{"id":5494,"date":"2026-09-23T18:31:12","date_gmt":"2026-09-23T10:31:12","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=5494"},"modified":"2026-09-24T16:33:44","modified_gmt":"2026-09-24T08:33:44","slug":"interpretation_paper_persistence_guided_prescribed_topological_simplification","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2026\/09\/23\/interpretation_paper_persistence_guided_prescribed_topological_simplification\/","title":{"rendered":"\u300aPersistence-guided Prescribed Topological Simplification\u300b\u8bba\u6587\u89e3\u8bfb"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u6700\u8fd1\u5728\u901bSiggraph 2026\u7684\u6587\u7ae0\u65f6\u6dd8\u5230\u4e86\u4e00\u7bc7\u5c06\u6301\u7eed\u540c\u8c03\u4e0e\u56fe\u5f62\u5b66\u7ed3\u5408\u5f97\u6781\u4e3a\u5de7\u5999\u7684\u6587\u7ae0Rong L, Ju T. Persistence-guided Prescribed Topological Simplification[J]. ACM Transactions on Graphics (TOG), 2026, 45(4): 1-16. \u4e0e\u5bf913\u5e74\u524d\u7684\u4e00\u7bc7\u6587\u7ae0Dey T K, Fan F, Wang Y. An efficient computation of handle and tunnel loops via Reeb graphs[J]. ACM Transactions on Graphics (TOG), 2013, 32(4): 1-10.\u7684\u611f\u89c9\u7c7b\u4f3c, \u53ef\u4ee5\u8bf4\u90fd\u8ba9\u6211\u7231\u4e0d\u91ca\u624b. \u524d\u51e0\u5e74\u672c\u6765\u662f\u6709\u7c7b\u4f3c\u7684\u60f3\u6cd5, \u4ee5\u6301\u7eed\u540c\u8c03\u7684\u89c2\u70b9\u89e3\u91ca\u51cf\u9762\u7684\u8fc7\u7a0b, \u4f46\u5948\u4f55\u5bf9\u6301\u7eed\u540c\u8c03\u7684\u7406\u89e3\u4e0d\u591f\u6df1\u523b, \u5f53\u65f6\u4e5f\u6ca1\u7ee7\u7eed\u843d\u5730. \u6b64\u6b21\u770b\u5230\u8fd9\u7bc7\u6587\u7ae0, \u81ea\u7136\u8ba9\u6211\u8d76\u7d27\u62dc\u8bfb\u4e86\u8d77\u6765. \u672c\u6765\u4ee5\u4e3a\u5728AI\u7684\u8f85\u52a9\u4e0b, \u7814\u8bfb\u7684\u8fc7\u7a0b\u4f1a\u5341\u5206\u987a\u5229, \u7ed3\u679c\u6ca1\u60f3\u5230\u81ea\u5df1\u592a\u83dc, \u4f9d\u65e7\u662f\u8bfb\u5f97\u78d5\u78d5\u7eca\u7eca\u7684. \u6240\u5e78\u65ad\u65ad\u7eed\u7eed\u8017\u65f6\u4e00\u4e2a\u6708, \u603b\u7b97\u662f\u5543\u4e0b\u6765\u4e86, \u987a\u5e26\u5728UE\u4e2d\u4e5f\u5b9e\u73b0\u4e86\u4e00\u7248\u6548\u679c.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. Linxuan Rong and Tao Ju. [Persistence-guided Prescribed Topological Simplification](https:\/\/doi.org\/10.1145\/3811384). ACM Transactions on Graphics, 45(4), Article 165, 2026.<\/p>\n<p>\u8fd9\u7bc7\u8bba\u6587\u5141\u8bb8\u7528\u6237\u6307\u5b9a\u5f62\u72b6\u4e2d\u8981\u4fdd\u7559\u7684\u8fde\u901a\u5206\u652f, \u8d2f\u7a7f\u901a\u9053\u548c\u7a7a\u8154\u6570\u91cf. \u4f5c\u8005\u6309\u6301\u4e45\u5ea6\u9009\u62e9\u4fdd\u7559\u7684\u7279\u5f81, \u4ece\u9aa8\u67b6\u6784\u9020\u5207\u5272\u4e0e\u586b\u5145\u5019\u9009, \u518d\u7528\u52a0\u6743\u6700\u5c0f\u5272\u9009\u62e9\u4e00\u6279\u65e0\u51b2\u7a81\u64cd\u4f5c, \u51cf\u5c11\u51e0\u4f55\u6539\u52a8. \u9605\u8bfb\u65f6, \u6211\u53cd\u590d\u6838\u5bf9\u7684\u662f\u5019\u9009\u80fd\u5426\u53ea\u6d88\u9664\u9884\u671f\u7279\u5f81, \u4ee5\u53ca\u6700\u5c0f\u5272\u548c\u8868\u9762\u63d0\u53d6\u5404\u81ea\u4fdd\u8bc1\u4e86\u4ec0\u4e48. \u63a5\u4e0b\u6765\u7684\u5185\u5bb9\u914d\u5408\u539f\u8bba\u6587&#8221;\u98df\u7528&#8221; \u6548\u679c\u66f4\u4f73(\u25cf\u02c7\u2200\u02c7\u25cf)<\/p>\n<p><strong>1. \u76ee\u6807\u6570\u91cf\u4e0e\u4fee\u590d\u4f4d\u7f6e<\/strong><\/p>\n<p>\u8bba\u6587\u8f93\u5165\u4e00\u4e2a\u5f62\u72b6$S$\u548c\u76ee\u6807Betti\u6570$b_k \\le \\beta_k(S)$, \u5e0c\u671b\u8f93\u51fa$S&#8217;$\u6ee1\u8db3$\\beta_k(S&#8217;) = b_k$, \u540c\u65f6\u51cf\u5c11\u51e0\u4f55\u6539\u52a8. \u4e09\u7ef4\u4e2d\u7684$\\beta_0$\u8868\u793a\u8fde\u901a\u5206\u652f\u6570, $\\beta_1$\u8868\u793a\u8d2f\u7a7f\u901a\u9053\u6570, $\\beta_2$\u8868\u793a\u5c01\u95ed\u7a7a\u8154\u6570. \u7528\u6237\u53ef\u4ee5\u4fdd\u7559\u82e5\u5e72\u901a\u9053\u548c\u7a7a\u8154, \u4e0d\u5fc5\u628a\u76ee\u6807\u8bbe\u6210\u53ea\u6709\u4e00\u4e2a\u5206\u652f, \u6ca1\u6709\u901a\u9053\u548c\u7a7a\u8154\u7684\u5e73\u51e1\u62d3\u6251.<br \/>\n$\\\\$ Betti\u6570\u53ea\u89c4\u5b9a\u6570\u91cf. \u51cf\u5c11\u4e00\u4e2a\u901a\u9053, \u65e2\u53ef\u4ee5\u5207\u5f00\u5b9e\u4f53\u7684\u4e00\u90e8\u5206, \u4e5f\u53ef\u4ee5\u586b\u4f4f\u901a\u9053. \u4e24\u79cd\u64cd\u4f5c\u7684\u51e0\u4f55\u4ee3\u4ef7\u4e0d\u540c, \u591a\u4e2a\u64cd\u4f5c\u4e4b\u95f4\u8fd8\u53ef\u80fd\u51b2\u7a81. \u7b97\u6cd5\u9700\u8981\u4ece\u8fd9\u4e9b\u4fee\u6539\u65b9\u5f0f\u4e2d\u9009\u51fa\u80fd\u4e00\u8d77\u6267\u884c\u7684\u4e00\u6279.<br \/>\n$\\\\$ \u4f5c\u8005\u4fdd\u7559\u6bcf\u7c7b\u7279\u5f81\u4e2d\u6301\u4e45\u5ea6\u6700\u9ad8\u7684\u6307\u5b9a\u6570\u91cf. \u6301\u4e45\u5ea6\u63d0\u4f9b\u6392\u5e8f\u4f9d\u636e, \u5374\u65e0\u6cd5\u533a\u5206\u91cd\u8981\u7ed3\u6784\u4e0e\u5c3a\u5ea6\u8f83\u5927\u7684\u91cd\u5efa\u9519\u8bef. \u7528\u6237\u4ecd\u9700\u6839\u636e\u5e94\u7528\u786e\u5b9a\u76ee\u6807\u6570\u91cf\u548c\u4ee3\u4ef7\u5ea6\u91cf.<br \/>\n$\\\\$ \u8bba\u6587\u53ea\u7b80\u5316\u5f53\u524d\u5f62\u72b6\u5bf9\u5e94\u7684\u4e00\u4e2a\u8fc7\u6ee4\u524d\u7f00. \u5728\u5176\u5b83\u65f6\u523b\u5b58\u5728, \u5f53\u524d\u5f62\u72b6\u4e2d\u4e0d\u5b58\u5728\u7684\u7279\u5f81, \u53ef\u4ee5\u4fdd\u7559\u5728\u8fc7\u6ee4\u91cc. \u540e\u9762\u7684\u8fc1\u79fb\u8bc1\u660e\u4e5f\u53ea\u8981\u6c42\u4fdd\u62a4\u5f53\u524d\u5f62\u72b6\u4e2d\u5e94\u4fdd\u7559\u7684\u7279\u5f81.<\/p>\n<p><strong>2. \u6574\u4f53\u6d41\u7a0b\u4e0e\u4e2d\u95f4\u4ea7\u7269<\/strong><\/p>\n<p>\u8bba\u6587\u7b2c4\u8282\u628a\u65b9\u6cd5\u5206\u6210\u8f93\u5165\u8f6c\u6362, \u8fc7\u6ee4\u7b80\u5316\u548c\u8868\u9762\u63d0\u53d6\u4e09\u6b65. \u5176\u4e2d\u8fc7\u6ee4\u7b80\u5316\u662f\u4e00\u4e2a\u5faa\u73af: \u8ba1\u7b97\u6301\u7eed\u914d\u5bf9\u4e0e\u4ee3\u8868\u94fe, \u6784\u9020\u5f62\u72b6\u548c\u80cc\u666f\u7684\u9aa8\u67b6, \u751f\u6210\u5019\u9009, \u7528\u56fe\u5272\u9009\u62e9\u5019\u9009, \u518d\u8fc1\u79fb\u9009\u4e2d\u7684\u80de.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/06-pipeline.png\" alt=\"\" width=\"2200\" height=\"960\" class=\"aligncenter size-full wp-image-5498\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/06-pipeline.png 2200w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/06-pipeline-300x131.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/06-pipeline-768x335.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/06-pipeline-1536x670.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/06-pipeline-2048x894.png 2048w\" sizes=\"(max-width: 2200px) 100vw, 2200px\" \/><\/p>\n<p>\u7b97\u6cd5\u7528\u9aa8\u67b6\u5bfb\u627e\u53ef\u4ee5\u5220\u9664\u7684\u51fa\u751f\u80de\u6216\u53ef\u4ee5\u586b\u5165\u7684\u6b7b\u4ea1\u80de, \u518d\u786e\u5b9a\u5b83\u4eec\u5728\u539f\u5f62\u72b6\u4e2d\u5bf9\u5e94\u7684\u4fee\u6539\u8303\u56f4. \u4f5c\u8005\u4e5f\u5728\u5b9e\u9a8c\u4e2d\u753b\u51fa\u9aa8\u67b6, \u5e2e\u52a9\u89c2\u5bdf\u901a\u9053\u548c\u7a7a\u8154. \u9aa8\u67b6\u5728\u8fd9\u91cc\u662f\u4e2d\u95f4\u4ea7\u7269, \u6700\u7ec8\u8f93\u51fa\u4e3a\u7b80\u5316\u540e\u7684\u5f62\u72b6\u53ca\u5176\u8868\u9762\u7f51\u683c.<br \/>\n$\\\\$ \u6bcf\u8f6e\u56fe\u5272\u5728\u5df2\u6709\u5019\u9009\u4e2d\u627e\u5230\u6570\u91cf\u6700\u591a, \u603b\u4ee3\u4ef7\u6700\u4f4e\u7684\u65e0\u51b2\u7a81\u96c6\u5408, \u540e\u7eed\u8fed\u4ee3\u518d\u5904\u7406\u5269\u4f59\u6b21\u8981\u7279\u5f81. \u8fd9\u4e2a\u6700\u4f18\u6027\u9650\u5b9a\u5728\u5f53\u524d\u5019\u9009\u96c6\u5408\u5185. \u4f5c\u8005\u6ca1\u6709\u8bc1\u660e\u6574\u4e2a\u8fc7\u7a0b\u80fd\u627e\u5230\u51e0\u4f55\u4fee\u6539\u7684\u5168\u5c40\u6700\u4f18\u89e3, \u4e5f\u6ca1\u6709\u4fdd\u8bc1\u4efb\u610f\u8f93\u5165\u90fd\u80fd\u8fbe\u5230\u76ee\u6807Betti\u6570.