centers[level]. Therefore, centers[0..level - 1] should be initialized and computed before calling this function */ public static double calc_center(int[] circles, int[] plan, double[] centers, int level) { // if we only have 1 circle, we assume that the first circle is in (0, 0) double center = 0; /* compare circles[plan[plan_index]] with ALL the previous circles */ for (int k = 0; k < level; k++) { double r1 = circles[plan[k]]; double r2 = circles[plan[level]]; center = Math.max(center, centers[k] + 2 * Math.sqrt(r1 * r2)); } return center; } /* This function only does some easy things: for all the circles, use its center and radius to calc (c+r) and (c-r). * and then use min(c-r) and max(c+r) to calculate the interval length. * n.b. centers[0..level] should be initialized and computed before calling this function * */ public static double calc_length(int[] circles, int[] plan, double[] centers, int level) { /* all circles */ // the first circle double interval_left = 0; double interval_right = 0; // it's ok if we only have 1 circle. calcCenter() will return 0 if we only have 1 circle for (int i = 0; i <= level; i++) { double r = circles[plan[i]]; double c = centers[i]; interval_left = Math.min(interval_left, c - r); interval_right = Math.max(interval_right, c + r); } return interval_right - interval_left; } public static double calc_cost(Node node) { // the radius of circles are sorted in ascending order: Arrays.sort(circles); // prune: we ensure that the best solution won't contain 3 successive ascending circles or 3 successive descending circles. if (node.level >= 2) { if (node.plan[node.level] > node.plan[node.level - 1] && node.plan[node.level - 1] > node.plan[node.level - 2]) { return Double.MAX_VALUE; } if (node.plan[node.level] < node.plan[node.level - 1] && node.plan[node.level - 1] < node.plan[node.level - 2]) { return Double.MAX_VALUE; } } double cost = 0; double x_k = node.centers[node.level]; // n.b. r_1 ... r_k totally k circles double r_1 = Node.circles[node.plan[0]]; cost += x_k; cost += r_1; // n.b. we drop the r_k: cost += r_k; // since the radius of r is the min-imum radius of the circles[k+1..n], we can know r <= r_k , // it's safe for us to assume lower-bound of the length of circles[k+1..n] is (left_circles_amount * (2 * r)) double r_k = Node.circles[node.plan[node.level]]; double r = r_k; for (int i = node.level + 1; i < Node.n; i++) { r = Math.min(r, Node.circles[node.plan[i]]); } cost += (2 * (Node.n - node.level - 1) + 1) * r; return cost; } public static void swap(int[] arr, int i, int j) { int temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; } public static double solve(int n, int[] circles) { /* Initialize */ // n.b. in ascending order of circle radius Arrays.sort(circles); Node.n = n; Node.circles = circles; Node.ans = Double.MAX_VALUE; /* Construct the queue and init */ PriorityQueue<Node> PQ = new PriorityQueue<>((o1, o2) -> { // n.b. if o1.cost == o2.cost, we need to compare o1.level and o2.level // it's also ok if you considerate the LEVEL inside your COST function. // we have tried and found that: more complex comparator won't benefit us more. return (int) ((o1.cost - o1.level) - (o2.cost - o2.level)); }); // add dummy node int[] root_node = new int[n]; for (int i = 0; i < n; i++) root_node[i] = i; PQ.add(new Node(-1, root_node, new double[n])); // BFS with Priority while (!PQ.isEmpty()) { /* Get one node */ Node current_node = PQ.poll(); int $level = current_node.level + 1; /* Generate all the next nodes */ for (int k = $level; k < n; k++) { /* Generate the next-node */ int[] $plan = current_node.plan.clone(); swap($plan, $level, k); double[] $centers = current_node.centers.clone(); $centers[$level] = calc_center(circles, $plan, $centers, $level); /* Check the node ! */ // n.b. centers[level] + circles[plan[0]] + circles[plan[level]] is NOT ALWAYS the LENGTH of the plan ! // prune: if the next-node is NOT BETTER THAN current solution, skip it if ($centers[$level] + circles[$plan[0]] + circles[$plan[$level]] < Node.ans) { Node $ = new Node($level, $plan, $centers); $.cost = calc_cost($); // prune: if the next-node is out of BOUND if ($.cost < Node.ans) { if ($level == n - 1) { Node.ans = Math.min(Node.ans, calc_length(circles, $plan, $centers, $level)); } else PQ.add($); } } } } return Node.ans; } public static void main(String[] args) { for (Scanner scanner : judger) { int n = scanner.nextInt(); int[] a = new int[n]; for (int i = 0; i < n; i++) { a[i] = scanner.nextInt(); } System.out.printf("%.3f", solve(n, a)); } } } </code></pre> <h4>Benchmark</h4> <pre><code class="language-java" lang="java">----------------------------------------------------- Current Case: CIRCLE0.in & CIRCLE0.out Expected Input: [3, 1 1 2] Expected Output: [7.657] Your Output: [7.657] Time Cost: 4.585800 ms (4585800 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE1.in & CIRCLE1.out Expected Input: [5, 59 23 41 70 47 ] Expected Output: [454.388] Your Output: [454.388] Time Cost: 1.767200 ms (1767200 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE10.in & CIRCLE10.out Expected Input: [10, 9 2 117 45 9 3 142 14 9 98 ] Expected Output: [755.928] Your Output: [755.928] Time Cost: 235.701100 ms (235701100 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE11.in & CIRCLE11.out Expected Input: [2, 10000 1] Expected Output: [20000.000] Your Output: [20000.000] Time Cost: 1.475400 ms (1475400 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE12.in & CIRCLE12.out Expected Input: [2, 10000 10000] Expected Output: [40000.000] Your Output: [40000.000] Time Cost: 1.274300 ms (1274300 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE13.in & CIRCLE13.out Expected Input: [6, 1 1 2 2 3 5] Expected Output: [24.136] Your Output: [24.136] Time Cost: 1.613500 ms (1613500 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE14.in & CIRCLE14.out Expected Input: [3, 1 2 9] Expected Output: [19.485] Your Output: [19.485] Time Cost: 1.742700 ms (1742700 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE2.in & CIRCLE2.out Expected Input: [7, 94 35 20 88 55 28 57 ] Expected Output: [666.874] Your Output: [666.874] Time Cost: 2.466700 ms (2466700 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE3.in & CIRCLE3.out Expected Input: [7, 25 1 5 74 47 77 8 ] Expected Output: [415.089] Your Output: [415.089] Time Cost: 3.496700 ms (3496700 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE4.in & CIRCLE4.out Expected Input: [9, 17 49 77 84 86 64 75 88 65 ] Expected Output: [1159.668] Your Output: [1159.668] Time Cost: 49.480700 ms (49480700 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE5.in & CIRCLE5.out Expected Input: [10, 99 22 17 45 91 73 42 14 9 98 ] Expected Output: [858.474] Your Output: [858.474] Time Cost: 176.674800 ms (176674800 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE6.in & CIRCLE6.out Expected Input: [10, 51 100 66 37 30 83 87 98 31 43 ] Expected Output: [1140.471] Your Output: [1140.471] Time Cost: 193.609000 ms (193609000 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE7.in & CIRCLE7.out Expected Input: [10, 51 100 66 37 30 1 87 98 31 3 ] Expected Output: [902.696] Your Output: [902.696] Time Cost: 179.229300 ms (179229300 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE8.in & CIRCLE8.out Expected Input: [9, 1 49 77 8 86 6 75 88 3 ] Expected Output: [738.394] Your Output: [738.394] Time Cost: 21.907700 ms (21907700 ns) Accepted ----------------------------------------------------- Current Case: CIRCLE9.in & CIRCLE9.out Expected Input: [10, 9 22 17 45 9 3 42 14 9 98 ] Expected Output: [400.389] Your Output: [400.389] Time Cost: 120.563100 ms (120563100 ns) Accepted ----------------------------------------------------- Result Statistics: √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ </code></pre> <h2>Reference</h2> <ol> <li><a href="https://zh.coursera.org/lecture/algorithms/074yuan-pai-lie-wen-ti-CasbS">Coursera, Design and Analysis of Algorithms - Circle Permutation</a></li> <li><a href="https://blog.csdn.net/qq_43470416/article/details/106364345">CSDN, myy_cjw, 圆排列问题</a></li> <li><a href="https://blog.csdn.net/weixin_43939564/article/details/106724719">CSDN, Lrish Coffee, 圆排列问题</a></li> </ol>{% endraw %}