<\/p>\n<p><strong>3. \u4ece\u7f51\u683c\u6216\u9690\u5f0f\u573a\u5efa\u7acb\u8fc7\u6ee4<\/strong><\/p>\n<p>\u8fc7\u6ee4\u7531\u73af\u5883\u80de\u590d\u5f62$\\mathbb X$, \u80de\u7684\u5168\u5e8f$\\pi$\u548c\u65f6\u95f4\u51fd\u6570$f$\u7ec4\u6210. $\\pi$\u8981\u6c42\u9762\u5148\u4e8e\u4e0a\u80de, \u56e0\u800c\u6bcf\u4e2a\u524d\u7f00\u90fd\u662f\u80de\u590d\u5f62; $f$\u6cbf$\\pi$\u4e0d\u51cf. \u540c\u4e00\u65f6\u95f4\u53ef\u4ee5\u5305\u542b\u591a\u4e2a\u80de, \u4f46\u6392\u5e8f\u65f6\u4ecd\u8981\u6ee1\u8db3\u9762\u5173\u7cfb.<br \/>\n$\\\\$ \u4e0b\u6587\u6cbf\u7528\u8bba\u6587\u7684\u73af\u5883\u4e0e\u7cfb\u6570\u7ea6\u5b9a: $\\mathbb X$\u5177\u6709\u5e73\u51e1\u62d3\u6251, \u4efb\u610f\u80de\u7684\u65f6\u95f4\u90fd\u4e0d\u4e3a\u96f6, \u94fe\u7684\u52a0\u6cd5\u6309\u6a212\u8ba1\u7b97. \u5bf9\u5076\u5173\u7cfb\u4e5f\u91c7\u7528\u8bba\u6587\u7b2c3.3\u8282\u7684\u80cc\u666f\u6784\u9020, \u4e0d\u63a8\u5e7f\u5230\u4efb\u610f\u80de\u590d\u5f62. \u76ee\u6807\u4e3a\u975e\u7a7a\u5f62\u72b6\u65f6, \u8fde\u901a\u5206\u652f\u6570\u81f3\u5c11\u4e3a1.<br \/>\n$\\\\$ \u5bf9\u7f51\u683c\u4e0a\u7684\u9690\u5f0f\u573a$g$, \u8bba\u6587\u7b2c6.1\u8282\u5728\u9876\u70b9\u5904\u53d6$f(v) = g(v)$, \u5176\u5b83\u80de\u7684\u65f6\u95f4\u53d6\u5176\u9876\u70b9\u503c\u7684\u6700\u5927\u503c:<br \/>\n$$<br \/>\nf(\\sigma) = \\max_{v\\text{\u4e3a}\\sigma\\text{\u7684\u9876\u70b9}}g(v).<br \/>\n$$\u8fd9\u4e2a\u5b9a\u4e49\u4fdd\u8bc1\u9762\u7684\u65f6\u95f4\u4e0d\u5927\u4e8e\u4e0a\u80de\u7684\u65f6\u95f4. \u5c06\u80de\u6309\u65f6\u95f4\u6392\u5e8f, \u5e76\u5728\u5e76\u5217\u65f6\u8ba9\u9762\u5148\u8fdb\u5165, \u5c31\u5f97\u5230\u5408\u6cd5\u8fc7\u6ee4. \u5f62\u72b6\u53d6\u8d1f\u65f6\u95f4\u5b50\u590d\u5f62:<br \/>\n$$<br \/>\n\\mathbb X_f = \\{\\sigma \\in \\mathbb X \\mid f(\\sigma) < 0\\}.\n$$\u56e0\u6b64\u4e00\u4e2a\u80de\u53ea\u6709\u5728\u5168\u90e8\u9876\u70b9\u90fd\u4e3a\u8d1f\u65f6\u624d\u8fdb\u5165\u521d\u59cb\u5f62\u72b6. \u56db\u9762\u4f53\u7f51\u683c\u4e2d\u7684\u8fd9\u79cd\u8d1f\u5b50\u590d\u5f62\u4e0e\u76f8\u5e94\u5206\u7247\u7ebf\u6027\u573a\u7684\u8d1f\u533a\u57df\u5177\u6709\u5bf9\u5e94\u7684\u62d3\u6251; \u7acb\u65b9\u7f51\u683c\u5219\u91c7\u7528\u8d1f\u9876\u70b9\u76846\u8fde\u901a\u7ea6\u5b9a. \u8ba8\u8bba\u4e09\u7ef4\u5b9e\u4f53\u7684Betti\u6570\u65f6, \u4e0d\u80fd\u628a\u8d1f\u533a\u57df\u4e0e\u5b83\u7684\u4e8c\u7ef4\u96f6\u7b49\u503c\u8fb9\u754c\u6df7\u4e3a\u4e00\u8c08.\n$\\\\$ \u4e09\u89d2\u7f51\u683c\u8f93\u5165\u53ef\u4ee5\u5148\u8f6c\u6362\u4e3a\u6709\u7b26\u53f7\u8ddd\u79bb\u4f53, \u4e5f\u53ef\u4ee5\u5148\u6784\u9020\u8d34\u5408\u8868\u9762\u7684\u56db\u9762\u4f53\u7f51\u683c, \u518d\u8ba1\u7b97\u9876\u70b9\u7684\u6709\u7b26\u53f7\u8ddd\u79bb. \u524d\u4e00\u79cd\u65b9\u5f0f\u53ef\u80fd\u4e22\u5931\u7ec6\u5c0f\u51e0\u4f55; \u540e\u4e00\u79cd\u65b9\u5f0f\u66f4\u4f9d\u8d56\u56db\u9762\u4f53\u5256\u5206\u7684\u8d28\u91cf\u4e0e\u89c4\u6a21. \u4f5c\u8005\u8fd8\u7528\u5c0f\u504f\u79fb\u907f\u5f00\u6070\u597d\u4e3a\u96f6\u7684\u9876\u70b9\u503c, \u4ee5\u4fbf\u7528\u4e25\u683c\u6b63\u8d1f\u53f7\u533a\u5206\u5f62\u72b6\u4e0e\u80cc\u666f.\n$\\\\$ \u8fc1\u79fb\u540e, \u9ad8\u7ef4\u80de\u7684\u65f6\u95f4\u53ef\u80fd\u4e0d\u518d\u7b49\u4e8e\u5176\u9876\u70b9\u65f6\u95f4\u7684\u6700\u5927\u503c. \u8868\u9762\u63d0\u53d6\u65f6\u9700\u8981\u5904\u7406\u8fd9\u4e2a\u5dee\u5f02, \u4e0d\u80fd\u53ea\u8bfb\u53d6\u539f\u9876\u70b9\u503c.\n\n<strong>4. \u7528\u6d3b\u8dc3\u914d\u5bf9\u9009\u62e9\u8981\u4fdd\u7559\u7684\u7279\u5f81<\/strong><\/p>\n<p>\u6301\u7eed\u914d\u5bf9$\\{b,d\\}$\u8bb0\u5f55\u4e00\u4e2a\u540c\u8c03\u7c7b\u7684\u51fa\u751f\u4e0e\u6b7b\u4ea1. \u5bf9$k$\u7ef4\u7279\u5f81, \u51fa\u751f\u80de$b$\u4e3a$k$\u7ef4, \u6b7b\u4ea1\u80de$d$\u4e3a$k + 1$\u7ef4; \u6301\u4e45\u5ea6\u4e3a$f(d) &#8211; f(b)$. \u5bf9\u59cb\u7ec8\u4e0d\u6b7b\u4ea1\u7684\u7c7b, \u91c7\u7528\u65e0\u9650\u6b7b\u4ea1\u65f6\u95f4\u7684\u7ea6\u5b9a.<br \/>\n$\\\\$ \u5f53\u524d\u5f62\u72b6\u4f4d\u4e8e\u96f6\u65f6\u523b, \u6240\u4ee5\u53ea\u6709\u6ee1\u8db3$f(b) < 0 < f(d)$\u7684\u914d\u5bf9\u4ee3\u8868\u5176\u4e2d\u5c1a\u5b58\u7684\u7279\u5f81. \u8bba\u6587\u79f0\u8fd9\u4e9b\u914d\u5bf9\u4e3a\u6d3b\u8dc3\u914d\u5bf9. \u6bcf\u4e00\u7ef4\u4e2d, \u6d3b\u8dc3\u914d\u5bf9\u6309\u6301\u4e45\u5ea6\u6392\u5e8f, \u524d$b_k$\u4e2a\u4e3a\u4e3b\u8981\u914d\u5bf9(Major), \u5269\u4f59\u5f85\u6d88\u9664\u7684\u4e3a\u6b21\u8981\u914d\u5bf9(Minor). \u4e3b\u8981\u4e0e\u6b21\u8981\u96c6\u5408\u7684\u5206\u754c\u82e5\u5b58\u5728\u6301\u4e45\u5ea6\u5e76\u5217, \u8fd8\u9700\u6838\u5bf9\u5e76\u5217\u89c4\u5219\u4e0e\u8fc1\u79fb\u8bc1\u660e\u662f\u5426\u76f8\u5bb9; \u4e0d\u80fd\u4ec5\u51ed\u4efb\u610f\u56fa\u5b9a\u6392\u5e8f\u5c31\u65ad\u8a00\u4e3b\u8981\u914d\u5bf9\u7684\u8eab\u4efd\u4fdd\u6301\u4e0d\u53d8.\n$\\\\$ \u5bf9\u6b21\u8981\u914d\u5bf9, \u53ef\u4ee5\u628a\u51fa\u751f\u80de\u4ece\u8d1f\u65f6\u95f4\u79fb\u5230\u6b63\u65f6\u95f4, \u6216\u628a\u6b7b\u4ea1\u80de\u4ece\u6b63\u65f6\u95f4\u79fb\u5230\u8d1f\u65f6\u95f4. \u82e5\u8fc1\u79fb\u540e\u7684\u914d\u5bf9\u4e24\u7aef\u4f4d\u4e8e\u96f6\u65f6\u523b\u540c\u4fa7, \u5b83\u5c31\u4e0d\u518d\u4ee3\u8868\u5f53\u524d\u5f62\u72b6\u4e2d\u7684\u7279\u5f81. \u8fd9\u4e9b\u80de\u4ecd\u5728\u73af\u5883\u590d\u5f62\u91cc, \u4e5f\u53ef\u80fd\u7ee7\u7eed\u6784\u6210\u975e\u6d3b\u8dc3\u7684\u6301\u7eed\u914d\u5bf9.\n$\\\\$ \u5355\u72ec\u6539\u4e00\u4e2a\u80de\u7684\u7b26\u53f7\u672a\u5fc5\u5408\u6cd5. \u5220\u9664\u9762\u5374\u4fdd\u7559\u5b83\u7684\u4e0a\u80de, \u6216\u6dfb\u52a0\u4e00\u4e2a\u7f3a\u5c11\u9762\u7684\u80de, \u90fd\u4f1a\u7834\u574f\u80de\u590d\u5f62\u6761\u4ef6. \u628a\u76f8\u5173\u80de\u4e00\u8d77\u8fc1\u79fb\u4e5f\u53ef\u80fd\u8bef\u6539\u5176\u5b83\u7279\u5f81, \u56e0\u6b64\u6784\u9020\u5019\u9009\u65f6\u8fd8\u8981\u63a7\u5236\u5176\u62d3\u6251\u6548\u679c.\n\n<strong>5. \u4fdd\u62a4\u4ee3\u8868\u5faa\u73af\u7684\u9aa8\u67b6\u5316<\/strong><\/p>\n<p>\u80de$\\sigma$\u5728\u5f53\u524d\u590d\u5f62\u4e2d\u6ca1\u6709\u4e0a\u80de\u65f6, \u79f0\u4e3a\u5b64\u7acb\u80de. \u8fd9\u91cc\u7684\u5b64\u7acb\u6307\u9762\u5173\u7cfb: \u4e00\u4e2a\u4e0e\u90bb\u5c45\u5171\u4eab\u8fb9, \u4f46\u6ca1\u6709\u4e09\u7ef4\u4e0a\u80de\u7684\u4e09\u89d2\u5f62, \u4ecd\u662f\u5b64\u7acb\u7684\u4e8c\u7ef4\u80de. \u82e5\u5b83\u7684\u4e00\u5f20\u9762$\\phi$\u53ea\u6709$\\sigma$\u8fd9\u4e2a\u4e0a\u80de, \u4e8c\u8005\u6784\u6210\u7b80\u5355\u5bf9$\\{\\phi,\\sigma\\}$. \u540c\u65f6\u5220\u9664\u5b83\u4eec\u662f\u4e00\u6b21\u4fdd\u6301\u540c\u4f26\u7c7b\u578b\u7684\u521d\u7b49\u584c\u7f29.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/01-simple-pair.png\" alt=\"\" width=\"2200\" height=\"720\" class=\"aligncenter size-full wp-image-5500\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/01-simple-pair.png 2200w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/01-simple-pair-300x98.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/01-simple-pair-768x251.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/01-simple-pair-1536x503.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/01-simple-pair-2048x670.png 2048w\" sizes=\"(max-width: 2200px) 100vw, 2200px\" \/><\/p>\n<p>\u4f5c\u8005\u53cd\u590d\u5220\u9664\u7b80\u5355\u5bf9\u6765\u8ba1\u7b97\u5f62\u72b6\u9aa8\u67b6$K$, \u4f46\u989d\u5916\u7981\u6b62\u5220\u9664\u4efb\u4f55\u6d3b\u8dc3\u914d\u5bf9\u751f\u6210\u5143\u4e0a\u7684\u80de. \u751f\u6210\u5143(Generator) \u662f\u8be5\u540c\u8c03\u7c7b\u7684\u4ee3\u8868\u5faa\u73af, \u5305\u542b\u5176\u51fa\u751f\u80de$b$. \u4fdd\u62a4\u751f\u6210\u5143\u5c31\u4fdd\u8bc1\u6d3b\u8dc3\u51fa\u751f\u80de\u7559\u5728\u9aa8\u67b6\u4e2d, \u540c\u65f6\u8ba9\u5220\u9664\u7684\u7b80\u5355\u5bf9\u6ee1\u8db3\u9644\u5f55A.5\u7684\u6761\u4ef6.<br \/>\n$\\\\$ \u6b21\u8981\u914d\u5bf9\u7684\u751f\u6210\u5143\u4e5f\u8981\u4fdd\u7559, \u56e0\u4e3a\u540e\u9762\u9700\u8981\u5bf9\u5c1a\u672a\u9009\u4e2d\u7684\u6d3b\u8dc3\u914d\u5bf9\u7ee7\u7eed\u4f7f\u7528\u539f\u4ee3\u8868\u94fe. \u7b80\u5355\u5bf9\u584c\u7f29\u672c\u8eab\u4fdd\u6301\u540c\u4f26\u7c7b\u578b, \u5373\u4f7f\u4e0d\u4fdd\u62a4\u751f\u6210\u5143\u4e5f\u4e0d\u4f1a\u6539\u53d8Betti\u6570; \u4f46\u5b83\u53ef\u80fd\u79fb\u9664\u539f\u4ee3\u8868\u94fe\u4e2d\u7684\u80de. \u4fdd\u62a4\u751f\u6210\u5143\u63d0\u4f9b\u7684\u662fA.5\u6240\u9700\u7684\u4ee3\u8868\u94fe\u907f\u8ba9\u6761\u4ef6, \u6bd4\u5355\u7eaf\u4fdd\u6301\u540c\u4f26\u66f4\u5f3a.<br \/>\n$\\\\$ \u9aa8\u67b6\u53ef\u80fd\u5305\u542b\u8fb9\u548c\u9762, \u5206\u522b\u652f\u6491\u901a\u9053\u548c\u7a7a\u8154\u7b49\u7279\u5f81. \u4fdd\u62a4\u4ee3\u8868\u5faa\u73af\u7684\u9650\u5236\u4e5f\u53ef\u80fd\u963b\u6b62\u8fdb\u4e00\u6b65\u584c\u7f29. \u56e0\u6b64, \u8fd9\u91cc\u5f97\u5230\u7684\u662f\u7528\u4e8e\u6784\u9020\u5019\u9009\u7684\u9aa8\u67b6, \u4e0d\u4fdd\u8bc1\u552f\u4e00\u6216\u5168\u5c40\u6700\u5c0f.<\/p>\n<p><strong>6. \u751f\u6210\u96c6\u5408\u5982\u4f55\u5b9a\u4e49\u5207\u5272\u4e0e\u586b\u5145?<\/strong><\/p>\n<p>\u8bbe$K$\u7531$X$\u6309\u987a\u5e8f\u5220\u9664\u4e00\u6279\u7b80\u5355\u5bf9\u5f97\u5230, \u51fa\u751f\u80de$b$\u5728$K$\u4e2d\u5b64\u7acb. \u751f\u6210\u96c6\u5408$G_{X,K}(b)$\u53d6\u81ea\u5df2\u5220\u9664\u7684\u80de\u4e0e$b$, \u662f\u5176\u4e2d\u5305\u542b$b$\u5e76\u5bf9$X$\u4e2d\u7684\u4e0a\u80de\u5c01\u95ed\u7684\u6700\u5c0f\u96c6\u5408. \u76f4\u89c2\u4e0a, \u4ece\u9aa8\u67b6\u91cc\u8981\u5220\u7684$b$\u51fa\u53d1, \u628a\u539f\u5f62\u72b6\u4e2d\u4f9d\u8d56\u5b83\u7684\u80de\u4e00\u8d77\u7eb3\u5165\u5220\u9664\u8303\u56f4.<br \/>\n$\\\\$ \u5220\u9664\u751f\u6210\u96c6\u5408\u540e, $X \\setminus G_{X,K}(b)$\u4ecd\u662f\u80de\u590d\u5f62, \u5e76\u4e0e$K \\setminus \\{b\\}$\u540c\u4f26\u7b49\u4ef7. \u4f5c\u8005\u636e\u6b64\u628a$G_{\\mathbb X_f,K}(b)$\u5b9a\u4e49\u4e3a$b$\u7684\u5207\u5272\u5019\u9009. \u751f\u6210\u96c6\u5408\u672c\u8eab\u901a\u5e38\u4e0d\u5411\u4e0b\u5c01\u95ed, \u65e0\u9700\u6784\u6210\u80de\u590d\u5f62; \u5b83\u8868\u793a\u64cd\u4f5c\u8303\u56f4, \u4e0e\u8868\u793a\u5faa\u73af\u7684\u751f\u6210\u5143\u4e0d\u540c.<br \/>\n$\\\\$ \u586b\u5145\u5728\u80cc\u666f\u4fa7\u7528\u540c\u4e00\u5957\u6784\u9020. \u666e\u901a\u8865\u96c6$\\bar X = \\mathbb X \\setminus X$\u53ef\u80fd\u7f3a\u5c11\u67d0\u4e9b\u80de\u7684\u9762, \u56e0\u800c\u4e0d\u4e00\u5b9a\u662f\u80de\u590d\u5f62. \u5bf9\u5076\u53cd\u8f6c\u9762\u4e0e\u4e0a\u80de\u5173\u7cfb, \u4f7f\u5bf9\u5076\u8865\u96c6\u6210\u4e3a\u80cc\u666f\u80de\u590d\u5f62:<br \/>\n$$<br \/>\n\\widetilde X = \\widehat{\\bar X} = \\widehat{\\mathbb X} \\setminus \\widehat X.<br \/>\n$$\u5f62\u72b6\u8fc7\u6ee4$\\{\\mathbb X,\\pi,f\\}$\u5bf9\u5e94\u80cc\u666f\u8fc7\u6ee4$\\{\\widehat{\\mathbb X},-\\pi,-f\\}$, \u5176\u4e2d$-\\pi$\u8868\u793a\u9006\u5e8f. \u6309\u8bba\u6587\u4f7f\u7528\u7684\u6301\u7eed\u540c\u8c03\u4e0e\u6301\u7eed\u76f8\u5bf9\u4e0a\u540c\u8c03\u5bf9\u5076, \u6709\u9650\u914d\u5bf9\u6ee1\u8db3$\\{b,d\\} \\leftrightarrow \\{\\widehat d,\\widehat b\\}$. \u82e5\u73af\u5883\u590d\u5f62\u7684\u7ef4\u6570\u4e3a$n$, $k$\u7ef4\u5f62\u72b6\u7279\u5f81\u5bf9\u5e94$n &#8211; k &#8211; 1$\u7ef4\u80cc\u666f\u7279\u5f81.<br \/>\n$\\\\$ \u4f5c\u8005\u5728\u80cc\u666f\u9aa8\u67b6\u5316\u65f6\u4fdd\u62a4\u4f59\u751f\u6210\u5143(Co-generator), \u5373\u80cc\u666f\u4fa7\u7684\u4ee3\u8868\u5faa\u73af. \u5bf9\u5f62\u72b6\u914d\u5bf9$\\{b,d\\}$, \u5b83\u5305\u542b$\\widehat d$. \u82e5$\\widehat d$\u5728\u80cc\u666f\u9aa8\u67b6\u4e2d\u5b64\u7acb, \u5c31\u6784\u9020\u5176\u751f\u6210\u96c6\u5408\u5e76\u8f6c\u56de\u539f\u590d\u5f62, \u5f97\u5230$d$\u7684\u586b\u5145\u5019\u9009. \u586b\u5145\u539f\u5f62\u72b6, \u7b49\u4ef7\u4e8e\u4ece\u5bf9\u5076\u80cc\u666f\u5220\u9664\u8fd9\u7ec4\u80de.<br \/>\n$\\\\$ \u51fa\u751f\u80de\u6216\u5bf9\u5076\u6b7b\u4ea1\u80de\u82e5\u5728\u76f8\u5e94\u9aa8\u67b6\u4e2d\u4e0d\u5b64\u7acb, \u5c31\u65e0\u6cd5\u6309\u4e0a\u8ff0\u89c4\u5219\u751f\u6210\u5207\u5272\u6216\u586b\u5145. \u4f8b\u5982, \u4e3a\u4e86\u4fdd\u7559\u5176\u5b83\u6d3b\u8dc3\u7279\u5f81\u7684\u4ee3\u8868\u5faa\u73af, \u7b97\u6cd5\u53ef\u80fd\u5fc5\u987b\u4fdd\u7559\u76ee\u6807\u7684\u4e0a\u80de. \u8fd9\u4e2a\u914d\u5bf9\u5728\u672c\u8f6e\u4fbf\u53ef\u80fd\u7f3a\u5c11\u4e00\u79cd, \u751a\u81f3\u4e24\u79cd\u5019\u9009.<\/p>\n<p><strong>7. \u5408\u6cd5\u8fc1\u79fb\u4e0e\u547d\u98985.1<\/strong><\/p>\n<p>\u8bbe$C$\u4e3a\u9009\u4e2d\u5207\u5272\u7684\u80de\u5e76\u96c6, $F$\u4e3a\u9009\u4e2d\u586b\u5145\u7684\u80de\u5e76\u96c6. \u5207\u5272\u53d6\u81ea\u8d1f\u65f6\u95f4\u80de, \u586b\u5145\u53d6\u81ea\u6b63\u65f6\u95f4\u80de, \u4e8c\u8005\u539f\u672c\u5c31\u6ca1\u6709\u5171\u540c\u80de. \u8bba\u6587\u6240\u8bf4\u7684&#8221;\u4e92\u4e0d\u76f8\u4ea4&#8221; \u8fd8\u8981\u6c42: $C$\u4e2d\u4efb\u4f55\u80de\u90fd\u4e0d\u80fd\u6709\u4e0a\u80de\u5c5e\u4e8e$F$. \u8fd9\u662f\u9762\u5173\u7cfb\u7ea6\u675f, \u6bd4\u96c6\u5408\u4ea4\u96c6\u4e3a\u7a7a\u66f4\u5f3a.<br \/>\n$\\\\$ \u4ee4$N$, $P$\u5206\u522b\u4e3a\u539f\u8fc7\u6ee4\u4e2d\u7684\u8d1f\u65f6\u95f4\u4e0e\u6b63\u65f6\u95f4\u5e8f\u5217. \u8fc1\u79fb\u6309\u4ee5\u4e0b\u987a\u5e8f\u7a33\u5b9a\u91cd\u6392, \u5404\u7ec4\u5185\u90e8\u4fdd\u7559\u539f\u76f8\u5bf9\u987a\u5e8f:<br \/>\n$$<br \/>\n\\pi&#8217; = (N \\setminus C)\\;|\\;F\\;|\\;C\\;|\\;(P \\setminus F).<br \/>\n$$\u7ed9$F$\u5206\u914d\u9760\u8fd1\u96f6\u7684\u8d1f\u65f6\u95f4, \u7ed9$C$\u5206\u914d\u9760\u8fd1\u96f6\u7684\u6b63\u65f6\u95f4, \u5e76\u4fdd\u6301\u65f6\u95f4\u6cbf$\\pi&#8217;$\u4e0d\u51cf. \u5b9e\u73b0\u53ef\u4ee5\u76f4\u63a5\u6784\u9020\u8fd9\u4e2a\u7ed3\u679c, \u65e0\u9700\u6a21\u62df\u9644\u5f55\u8bc1\u660e\u91cc\u7684\u6bcf\u6b21\u76f8\u90bb\u4ea4\u6362.<br \/>\n$\\\\$ \u5207\u5272\u5305\u542b\u76ee\u6807\u7684\u8d1f\u65f6\u95f4\u4e0a\u80de, \u586b\u5145\u5305\u542b\u76ee\u6807\u7f3a\u5c11\u7684\u6b63\u65f6\u95f4\u9762, \u800c\u4e92\u4e0d\u76f8\u4ea4\u6761\u4ef6\u6392\u9664\u4e86\u5220\u6389\u9762\u5374\u6dfb\u52a0\u5176\u4e0a\u80de\u7684\u51b2\u7a81. \u8fd9\u4e9b\u95ed\u5305\u4e0e\u9762\u5173\u7cfb\u6761\u4ef6\u4fdd\u8bc1\u4e86\u91cd\u6392\u7684\u5408\u6cd5\u6027, \u4e0d\u80fd\u7701\u7565\u540e\u53ea\u505a\u6279\u91cf\u7b26\u53f7\u7ffb\u8f6c.<br \/>\n$\\\\$ \u547d\u98985.1\u8bc1\u660e: \u9009\u4e2d$m$\u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u6b21\u8981\u914d\u5bf9\u5019\u9009, \u8fc1\u79fb\u540e\u4fdd\u6301\u5408\u6cd5\u8fc7\u6ee4\u4e0e\u539f\u6709\u4e3b\u8981\u914d\u5bf9, \u5e76\u6070\u597d\u51cf\u5c11$m$\u4e2a\u6b21\u8981\u6d3b\u8dc3\u914d\u5bf9. \u540e\u9762\u7684\u4f18\u5316\u7528\u5019\u9009\u6570\u91cf\u8861\u91cf\u6d88\u9664\u7684\u7279\u5f81\u6570, \u4f9d\u636e\u5c31\u5728\u8fd9\u91cc.<\/p>\n<p><strong>8. \u628a\u5019\u9009\u9009\u62e9\u53d8\u6210\u4e8c\u5206\u56fe\u95ee\u9898<\/strong><\/p>\n<p>\u628a\u6bcf\u4e2a\u5207\u5272\u5019\u9009\u8868\u793a\u4e3a$V_c$\u4e2d\u7684\u4e00\u4e2a\u9876\u70b9, \u6bcf\u4e2a\u586b\u5145\u5019\u9009\u8868\u793a\u4e3a$V_f$\u4e2d\u7684\u4e00\u4e2a\u9876\u70b9. \u82e5\u4e00\u5bf9\u5207\u5272\u4e0e\u586b\u5145\u4e0d\u6ee1\u8db3\u4e92\u4e0d\u76f8\u4ea4\u6761\u4ef6, \u5c31\u5728\u5b83\u4eec\u4e4b\u95f4\u8fde\u8fb9. \u540c\u4fa7\u5019\u9009\u4e4b\u95f4\u4e0d\u8fde\u8fb9, \u5f97\u5230\u4e8c\u5206\u51b2\u7a81\u56fe$G = (V_c \\cup V_f,E_G)$.<br \/>\n$\\\\$ \u72ec\u7acb\u96c6\u5bf9\u5e94\u4e00\u7ec4\u65e0\u51b2\u7a81\u5019\u9009. \u6839\u636e\u547d\u98985.1, \u8981\u5728\u672c\u8f6e\u6d88\u9664\u6700\u591a\u6b21\u8981\u7279\u5f81, \u5c31\u5e94\u9009\u6700\u5927\u72ec\u7acb\u96c6. \u6587\u4e2d\u7684MIS\u6307Maximum Independent Set, \u4e0d\u662f\u4ec5\u4ec5\u65e0\u6cd5\u518d\u52a0\u5165\u9876\u70b9\u7684\u6781\u5927\u72ec\u7acb\u96c6. \u6570\u91cf\u76f8\u540c\u7684\u6700\u5927\u72ec\u7acb\u96c6\u53ef\u80fd\u6709\u591a\u4e2a, \u4e0b\u4e00\u6b65\u518d\u7528\u51e0\u4f55\u4ee3\u4ef7\u533a\u5206\u5b83\u4eec.<br \/>\n$\\\\$ \u4f5c\u8005\u5148\u6784\u9020\u5355\u4f4d\u5bb9\u91cf\u7f51\u7edc$H$: \u6dfb\u52a0\u6e90\u70b9$s$\u548c\u6c47\u70b9$t$, \u4ee4$s \\to c$\u4e0e$f \\to t$\u5bb9\u91cf\u4e3a1, \u51b2\u7a81\u5f27$c \\to f$\u5bb9\u91cf\u4e3a$\\infty$. \u5bf9\u6709\u9650\u5bb9\u91cf\u5272$(S,T)$, \u53d6$s \\in S$, $t \\in T$, \u5b9a\u4e49\u8986\u76d6$U$\u4e0e\u72ec\u7acb\u96c6$I$:<br \/>\n$$<br \/>\nU = (V_c \\cap T) \\cup (V_f \\cap S),\\qquad<br \/>\nI = (V_c \\cap S) \\cup (V_f \\cap T).<br \/>\n$$\u82e5\u67d0\u6761\u51b2\u7a81\u8fb9\u7684\u4e24\u7aef\u90fd\u4e0d\u5728$U$, \u5c31\u6709$c \\in S,f \\in T$, \u5272\u4e2d\u5c06\u5305\u542b\u65e0\u7a77\u5bb9\u91cf\u5f27. \u56e0\u6b64$U$\u8986\u76d6\u6240\u6709\u51b2\u7a81\u8fb9, \u4e14\u6709\u9650\u5272\u8fb9\u4e0e$U$\u7684\u9876\u70b9\u4e00\u4e00\u5bf9\u5e94, \u5272\u5bb9\u91cf\u4e3a$|U|$. \u53cd\u8fc7\u6765, \u6bcf\u4e2a\u9876\u70b9\u8986\u76d6\u4e5f\u80fd\u6784\u9020\u540c\u5bb9\u91cf\u7684\u5272. \u6700\u5c0f\u5272\u4e8e\u662f\u5bf9\u5e94\u6700\u5c0f\u9876\u70b9\u8986\u76d6, \u5176\u8865\u96c6\u5c31\u662f\u6700\u5927\u72ec\u7acb\u96c6.<br \/>\n$\\\\$ \u53cd\u5411\u6784\u9020\u4e0d\u80fd\u7701\u7565, \u5426\u5219\u53ea\u80fd\u8bf4\u660e\u5272\u7ed9\u51fa\u4e86\u67d0\u4e2a\u72ec\u7acb\u96c6, \u8fd8\u4e0d\u80fd\u8bc1\u660e\u5b83\u6700\u5927. \u7ed9\u5b9a\u4efb\u610f\u9876\u70b9\u8986\u76d6$U$, \u5c06$V_c \\setminus U$\u4e0e$V_f \\cap U$\u653e\u5230\u6e90\u70b9\u4fa7, \u5c06\u5176\u4f59\u5019\u9009\u653e\u5230\u6c47\u70b9\u4fa7, \u518d\u5206\u522b\u52a0\u5165$s$, $t$. \u82e5\u5b58\u5728\u4ece\u6e90\u70b9\u4fa7\u5230\u6c47\u70b9\u4fa7\u7684\u51b2\u7a81\u5f27$c \\to f$, \u5219$c$, $f$\u5747\u4e0d\u5c5e\u4e8e$U$, \u4e0e\u8986\u76d6\u6761\u4ef6\u77db\u76fe. \u6240\u4ee5\u5272\u53ea\u5305\u542b\u4e0e$U$\u5bf9\u5e94\u7684\u5355\u4f4d\u5bb9\u91cf\u8fb9, \u5bb9\u91cf\u6070\u4e3a$|U|$. \u73b0\u5728\u5047\u8bbe\u6700\u5c0f\u5272\u7ed9\u51fa\u7684\u72ec\u7acb\u96c6$I$\u4e0d\u662f\u6700\u5927\u7684, \u5219\u66f4\u5927\u72ec\u7acb\u96c6\u7684\u8865\u96c6\u5c31\u662f\u66f4\u5c0f\u8986\u76d6, \u7531\u8fd9\u4e00\u6784\u9020\u4f1a\u5f97\u5230\u66f4\u5c0f\u7684\u5272, \u4ea7\u751f\u77db\u76fe.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/04-cut-cover.png\" alt=\"\" width=\"2200\" height=\"1330\" class=\"aligncenter size-full wp-image-5506\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/04-cut-cover.png 2200w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/04-cut-cover-300x181.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/04-cut-cover-768x464.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/04-cut-cover-1536x929.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/04-cut-cover-2048x1238.png 2048w\" sizes=\"(max-width: 2200px) 100vw, 2200px\" \/><\/p>\n<p>$\\\\$ \u539f\u56fe\u4e2d\u7684\u7d2b\u8272\u66f2\u7ebf\u53ea\u8868\u793a\u5206\u533a. \u9009\u70b9\u65f6\u53d6\u5de6\u4fa7\u5019\u9009\u4e2d\u4f4d\u4e8e\u6e90\u70b9\u4fa7\u7684\u70b9, \u52a0\u4e0a\u53f3\u4fa7\u5019\u9009\u4e2d\u4f4d\u4e8e\u6c47\u70b9\u4fa7\u7684\u70b9. \u6309\u539f\u56fe\u4ece\u4e0a\u5230\u4e0b\u7f16\u53f7, \u72ec\u7acb\u96c6\u4e3a$\\{c_3, f_1, f_2, f_3\\}$, \u8986\u76d6\u4e3a$\\{c_1, c_2, f_4\\}$. \u53e6\u5916, \u6709\u5411\u5272\u53ea\u8ba1\u7b97$S \\to T$\u7684\u5f27, \u4e0d\u80fd\u4ec5\u51ed\u7ebf\u6761\u7a7f\u8fc7\u5206\u754c\u5c31\u628a\u5b83\u7b97\u5165\u5bb9\u91cf.<\/p>\n<p><strong>9. \u5728\u6570\u91cf\u6700\u591a\u7684\u65b9\u6848\u4e2d\u6700\u5c0f\u5316\u4ee3\u4ef7<\/strong><\/p>\n<p>\u8bba\u6587\u5141\u8bb8\u7528\u6237\u4e3a\u6bcf\u4e2a\u5019\u9009\u6307\u5b9a\u51e0\u4f55\u4ee3\u4ef7$w(v)$. \u7b2c7\u8282\u9ed8\u8ba4\u5bf9\u5207\u5272\u53d6$|f(b)|$, \u5bf9\u586b\u5145\u53d6$|f(d)|$, \u504f\u597d\u65f6\u95f4\u51fd\u6570\u610f\u4e49\u4e0a\u66f4\u9760\u8fd1\u5f62\u72b6\u7684\u64cd\u4f5c. \u8fd9\u4e2a\u4ee3\u4ef7\u4e0e\u6301\u4e45\u5ea6$f(d) &#8211; f(b)$\u4e0d\u540c: \u6301\u4e45\u5ea6\u7528\u4e8e\u51b3\u5b9a\u7279\u5f81\u662f\u5426\u5e94\u4fdd\u7559, $w(v)$\u7528\u4e8e\u6bd4\u8f83\u6d88\u9664\u6b21\u8981\u7279\u5f81\u7684\u64cd\u4f5c.<br \/>\n$\\\\$ \u4f5c\u8005\u628a$H$\u4e2d\u6709\u9650\u8fb9\u7684\u5bb9\u91cf\u4ece1\u6539\u4e3a$W &#8211; w(v)$, \u5176\u4e2d$W$\u4e3a\u5168\u90e8\u5019\u9009\u7684\u603b\u4ee3\u4ef7, \u5f97\u5230$H^*$. \u8282\u70b9, \u8fb9, \u65b9\u5411\u4e0e\u51b2\u7a81\u5f27\u5bb9\u91cf\u4fdd\u6301\u4e0d\u53d8. \u5bf9\u6709\u9650\u5272$E$, \u8bb0\u8986\u76d6\u4e3a$U_E$, \u72ec\u7acb\u96c6\u4e3a$I_E$, \u6709:<br \/>\n$$<br \/>\n\\operatorname{cap}_{H^*}(E)<br \/>\n= |U_E|W &#8211; w(U_E)<br \/>\n= (|U_E| &#8211; 1)W + w(I_E),<br \/>\n$$\u5176\u4e2d, $w(A)$\u4e3a\u96c6\u5408$A$\u4e2d\u9876\u70b9\u4ee3\u4ef7\u4e4b\u548c. \u8986\u76d6\u5927\u5c0f\u76f8\u540c\u65f6, \u7b2c\u4e00\u9879\u4e3a\u5e38\u6570, \u6700\u5c0f\u5316\u5272\u5bb9\u91cf\u5c31\u7b49\u4e8e\u6700\u5c0f\u5316\u6240\u9009\u72ec\u7acb\u96c6\u7684\u4ee3\u4ef7. \u76f4\u63a5\u628a\u6709\u9650\u5bb9\u91cf\u8bbe\u6210$w(v)$, \u4f18\u5316\u7684\u5c06\u662f\u8986\u76d6\u7684\u4ee3\u4ef7, \u4e0d\u662f\u4fdd\u7559\u7684\u72ec\u7acb\u96c6.<br \/>\n$\\\\$ \u5bf9\u6b63\u4ee3\u4ef7\u4e14\u81f3\u5c11\u6709\u4e00\u6761\u51b2\u7a81\u8fb9\u7684\u56fe, \u547d\u98985.2\u7684\u8bc1\u660e\u5148\u8bf4\u660e$H^*$\u7684\u6700\u5c0f\u5272\u4e5f\u662f$H$\u7684\u6700\u5c0f\u5272, \u4fdd\u8bc1\u6240\u9009\u6570\u91cf\u6700\u5927. \u51b2\u7a81\u8fb9\u4f7f\u9876\u70b9\u8986\u76d6\u975e\u7a7a; \u82e5\u53e6\u4e00\u4e2a\u8986\u76d6\u5c11\u4e00\u4e2a\u9876\u70b9, \u5bb9\u91cf\u4e2d\u5c11\u6389\u7684$W$\u8db3\u4ee5\u4e25\u683c\u538b\u8fc7\u4ee3\u4ef7\u9879\u7684\u53d8\u5316. \u7136\u540e\u5728$H$\u7684\u6700\u5c0f\u5272\u4e4b\u95f4\u6bd4\u8f83, \u56e0\u4e3a\u8986\u76d6\u5927\u5c0f\u76f8\u540c, \u5c31\u53ef\u4ee5\u6d88\u53bb\u5e38\u6570\u9879, \u5f97\u5230\u603b\u4ee3\u4ef7\u6700\u5c0f.<br \/>\n$\\\\$ \u8fd9\u91cc\u4e25\u683c\u5c0f\u4e8e\u96f6\u7684\u6765\u6e90\u503c\u5f97\u5355\u72ec\u5199\u51fa. \u8bbe$U^*$\u662f$H^*$\u6700\u5c0f\u5272\u7684\u8986\u76d6, \u53cd\u8bbe\u53e6\u6709\u8986\u76d6$U$\u6ee1\u8db3$|U| \\le |U^*| &#8211; 1$, \u5e76\u7528\u4e0a\u4e00\u8282\u7684\u6784\u9020\u5f97\u5230\u5272$E$. \u8bb0$U^*$\u5bf9\u5e94\u7684\u5272\u4e3a$E^*$, \u5219:<br \/>\n$$<br \/>\n\\operatorname{cap}_{H^*}(E) &#8211; \\operatorname{cap}_{H^*}(E^*)<br \/>\n\\le -W &#8211; w(U) + w(U^*)<br \/>\n\\le -w(U) < 0.\n$$\u5176\u4e2d\u7b2c\u4e8c\u4e2a\u4e0d\u7b49\u53f7\u4f7f\u7528$w(U^*) \\le W$, \u6700\u540e\u4e00\u4e2a\u4e25\u683c\u4e0d\u7b49\u53f7\u4f7f\u7528$U$\u975e\u7a7a\u4e14\u9876\u70b9\u4ee3\u4ef7\u5747\u4e3a\u6b63. \u8fd9\u624d\u4e0e$E^*$\u6700\u5c0f\u77db\u76fe. \u5f97\u5230\u8986\u76d6\u6570\u91cf\u6700\u5c11\u540e, \u540e\u7eed\u6bd4\u8f83\u53ea\u5728\u76f8\u540c\u8986\u76d6\u6570\u91cf\u7684\u5272\u4e4b\u95f4\u8fdb\u884c, \u5e76\u4e0d\u9700\u8981\u53cd\u8fc7\u6765\u8bc1\u660e$H$\u7684\u6bcf\u4e2a\u6700\u5c0f\u5272\u90fd\u662f$H^*$\u7684\u6700\u5c0f\u5272.\n$\\\\$ \u65e0\u51b2\u7a81\u8fb9\u65f6\u76f4\u63a5\u9009\u62e9\u5168\u90e8\u5019\u9009\u5373\u53ef. \u539f\u6587\u9644\u5f55B\u5173\u4e8e\u8986\u76d6\u975e\u7a7a\u7684\u8bba\u8bc1\u6ca1\u6709\u8986\u76d6\u8fd9\u4e2a\u9000\u5316\u60c5\u51b5: \u82e5\u56fe\u53ea\u6709\u4e00\u4e2a\u5019\u9009$v$, \u5c31\u6709$W - w(v) = 0$, \u67d0\u4e2a\u96f6\u5bb9\u91cf\u6700\u5c0f\u5272\u4e5f\u53ef\u80fd\u4e0d\u9009$v$. \u56e0\u6b64\u5b9e\u73b0\u5e94\u5355\u72ec\u5904\u7406\u65e0\u51b2\u7a81\u56fe, \u6216\u91c7\u7528\u4e0b\u8ff0\u66f4\u5927\u7684\u5bb9\u91cf\u5e38\u6570; \u4e0d\u80fd\u628a\u539f\u6587\u516c\u5f0f\u7406\u89e3\u4e3a\u5bf9\u4efb\u610f\u9000\u5316\u56fe\u90fd\u65e0\u9700\u5904\u7406.\n$\\\\$ \u6211\u66fe\u7591\u60d1\u7b2c\u4e8c\u6b65\u662f\u5426\u8fd8\u9700\u8981$H$. \u5b83\u4ecd\u7528\u4e8e\u754c\u5b9a\"\u6570\u91cf\u540c\u6837\u6700\u591a\" \u7684\u6bd4\u8f83\u8303\u56f4, \u4f46\u6bd4\u8f83\u5bb9\u91cf\u65f6\u4f7f\u7528$H^*$. \u5b9e\u73b0\u53ea\u9700\u6c42\u89e3$H^*$, \u65e0\u9700\u5148\u6c42\u89e3$H$. \u9006\u547d\u9898\u4e5f\u4e0d\u6210\u7acb: $H$\u7684\u67d0\u4e2a\u6700\u5c0f\u5272\u53ea\u4fdd\u8bc1\u6570\u91cf\u6700\u591a, \u672a\u5fc5\u6ee1\u8db3\u7b2c\u4e8c\u7ea7\u4ee3\u4ef7\u76ee\u6807.\n\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/05-weighted-cut.png\" alt=\"\" width=\"2200\" height=\"980\" class=\"aligncenter size-full wp-image-5507\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/05-weighted-cut.png 2200w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/05-weighted-cut-300x134.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/05-weighted-cut-768x342.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/05-weighted-cut-1536x684.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/05-weighted-cut-2048x912.png 2048w\" sizes=\"(max-width: 2200px) 100vw, 2200px\" \/><\/p>\n<p>$\\\\$ $W &#8211; w(v)$\u53ef\u4ee5\u79f0\u4e3a\u8865\u6743\u5bb9\u91cf. \u5b83\u5728\u5efa\u56fe\u65f6\u786e\u5b9a, \u4e0e\u6700\u5927\u6d41\u8fd0\u884c\u4e2d\u7684\u6b8b\u4f59\u5bb9\u91cf\u4e0d\u540c. \u540e\u8005\u53d6\u51b3\u4e8e\u5f53\u524d\u6d41\u91cf, \u6b63\u5411\u5f27\u5269\u4f59\u91cf\u4e3a$c(u,v) &#8211; x(u,v)$, \u53cd\u5411\u6b8b\u4f59\u5f27\u7528\u4e8e\u64a4\u56de\u6d41\u91cf. \u5bf9\u975e\u8d1f\u4ee3\u4ef7, \u53ef\u7528$M > W$\u6784\u9020$M &#8211; w(v)$, \u4e25\u683c\u4fdd\u8bc1\u6570\u91cf\u4f18\u5148, \u540c\u65f6\u5904\u7406\u96f6\u4ee3\u4ef7\u4e0e\u4e0a\u8ff0\u9000\u5316\u60c5\u51b5. \u8fd9\u662f\u5b9e\u73b0\u6269\u5c55, \u4e0d\u9002\u7528\u4e8e\u672a\u7ecf\u91cd\u65b0\u63a8\u5bfc\u7684\u8d1f\u4ee3\u4ef7\u8f93\u5165.<\/p>\n<p><strong>10. \u4e3a\u4ec0\u4e48\u4e00\u8f6e\u56fe\u5272\u4e4b\u540e\u8fd8\u8981\u8fed\u4ee3?<\/strong><\/p>\n<p>\u56fe\u5272\u80fd\u627e\u5230\u7ed9\u5b9a\u5019\u9009\u4e2d\u7684\u6700\u4f18\u96c6\u5408, \u4f46\u6709\u4e9b\u6b21\u8981\u914d\u5bf9\u53ef\u80fd\u6ca1\u6709\u5019\u9009, \u73b0\u6709\u5019\u9009\u4e5f\u53ef\u80fd\u76f8\u4e92\u51b2\u7a81. \u4f5c\u8005\u89c2\u5bdf\u5230, \u4e0d\u5b64\u7acb\u7684\u51fa\u751f\u80de\u6216\u5bf9\u5076\u6b7b\u4ea1\u80de\u5e38\u4f4d\u4e8e\u5176\u5b83\u6d3b\u8dc3\u914d\u5bf9\u7684\u4ee3\u8868\u5faa\u73af\u4e0a. \u4e3a\u4fdd\u62a4\u8fd9\u4e9b\u5faa\u73af, \u9aa8\u67b6\u5316\u4e0d\u80fd\u628a\u76ee\u6807\u9644\u8fd1\u7684\u80de\u7ee7\u7eed\u584c\u7f29\u6389.<br \/>\n$\\\\$ \u6d88\u9664\u4e00\u6279\u6b21\u8981\u914d\u5bf9\u540e, \u4e0b\u4e00\u8f6e\u9700\u8981\u4fdd\u62a4\u7684\u4ee3\u8868\u5faa\u73af\u51cf\u5c11. \u4f5c\u8005\u91cd\u65b0\u8ba1\u7b97\u6301\u7eed\u540c\u8c03\u548c\u4e24\u4fa7\u9aa8\u67b6, \u5c1d\u8bd5\u751f\u6210\u4e0a\u4e00\u8f6e\u7f3a\u5931\u7684\u5207\u5272\u6216\u586b\u5145. \u5728\u65e7\u51b2\u7a81\u56fe\u4e0a\u91cd\u590d\u6c42\u6700\u5c0f\u5272, \u65e0\u6cd5\u63d0\u4f9b\u8fd9\u4e9b\u65b0\u5019\u9009.<br \/>\n$\\\\$ \u7b97\u6cd51\u5728\u5019\u9009\u4e3a\u7a7a\u65f6\u505c\u6b62. \u5019\u9009\u975e\u7a7a\u65f6, \u81f3\u5c11\u80fd\u9009\u62e9\u4e00\u4e2a\u9876\u70b9; \u547d\u98985.1\u4fdd\u8bc1\u6b21\u8981\u914d\u5bf9\u6570\u4e0b\u964d, \u6240\u4ee5\u5faa\u73af\u4f1a\u7ec8\u6b62. \u82e5\u7ec8\u6b62\u65f6\u5df2\u6ca1\u6709\u6b21\u8981\u914d\u5bf9, \u5c31\u8fbe\u5230\u4e86\u76ee\u6807Betti\u6570. \u82e5\u4ecd\u6709\u6b21\u8981\u914d\u5bf9\u4f46\u65e0\u6cd5\u751f\u6210\u5019\u9009, \u7b97\u6cd5\u4e5f\u4f1a\u505c\u6b62, \u8fd9\u662f\u7406\u8bba\u4fdd\u8bc1\u7684\u8fb9\u754c.<br \/>\n$\\\\$ \u82e5\u6bcf\u4e2a\u6b21\u8981\u914d\u5bf9\u90fd\u6709\u5207\u5272\u548c\u586b\u5145\u5019\u9009, \u9009\u62e9\u5168\u90e8\u5207\u5272\u6216\u5168\u90e8\u586b\u5145\u5c31\u80fd\u6d88\u9664\u5168\u90e8\u6b21\u8981\u7279\u5f81. \u8bba\u6587\u7528\u8fd9\u4e00\u5145\u5206\u6761\u4ef6\u8bf4\u660e\u5019\u9009\u7f3a\u5931\u7684\u5f71\u54cd: \u56fe\u5272\u5373\u4f7f\u6c42\u51fa\u4e86\u672c\u8f6e\u6700\u4f18\u89e3, \u4e5f\u65e0\u6cd5\u9009\u62e9\u5c1a\u672a\u751f\u6210\u7684\u64cd\u4f5c.<\/p>\n<p><strong>11. \u4fee\u6539\u80de\u590d\u5f62\u4e4b\u540e, \u5982\u4f55\u63d0\u53d6\u8868\u9762?<\/strong><\/p>\n<p>\u7b80\u5316\u540e\u7684\u8d1f\u80de\u590d\u5f62\u53ef\u80fd\u5305\u542b\u9762\u548c\u8fb9\u7b49\u4f4e\u7ef4\u7ed3\u6784, \u5b83\u7684\u8fb9\u754c\u4e0d\u4e00\u5b9a\u76f4\u63a5\u7ed9\u51fa\u6240\u9700\u7684\u6d41\u5f62\u4e09\u89d2\u7f51\u683c. \u7b2c6.2\u8282\u56e0\u6b64\u5355\u72ec\u5904\u7406\u8f93\u51fa: \u5728\u4fdd\u7559\u8d1f\u5b50\u590d\u5f62\u62d3\u6251\u7684\u524d\u63d0\u4e0b, \u628a\u4fee\u6539\u540e\u7684\u80de\u65f6\u95f4\u8f6c\u6210\u53ef\u4f9b\u7b49\u503c\u9762\u7b97\u6cd5\u8bfb\u53d6\u7684\u4fe1\u606f.<br \/>\n$\\\\$ \u76f4\u63a5\u7528\u9876\u70b9\u503c\u8fd0\u884cMarching Tetrahedra\u4f1a\u6f0f\u6389\u9ad8\u7ef4\u80de\u7684\u53d8\u5316. \u4f8b\u5982\u4e00\u4e2a\u56db\u9762\u4f53\u7684\u9876\u70b9, \u8fb9\u548c\u9762\u90fd\u4e3a\u8d1f, \u53ea\u6709\u4e09\u7ef4\u5185\u90e8\u80de\u4e3a\u6b63. \u8d1f\u5b50\u590d\u5f62\u662f\u5c01\u95ed\u56db\u9762\u4f53\u58f3, \u5305\u542b\u4e00\u4e2a\u7a7a\u8154; \u4f46\u7b49\u503c\u9762\u7b97\u6cd5\u53ea\u770b\u5230\u56db\u4e2a\u8d1f\u9876\u70b9, \u65e0\u6cd5\u8bc6\u522b\u5185\u90e8\u7684\u6b63\u80de.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/07-refinement.png\" alt=\"\" width=\"2200\" height=\"920\" class=\"aligncenter size-full wp-image-5508\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/07-refinement.png 2200w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/07-refinement-300x125.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/07-refinement-768x321.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/07-refinement-1536x642.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/07-refinement-2048x856.png 2048w\" sizes=\"(max-width: 2200px) 100vw, 2200px\" \/><\/p>\n<p>$\\\\$ \u4f5c\u8005\u628a\u8d1f\u80de\u6216\u81f3\u5c11\u542b\u4e00\u4e2a\u6b63\u9876\u70b9\u7684\u80de\u79f0\u4e3a\u5e73\u51e1\u80de. \u5bf9\u5408\u6cd5\u65f6\u95f4\u51fd\u6570, \u8fd9\u7c7b\u80de\u7684\u7b26\u53f7\u80fd\u7531\u9876\u70b9\u7b26\u53f7\u786e\u5b9a. \u975e\u5e73\u51e1\u80de\u5219\u81ea\u8eab\u4e3a\u6b63, \u9876\u70b9\u5168\u8d1f, \u9700\u8981\u7ec6\u5206.<br \/>\n$\\\\$ \u7ec6\u5206\u4ece\u4f4e\u7ef4\u5411\u9ad8\u7ef4\u8fdb\u884c. \u5bf9\u975e\u5e73\u51e1\u80de, \u4ee5\u53ca\u542b\u6709\u5df2\u7ec6\u5206\u9762\u7684\u80de, \u5728\u80de\u7684\u8d28\u5fc3\u6dfb\u52a0\u9876\u70b9, \u518d\u5c06\u5b83\u4e0e\u8fb9\u754c\u9762\u8fde\u63a5. \u65b0\u5efa\u80de\u7ee7\u627f\u539f\u80de\u7684\u65f6\u95f4\u503c, \u539f\u6709\u8fb9\u754c\u80de\u4fdd\u7559\u81ea\u5df1\u7684\u503c. \u5bf9\u4e0a\u9762\u7684\u56db\u9762\u4f53, \u65b0\u7684\u6b63\u4e2d\u5fc3\u70b9\u628a\u539f\u672c\u53ea\u8bb0\u5f55\u5728\u4e09\u7ef4\u80de\u4e0a\u7684\u6b63\u503c\u5e26\u5165\u9876\u70b9\u6570\u636e.<br \/>\n$\\\\$ \u5411\u4e0a\u5904\u7406\u542b\u5df2\u7ec6\u5206\u9762\u7684\u80de, \u662f\u4e3a\u4e86\u8ba9\u76f8\u90bb\u90e8\u5206\u5171\u4eab\u4e00\u81f4\u7684\u5256\u5206. \u53ea\u5728\u6bcf\u4e2a\u4f53\u80de\u5185\u90e8\u72ec\u7acb\u63d2\u70b9, \u53ef\u80fd\u7559\u4e0b\u4e0d\u5339\u914d\u7684\u516c\u5171\u9762. \u7ec6\u5206\u7ed3\u675f\u540e, \u9876\u70b9\u5f0f\u63d0\u53d6\u624d\u80fd\u8bfb\u53d6\u80de\u7684\u6b63\u8d1f\u7ed3\u6784.<br \/>\n$\\\\$ \u56db\u9762\u4f53\u7f51\u683c\u7ec6\u5206\u540e\u4ecd\u7531\u56db\u9762\u4f53\u7ec4\u6210. \u7acb\u65b9\u7f51\u683c\u4f1a\u4ea7\u751f\u542b\u56db\u9762\u4f53, \u7acb\u65b9\u4f53\u548c\u68f1\u9525\u7684\u6df7\u5408\u590d\u5f62, \u8bba\u6587\u5206\u522b\u4f7f\u7528\u5bf9\u5e94\u7684\u63d0\u53d6\u65b9\u6cd5; \u5bf9\u7acb\u65b9\u4f53\u8fd8\u8c03\u6574\u67e5\u627e\u8868\u4ee5\u4fdd\u7559\u8d1f\u9876\u70b9\u76846\u8fde\u901a\u7ea6\u5b9a. \u56e0\u6b64\u8f93\u51fa\u7684\u62d3\u6251\u4fdd\u8bc1\u4f9d\u8d56\u7ec6\u5206\u89c4\u5219\u548c\u8fde\u901a\u6027\u7ea6\u5b9a, \u4e0d\u80fd\u66ff\u6362\u6210\u4efb\u610f\u7b49\u503c\u9762\u5b9e\u73b0\u540e\u4ecd\u9ed8\u8ba4\u6210\u7acb.<\/p>\n<p><strong>12. \u76ee\u6807\u63a7\u5236, \u4ee3\u4ef7\u9009\u62e9\u4e0e\u6279\u91cf\u5b9e\u9a8c<\/strong><\/p>\n<p>\u7b2c7\u8282\u5728\u540c\u4e00\u8f93\u5165\u4e0a\u6539\u53d8\u76ee\u6807\u6570\u91cf\u548c\u4ee3\u4ef7\u51fd\u6570, \u518d\u7528\u6279\u91cf\u76ee\u6807\u914d\u7f6e\u4e0e\u8f93\u5165\u6270\u52a8\u6d4b\u8bd5\u6210\u529f\u7387.<br \/>\n$\\\\$ $\\cdot$ \u76ee\u6807\u6570\u91cf\u63a7\u5236: Tree\u8f93\u5165\u5305\u542b1\u4e2a\u8fde\u901a\u5206\u652f, 17\u4e2a\u901a\u9053\u548c1\u4e2a\u7a7a\u8154. \u4f5c\u8005\u79fb\u9664\u7a7a\u8154, \u5206\u522b\u4fdd\u75590, 3, 5, 10\u4e2a\u901a\u9053, \u5e76\u7528\u6301\u7eed\u56fe\u68c0\u67e5\u7559\u4e0b\u7684\u662f\u5bf9\u5e94\u6570\u91cf\u7684\u9ad8\u6301\u4e45\u5ea6\u7279\u5f81. Bone\u5219\u5305\u542b102\u4e2a\u5206\u652f, 582\u4e2a\u901a\u9053\u548c57\u4e2a\u7a7a\u8154, \u7528\u4e8e\u6d4b\u8bd5\u591a\u7c7b\u7279\u5f81\u7684\u540c\u65f6\u63a7\u5236.<br \/>\n$\\\\$ $\\cdot$ \u5207\u5272\u4e0e\u586b\u5145\u7684\u9009\u62e9: Tree\u4e2d\u540c\u4e00\u4e2a\u5c40\u90e8\u4f4d\u7f6e, \u4f1a\u968f\u76ee\u6807\u6570\u91cf\u6539\u53d8\u800c\u91c7\u7528\u4e0d\u540c\u64cd\u4f5c. \u9009\u62e9\u53d6\u51b3\u4e8e\u672c\u8f6e\u5019\u9009\u4e4b\u95f4\u7684\u51b2\u7a81\u548c\u603b\u4ee3\u4ef7, \u6240\u4ee5\u4e0d\u80fd\u5bf9\u6bcf\u4e2a\u6b21\u8981\u7279\u5f81\u72ec\u7acb\u6311\u9009\u5176\u6700\u4fbf\u5b9c\u7684\u64cd\u4f5c, \u518d\u5047\u5b9a\u6240\u6709\u5c40\u90e8\u9009\u62e9\u80fd\u4e00\u8d77\u6267\u884c.<br \/>\n$\\\\$ $\\cdot$ \u4ee3\u4ef7\u51fd\u6570\u7684\u5f71\u54cd: Vessel\u5b9e\u9a8c\u6bd4\u8f83\u65f6\u95f4\u4ee3\u4ef7, \u80de\u6570\u91cf\u4ee3\u4ef7, \u4ee5\u53ca\u504f\u597d\u5207\u5272\u6216\u586b\u5145\u7684\u8bbe\u7f6e. \u80de\u6570\u91cf\u8f83\u5c11\u7684\u6539\u52a8\u4ecd\u53ef\u80fd\u4f4d\u4e8e\u4e0d\u5408\u9002\u7684\u4f4d\u7f6e. Helmet\u662f\u8584\u58f3, \u9ed8\u8ba4\u65f6\u95f4\u4ee3\u4ef7\u4e0b\u5207\u5272\u4e0e\u586b\u5145\u7684\u4ee3\u4ef7\u63a5\u8fd1, \u5207\u5272\u4f1a\u4ea7\u751f\u88c2\u7f1d. \u4f5c\u8005\u6539\u7528\u504f\u597d\u586b\u5145\u7684\u5ea6\u91cf\u540e, \u5728\u8fbe\u5230\u540c\u4e00\u76ee\u6807\u62d3\u6251\u7684\u540c\u65f6\u907f\u514d\u4e86\u8fd9\u4e9b\u88c2\u7f1d.<br \/>\n$\\\\$ $\\cdot$ \u76ee\u6807\u672a\u77e5\u65f6\u7684\u8f85\u52a9\u5224\u65ad: Skull\u5b9e\u9a8c\u901a\u8fc7\u6301\u4e45\u5ea6\u76f4\u65b9\u56fe\u9009\u62e9\u9608\u503c, \u5f97\u5230\u4fdd\u755933\u4e2a\u901a\u9053\u7684\u76ee\u6807. \u8fd9\u5c55\u793a\u4e86\u6301\u7eed\u540c\u8c03\u5982\u4f55\u5e2e\u52a9\u7528\u6237\u9009\u62e9\u6570\u91cf, \u5e76\u672a\u7ed9\u51fa\u81ea\u52a8\u8bc6\u522b\u771f\u5b9e\u89e3\u5256\u62d3\u6251\u7684\u4fdd\u8bc1.<br \/>\n$\\\\$ \u6279\u91cf\u5b9e\u9a8c\u5305\u62ecEnzyme\u4e0eRoot\u4e0a\u76841500\u591a\u4e2a\u76ee\u6807\u914d\u7f6e, \u4ee5\u53ca310\u4e2a\u52a0\u5165\u6270\u52a8\u540e\u62d3\u6251\u53d8\u590d\u6742\u7684Thingi10K\u6a21\u578b. \u4f5c\u8005\u62a5\u544a\u8fd9\u4e9b\u6d4b\u8bd5\u5747\u8fbe\u5230\u6307\u5b9a\u62d3\u6251; \u8fde\u540c\u5c55\u793a\u6848\u4f8b, \u8fd12000\u6b21\u6d4b\u8bd5\u4e2d\u53ea\u6709\u4e24\u6b21\u9700\u8981\u7b2c\u4e8c\u8f6e\u8fed\u4ee3. \u8fd9\u662f\u8be5\u6d4b\u8bd5\u96c6\u4e0a\u7684\u7ed3\u679c, \u4e0d\u80fd\u66ff\u4ee3\u4efb\u610f\u8f93\u5165\u4e0a\u7684\u6210\u529f\u6027\u8bc1\u660e.<br \/>\n$\\\\$ \u5bf9\u6bd4\u5b9e\u9a8c\u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8f93\u5165\u7ea6\u675f. Zeng\u7b49\u4eba\u7684\u65b9\u6cd5\u9700\u8981\u7528\u6237\u63d0\u4f9b\u6838\u5fc3\u4e0e\u90bb\u57df, \u4f5c\u8005\u4e3a\u5176\u8bbe\u7f6e\u4e86\u5e73\u51e1\u62d3\u6251\u76ee\u6807; Kissi\u7b49\u4eba\u7684\u51fd\u6570\u7b80\u5316\u65b9\u6cd5\u5219\u4f7f\u7528\u5b9e\u9a8c\u4e2d\u7684\u6700\u9ad8\u9608\u503c. \u6bd4\u8f83\u7ed3\u679c\u5e94\u8fde\u540c\u8fd9\u4e9b\u8bbe\u7f6e\u4e00\u8d77\u9605\u8bfb.<\/p>\n<p><strong>13. \u8ba1\u7b97\u5f00\u9500\u4e0e\u9002\u7528\u8fb9\u754c<\/strong><\/p>\n<p>\u4f5c\u8005\u4f7f\u7528PHAT\u8ba1\u7b97\u6301\u7eed\u540c\u8c03, \u5e76\u7528\u7a77\u5c3d\u7ea6\u5316\u83b7\u5f97\u89c4\u8303\u751f\u6210\u5143. \u4f5c\u8005\u89c2\u5bdf\u5230, \u76f8\u6bd4\u6807\u51c6\u7ea6\u5316\u7ed9\u51fa\u7684\u4ee3\u8868\u5faa\u73af, \u89c4\u8303\u751f\u6210\u5143\u80fd\u751f\u6210\u5f62\u72b6\u66f4\u5408\u9002\u7684\u5207\u5272\u548c\u586b\u5145. \u9009\u62e9\u4e0d\u540c\u7684\u4ee3\u8868\u94fe, \u4f1a\u901a\u8fc7\u9aa8\u67b6\u6539\u53d8\u5019\u9009\u7684\u51e0\u4f55\u8303\u56f4.<br \/>\n$\\\\$ \u88681\u4e2d, Bone\u7684\u8fc7\u6ee4\u7b80\u5316\u8017\u65f6157.46\u79d2, \u5176\u4e2dPHAT\u4e3a133.10\u79d2; Root\u5206\u522b\u4e3a1600.24\u79d2\u4e0e1448.60\u79d2. \u5728\u4f5c\u8005\u7684\u6d4b\u8bd5\u673a\u5668\u4e0a, \u6301\u7eed\u540c\u8c03\u8ba1\u7b97\u5360\u636e\u4e3b\u8981\u8fd0\u884c\u65f6\u95f4, \u5019\u9009\u9009\u62e9\u7684\u5f00\u9500\u5219\u5f88\u5c0f. \u8fd9\u4e9b\u6d4b\u91cf\u53cd\u6620\u8be5\u5b9e\u73b0\u7684\u74f6\u9888, \u4e0d\u6784\u6210\u6574\u4f53\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u8bc1\u660e.<br \/>\n$\\\\$ \u7f51\u683c\u5206\u8fa8\u7387\u9650\u5236\u4e86\u53ef\u89c1\u7279\u5f81\u4e0e\u5019\u9009\u4f4d\u7f6e, \u5207\u5272\u548c\u586b\u5145\u4e5f\u53ef\u80fd\u7559\u4e0b\u4e0d\u5149\u6ed1\u7684\u79bb\u6563\u5f62\u72b6. \u4f5c\u8005\u63d0\u51fa\u901a\u8fc7\u5c40\u90e8\u8fc7\u6ee4\u51cf\u5c11\u8ba1\u7b97\u91cf, \u5728\u7b80\u5316\u4e2d\u7ec6\u5316\u7f51\u683c\u6539\u5584\u51e0\u4f55\u7b49\u540e\u7eed\u65b9\u5411; \u5f53\u524d\u5b9e\u73b0\u5c1a\u672a\u63d0\u4f9b\u8fd9\u4e9b\u80fd\u529b.<br \/>\n$\\\\$ \u547d\u98985.1\u8d1f\u8d23\u4fdd\u8bc1\u5019\u9009\u7684\u62d3\u6251\u6548\u679c, \u547d\u98985.2\u8d1f\u8d23\u7ed9\u5b9a\u5019\u9009\u96c6\u5408\u4e0a\u7684\u9009\u62e9\u6700\u4f18, \u7b2c6\u8282\u5904\u7406\u8f93\u51fa\u8f6c\u6362. \u5176\u4e2d, \u6211\u82b1\u6700\u591a\u65f6\u95f4\u6838\u5bf9\u7684\u662f5.1\u5bf9\u8fc1\u79fb\u8fc7\u7a0b\u4e2d\u914d\u5bf9\u53d8\u5316\u7684\u9650\u5236.<\/p>\n<p><strong>14. \u8bc1\u660e\u8865\u8bb0: \u5b64\u7acb\u6027\u4e0eA.3, A.4<\/strong><\/p>\n<p>A.2(1) \u8bf4\u660e, \u7b80\u5355\u5bf9\u4e2d\u7684\u9ad8\u7ef4\u80de\u4e0d\u80fd\u662f\u51fa\u751f\u80de. \u8bbe\u7b80\u5355\u5bf9\u4e3a$\\{\\phi,\\sigma\\}$, \u5176\u4e2d$\\phi$\u662f$\\sigma$\u7684\u81ea\u7531\u9762. \u5728\u5f53\u524d\u590d\u5f62\u4e2d, \u53ea\u6709$\\sigma$\u7684\u8fb9\u754c\u5305\u542b$\\phi$. \u56e0\u6b64, \u65e0\u8bba\u7ed9\u5305\u542b$\\sigma$\u7684\u94fe\u6dfb\u52a0\u54ea\u4e9b\u540c\u7ef4\u80de, \u90fd\u65e0\u6cd5\u62b5\u6d88\u8fb9\u754c\u4e2d\u7684$\\phi$, \u4e5f\u5c31\u65e0\u6cd5\u5f97\u5230\u8fb9\u754c\u4e3a\u96f6\u7684\u5faa\u73af. \u800c\u51fa\u751f\u80de\u5fc5\u987b\u5305\u542b\u5728\u5b83\u6240\u4ea7\u751f\u7684\u751f\u6210\u5faa\u73af\u4e2d, \u6240\u4ee5$\\sigma$\u4e0d\u53ef\u80fd\u662f\u51fa\u751f\u80de.<br \/>\n$\\\\$ A.2(2) \u8bf4\u660e, \u82e5\u51fa\u751f\u80de$b$\u5728\u5f53\u524d\u8fc7\u6ee4\u524d\u7f00$X$\u4e2d\u5b64\u7acb, \u5219\u5176\u6b7b\u4ea1\u642d\u6863$d$\u4e0d\u5c5e\u4e8e$X$. \u8fd9\u91cc\u7684\u524d\u7f00\u662f\u6309\u8fc7\u6ee4\u987a\u5e8f\u5df2\u7ecf\u52a0\u5165\u7684\u5168\u90e8\u80de, \u6240\u4ee5\u7ed3\u8bba\u662f$b \\in X$, $d \\notin X$, \u800c\u4e0d\u662f\u8bf4$d$\u5728\u7a7a\u95f4\u4e0a\u79bb$b$\u5f88\u8fdc. \u53ef\u4ee5\u7528\u53cd\u8bc1\u6cd5\u7406\u89e3: \u82e5$d$\u5df2\u7ecf\u52a0\u5165, \u90a3\u4e48$b$\u51fa\u751f\u65f6\u4ea7\u751f\u7684\u751f\u6210\u5faa\u73af$z$\u5c31\u5df2\u7ecf\u6210\u4e3a\u67d0\u4e2a\u9ad8\u4e00\u7ef4\u94fe\u7684\u8fb9\u754c. \u7531\u4e8e$z$\u5305\u542b$b$, \u8fd9\u4e2a\u9ad8\u4e00\u7ef4\u94fe\u4e2d\u81f3\u5c11\u8981\u6709\u4e00\u4e2a\u80de, \u5176\u8fb9\u754c\u5305\u542b$b$. \u8fd9\u5c31\u8981\u6c42\u5f53\u524d\u590d\u5f62\u4e2d\u5b58\u5728$b$\u7684\u4e0a\u80de, \u4e0e$b$\u5b64\u7acb\u77db\u76fe. \u56e0\u800c, \u53ea\u8981$b$\u5728\u5f53\u524d\u590d\u5f62\u4e2d\u4ecd\u7136\u5b64\u7acb, \u5b83\u6240\u4ea7\u751f\u7684\u540c\u8c03\u7c7b\u5c31\u5c1a\u672a\u6b7b\u4ea1.<br \/>\n$\\\\$ A.3\u628a\u5b64\u7acb\u7684\u6b21\u8981\u51fa\u751f\u80de$b$\u9010\u6b65\u79fb\u5230\u8d1f\u65f6\u95f4\u672b\u5c3e, \u518d\u6539\u4e3a\u6b63\u65f6\u95f4. \u539f\u6587$\\sigma^-$, $\\sigma^+$\u5206\u522b\u662f\u6700\u540e\u4e00\u4e2a\u8d1f\u65f6\u95f4\u80de\u548c\u7d27\u968f\u5176\u540e\u7684\u6b63\u65f6\u95f4\u80de; \u6700\u540e\u53d6$0 < f'(b) < f(\\sigma^+)$, \u4e0d\u9700\u8981\u8d8a\u8fc7$\\sigma^+$. \u5b64\u7acb\u6027\u4fdd\u8bc1\u9014\u4e2d\u4e0d\u4f1a\u8de8\u8fc7\u81ea\u5df1\u7684\u8d1f\u65f6\u95f4\u4e0a\u80de, \u6240\u4ee5\u6bcf\u4e00\u6b65\u5408\u6cd5.\n\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/cases_swapping_cell-pair_modifies_p-pairs_generators_co_generators.png\" alt=\"\" width=\"1582\" height=\"935\" class=\"aligncenter size-full wp-image-5538\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/cases_swapping_cell-pair_modifies_p-pairs_generators_co_generators.png 1582w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/cases_swapping_cell-pair_modifies_p-pairs_generators_co_generators-300x177.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/cases_swapping_cell-pair_modifies_p-pairs_generators_co_generators-768x454.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/cases_swapping_cell-pair_modifies_p-pairs_generators_co_generators-1536x908.png 1536w\" sizes=\"(max-width: 1582px) 100vw, 1582px\" \/><\/p>\n<p>\u9014\u4e2d\u82e5\u7aef\u70b9\u91cd\u65b0\u914d\u5bf9, \u4f7f\u7528\u51fa\u751f\u80de\u4ea4\u6362Case 1.1.2. \u53e6\u4e00\u51fa\u751f\u80de$c$\u5728\u8fc7\u6ee4\u987a\u5e8f\u4e2d\u51fa\u751f\u66f4\u665a, \u6b7b\u4ea1\u66f4\u65e9. \u7531\u4e8e$f$\u53ea\u8981\u6c42\u4e0d\u51cf, \u8fd9\u8bf4\u660e\u5176\u6301\u4e45\u5ea6\u4e0d\u5927\u4e8e$b$\u7684\u6301\u4e45\u5ea6, \u4e0d\u4e00\u5b9a\u4e25\u683c\u66f4\u5c0f. \u4ea4\u6362\u540e$b$\u4ecd\u5b64\u7acb, A.2(2) \u53c8\u8981\u6c42\u5b83\u63a5\u8fc7\u7684\u6b7b\u4ea1\u642d\u6863\u4e3a\u6b63\u65f6\u95f4, \u56e0\u6b64$c$\u539f\u6765\u4e5f\u662f\u6d3b\u8dc3\u914d\u5bf9. \u82e5\u4e3b\u8981\u4e0e\u6b21\u8981\u4e4b\u95f4\u6ca1\u6709\u6301\u4e45\u5ea6\u5e76\u5217, \u5373\u53ef\u63a8\u51fa$c$\u4e5f\u662f\u6b21\u8981\u914d\u5bf9, \u6362\u642d\u6863\u4e0d\u89e6\u53ca\u4e3b\u8981\u914d\u5bf9. \u6700\u540e\u6539\u7b26\u53f7, \u6070\u597d\u8ba9$b$\u5f53\u524d\u7684\u4e00\u5bf9\u9000\u51fa\u6d3b\u8dc3\u96c6\u5408; \u8fb9\u754c\u5904\u7684\u5e76\u5217\u60c5\u5f62\u9700\u8981\u53e6\u884c\u7ea6\u5b9a\u548c\u8bba\u8bc1.<br \/>\n$\\\\$ \u6bcf\u6b21\u4ea4\u6362\u7684\u5408\u6cd5\u6027\u8fd8\u53ef\u4ee5\u62c6\u6210\u4e24\u4e2a\u68c0\u67e5. \u8bbe$b$\u540e\u9762\u7d27\u90bb\u7684\u8d1f\u65f6\u95f4\u80de\u4e3a$c$. $c$\u4e0d\u53ef\u80fd\u662f$b$\u7684\u9762, \u56e0\u4e3a\u539f\u987a\u5e8f\u5df2\u8981\u6c42\u9762\u5148\u4e8e$b$; $c$\u4e5f\u4e0d\u53ef\u80fd\u662f$b$\u7684\u4e0a\u80de, \u56e0\u4e3a$b$\u5728\u8d1f\u65f6\u95f4\u590d\u5f62\u4e2d\u5b64\u7acb. \u56e0\u800c\u4e8c\u8005\u6ca1\u6709\u9762\u5173\u7cfb, \u4ea4\u6362\u4e0d\u4f1a\u7834\u574f\u8fc7\u6ee4\u987a\u5e8f. \u518d\u628a$b$\u7684\u65f6\u95f4\u63d0\u9ad8\u5230$f(c)$, \u65f6\u95f4\u4ecd\u6cbf\u65b0\u987a\u5e8f\u4e0d\u51cf. \u6700\u540e\u5c06$b$\u6539\u6210\u5c0f\u4e8e\u9996\u4e2a\u6b63\u65f6\u95f4\u80de\u7684\u6b63\u503c\u65f6, \u5b83\u7684\u9762\u90fd\u5df2\u5728\u524d\u9762, \u6240\u6709\u4e0a\u80de\u4ecd\u5728\u540e\u9762, \u8fd9\u4e00\u6b65\u4e5f\u5408\u6cd5.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/02-birth-swaps.png\" alt=\"\" width=\"2200\" height=\"840\" class=\"aligncenter size-full wp-image-5509\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/02-birth-swaps.png 2200w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/02-birth-swaps-300x115.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/02-birth-swaps-768x293.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/02-birth-swaps-1536x586.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/02-birth-swaps-2048x782.png 2048w\" sizes=\"(max-width: 2200px) 100vw, 2200px\" \/><\/p>\n<p>$\\\\$ Case 1.1.1\u4ecd\u53ef\u80fd\u66f4\u65b0\u751f\u6210\u5143\u4e0e\u4f59\u751f\u6210\u5143, \u53ea\u662f\u7aef\u70b9\u4e0d\u53d8. \u6240\u4ee5A.3\u4fdd\u7559\u4e3b\u8981\u914d\u5bf9\u7684\u7ed3\u8bba, \u4e0d\u5e94\u6269\u5927\u4e3a\u4fdd\u7559\u6240\u6709\u4ee3\u8868\u94fe. \u6b7b\u4ea1\u987a\u5e8f\u5f71\u54cd\u7ed3\u679c, \u662f\u56e0\u4e3a\u7ea6\u5316\u65f6\u5148\u5230\u7684\u6b7b\u4ea1\u4e8b\u4ef6\u6d88\u53bb\u7684\u57fa\u5e95\u7ec4\u5408\u4e0d\u540c; \u8fd8\u5fc5\u987b\u6ee1\u8db3\u539f\u6587\u7684\u4ee3\u8868\u94fe\u5305\u542b\u6761\u4ef6, \u4e0d\u80fd\u53ea\u6bd4\u8f83\u4e24\u4e2a\u65f6\u95f4.<br \/>\n$\\\\$ \u53ef\u4ee5\u7528\u4e00\u4e2a\u6a212\u4e3b\u5143\u8ba1\u7b97\u770b\u6e05\u4e24\u79cd\u60c5\u51b5. \u8bbe\u76f8\u90bb\u51fa\u751f\u80de\u539f\u6765\u6ee1\u8db3$a < b$, \u4e24\u4e2a\u6b7b\u4ea1\u80de\u4e3a$a'$, $b'$. \u53ea\u770b\u51b3\u5b9a\u4e3b\u5143\u7684$a$, $b$\u4e24\u884c, \u628a\u5176\u5b83\u66f4\u65e9\u7684\u884c\u7565\u53bb: \u5047\u8bbe$a'$\u5217\u5728\u8fd9\u4e24\u884c\u4e0a\u7684\u7cfb\u6570\u4e3a$(1,0)$, $b'$\u5217\u4e3a$(1,1)$. \u8fd9\u8868\u793a$b'$\u5217\u540c\u65f6\u5305\u542b$a$\u548c$b$, \u5bf9\u5e94\u8fd9\u91cc\u9700\u8981\u7684\u5305\u542b\u5173\u7cfb. \u4e3b\u5143\u53d6\u5217\u4e2d\u987a\u5e8f\u6700\u9760\u540e\u7684\u975e\u96f6\u884c; \u4ea4\u6362$a$, $b$\u5c31\u6539\u53d8\u4e86\u8fd9\u4e24\u884c\u8c01\u66f4\u9760\u540e.\n$\\\\$ \u82e5$a' < b'$, \u5148\u5904\u7406$a'$\u5217\u65f6, \u4e3b\u5143\u65e0\u8bba\u4ea4\u6362\u524d\u540e\u90fd\u662f$a$. \u539f\u987a\u5e8f\u4e0b$b'$\u5217\u4ee5$b$\u4e3a\u4e3b\u5143; \u4ea4\u6362\u540e\u5b83\u6682\u65f6\u4ee5$a$\u4e3a\u4e3b\u5143, \u4f46\u52a0\u4e0a\u5df2\u5904\u7406\u7684$a'$\u5217\u4fbf\u5f97\u5230$(0,1)$, \u4e3b\u5143\u4ecd\u5f52$b$. \u6240\u4ee5\u914d\u5bf9\u4ecd\u662f$\\{a,a'\\}$\u548c$\\{b,b'\\}$, \u53ea\u662f\u7ea6\u5316\u4f7f\u7528\u7684\u94fe\u53d1\u751f\u53d8\u5316, \u5bf9\u5e94Case 1.1.1.\n$\\\\$ \u82e5$b' < a'$, \u5219\u5148\u5904\u7406\u542b\u4e24\u884c\u7684$b'$\u5217. \u4ea4\u6362\u524d\u5b83\u4ee5$b$\u4e3a\u4e3b\u5143, \u4ea4\u6362\u540e\u4ee5$a$\u4e3a\u4e3b\u5143. \u968f\u540e\u5904\u7406$a'$\u5217: \u4ea4\u6362\u524d\u4e3b\u5143\u4e3a$a$; \u4ea4\u6362\u540e\u9700\u52a0\u4e0a$b'$\u5217\u6d88\u53bb$a$, \u7559\u4e0b$b$. \u4e8e\u662f\u914d\u5bf9\u7531$\\{b,b'\\}$, $\\{a,a'\\}$\u53d8\u6210$\\{a,b'\\}$, $\\{b,a'\\}$, \u5bf9\u5e94Case 1.1.2. \u6b7b\u4ea1\u987a\u5e8f\u51b3\u5b9a\u4e86\u54ea\u4e00\u5217\u80fd\u5148\u5360\u7528\u4e3b\u5143, \u54ea\u4e00\u5217\u5fc5\u987b\u501f\u52a9\u5b83\u6d88\u5143. \u8fd9\u4e2a\u5c40\u90e8\u4f8b\u5b50\u89e3\u91ca\u4ea4\u6362\u673a\u5236, \u4e0d\u80fd\u66ff\u4ee3\u4e00\u822c\u60c5\u5f62\u4e2d\u7684\u751f\u6210\u5143\u5305\u542b\u6761\u4ef6.\n$\\\\$ A.4\u8ddf\u8e2a\u7aef\u70b9\u96c6\u5408: \u6b21\u8981\u51fa\u751f\u80de\u96c6\u5408\u5c11$b$, \u6b21\u8981\u6b7b\u4ea1\u80de\u96c6\u5408\u5c11\u6700\u7ec8\u4e0e$b$\u914d\u5bf9\u7684$d$, \u4e0d\u4e00\u5b9a\u662f\u5b83\u6700\u521d\u7684\u642d\u6863. \u8fd9\u4e2a$d$\u6216\u8005\u662f$b$\u7684\u4e0a\u80de, \u6216\u8005\u5176\u5bf9\u5076\u5728\u6700\u7ec8\u80cc\u666f\u4e2d\u4e0d\u5b64\u7acb. \u5728A.6\u4e2d\u6392\u9664\u8fd9\u4e24\u79cd\u60c5\u51b5\u540e, \u624d\u80fd\u786e\u8ba4\u540e\u7eed\u76ee\u6807\u4ecd\u5c5e\u4e8e\u6b21\u8981\u914d\u5bf9.\n\n<strong>15. \u8bc1\u660e\u8865\u8bb0: A.5\u4e3a\u4f55\u4fdd\u7559\u4ee3\u8868\u94fe?<\/strong><\/p>\n<p>A.5\u8981\u6c42\u7b80\u5355\u5bf9$\\{\\phi,\\sigma\\}$\u907f\u5f00\u539f\u6709\u5168\u90e8\u6d3b\u8dc3\u751f\u6210\u5143. \u5148\u8fc1\u79fb$\\sigma$, \u5b83\u5728\u79bb\u5f00\u5f62\u72b6\u524d\u59cb\u7ec8\u7b80\u5355, \u7531A.2(1) \u59cb\u7ec8\u4e3a\u6b7b\u4ea1\u80de. \u76f8\u5173\u66f4\u65b0\u53ea\u6709\u6b7b\u4ea1\u80de\u4ea4\u6362Case 2.1.1\u548c2.1.2, \u4e14\u6d89\u53ca\u7684\u6b7b\u4ea1\u80de\u90fd\u4e3a\u8d1f\u65f6\u95f4. \u7ed3\u5408\u5c40\u90e8\u89c4\u5219\u53ea\u66f4\u65b0\u76f8\u5173\u4e24\u5bf9\u53ca\u5176\u6307\u5b9a\u4ee3\u8868\u94fe, \u624d\u80fd\u63a8\u51fa\u539f\u6709\u6d3b\u8dc3\u914d\u5bf9\u4e0d\u53d8.<br \/>\n$\\\\$ \u8fd9\u91cc\u4e0d\u80fd\u53ea\u51ed&#8221;\u4e24\u4e2a\u914d\u5bf9\u4e0d\u6d3b\u8dc3&#8221; \u5c31\u65ad\u8a00\u5176\u5b83\u4ee3\u8868\u94fe\u4e0d\u53d8, \u8fd8\u8981\u6838\u5bf9\u66f4\u65b0\u8303\u56f4. \u8bbe\u4ea4\u6362\u7684\u6b7b\u4ea1\u80de\u4e3a$a$, $b$, \u5bf9\u5e94$\\{a^*,a\\}$\u548c$\\{b^*,b\\}$. \u56e0\u4e3a$a$, $b$\u90fd\u5728\u96f6\u65f6\u523b\u4e4b\u524d, \u5b83\u4eec\u7684\u51fa\u751f\u80de\u66f4\u65e9, \u6240\u4ee5\u8fd9\u4e24\u5bf9\u5747\u4e0d\u6d3b\u8dc3. Case 2.1.1\u53ea\u4fee\u6539\u7b2c\u4e8c\u5bf9\u7684\u751f\u6210\u5143\u548c\u7b2c\u4e00\u5bf9\u7684\u4f59\u751f\u6210\u5143; Case 2.1.2\u91cd\u65b0\u7ec4\u5408\u8fd9\u4e24\u5bf9\u7684\u7aef\u70b9, \u5e76\u66f4\u65b0\u5b83\u4eec\u7684\u4ee3\u8868\u94fe. \u4e24\u6761\u89c4\u5219\u90fd\u4e0d\u66f4\u65b0\u7b2c\u4e09\u5bf9. \u56e0\u800c\u539f\u6709\u6d3b\u8dc3\u914d\u5bf9\u53ca\u5176\u6307\u5b9a\u4ee3\u8868\u94fe\u4e0d\u53d7\u5f71\u54cd. \u8fd9\u4e2a\u7ed3\u8bba\u4f9d\u8d56\u8bba\u6587\u5217\u51fa\u7684\u5c40\u90e8\u66f4\u65b0\u89c4\u5219, \u4e0d\u662f\u4efb\u610f\u91cd\u65b0\u9009\u53d6\u540c\u8c03\u57fa\u5e95\u90fd\u5177\u6709\u7684\u6027\u8d28.<br \/>\n$\\\\$ $\\sigma$\u53d8\u6210\u6b63\u65f6\u95f4\u540e\u4e34\u65f6\u4ea7\u751f\u4e00\u4e2a\u6d3b\u8dc3\u914d\u5bf9. \u518d\u8fc1\u79fb$\\phi$: \u5b83\u5931\u53bb\u552f\u4e00\u4e0a\u80de, \u5df2\u7ecf\u5b64\u7acb. \u82e5\u5b83\u662f\u51fa\u751f\u80de, A.2(2) \u6392\u9664\u975e\u6d3b\u8dc3\u914d\u5bf9, \u800c\u751f\u6210\u5143\u907f\u8ba9\u6761\u4ef6\u6392\u9664\u539f\u6709\u6d3b\u8dc3\u914d\u5bf9, \u6240\u4ee5\u53ea\u80fd\u4e0e$\\sigma$\u914d\u5bf9; \u5426\u5219\u5b83\u6682\u65f6\u662f\u6b7b\u4ea1\u80de.<br \/>\n$\\\\$ $\\phi$\u4e3a\u6b7b\u4ea1\u80de\u65f6, Case 3.1\u53ef\u80fd\u8ba9\u5b83\u8f6c\u4e3a\u51fa\u751f\u80de, \u4f46\u4ecd\u53ea\u80fd\u914d$\\sigma$. \u914d\u4e0a$\\sigma$\u540e, Case 1.1.1\u8981\u6c42\u5b83\u8fdb\u5165\u67d0\u4e2a\u539f\u6709\u6d3b\u8dc3\u751f\u6210\u5143, \u4e0e\u524d\u63d0\u77db\u76fe; Case 1.1.2\u4f1a\u8ba9\u5b83\u914d\u8d1f\u65f6\u95f4\u6b7b\u4ea1\u80de, \u4e0e\u5b64\u7acb\u6027\u77db\u76fe. \u539f\u6709\u6d3b\u8dc3\u4ee3\u8868\u94fe\u56e0\u6b64\u4fdd\u6301\u4e0d\u53d8.<br \/>\n$\\\\$ \u6700\u540e$\\phi$\u4e0e$\\sigma$\u76f8\u90bb, $\\phi$\u662f$\\sigma$\u6700\u540e\u8fdb\u5165\u7684\u9762, $\\sigma$\u662f$\\phi$\u7b2c\u4e00\u4e2a\u8fdb\u5165\u7684\u4e0a\u80de. \u8fb9\u754c\u77e9\u9635\u7684$\\sigma$\u5217\u4ee5$\\phi$\u4e3a\u4e3b\u5143, \u6b64\u524d\u4e5f\u6ca1\u6709\u5217\u80fd\u6d88\u53bb\u8be5\u884c, \u6545\u4e8c\u8005\u6784\u6210\u914d\u5bf9. \u628a$\\phi$\u6539\u4e3a\u6b63\u65f6\u95f4\u5c31\u6d88\u9664\u4e34\u65f6\u6d3b\u8dc3\u914d\u5bf9. \u6b63\u5f0f\u5f15\u7406\u4fdd\u7559\u7684\u662f\u539f\u6709\u6d3b\u8dc3\u914d\u5bf9\u53ca\u5176\u4ee3\u8868\u94fe, \u4e0d\u5305\u62ec\u6240\u6709\u975e\u6d3b\u8dc3\u914d\u5bf9.<br \/>\n$\\\\$ &#8220;\u6b64\u524d\u6ca1\u6709\u5217\u80fd\u6d88\u53bb\u8be5\u884c&#8221; \u8fd8\u5305\u62ec\u6b64\u524d\u5df2\u7ecf\u7ea6\u5316\u8fc7\u7684\u5217. \u5728$\\sigma$\u4e4b\u524d, \u6ca1\u6709\u80de\u4ee5$\\phi$\u4e3a\u9762, \u56e0\u800c\u6240\u6709\u539f\u59cb\u8fb9\u754c\u5217\u5728$\\phi$\u884c\u7684\u7cfb\u6570\u90fd\u4e3a\u96f6. \u7ea6\u5316\u53ea\u5bf9\u8fd9\u4e9b\u5217\u505a\u7ebf\u6027\u7ec4\u5408, \u4e0d\u53ef\u80fd\u5728\u8fd9\u4e00\u884c\u4ea7\u751f\u975e\u96f6\u503c. \u800c$\\sigma$\u5217\u5728$\\phi$\u884c\u975e\u96f6, \u4e14\u66f4\u9760\u540e\u7684\u884c\u5747\u4e3a\u96f6, \u6240\u4ee5\u5b83\u7684\u4e3b\u5143\u65e0\u6cd5\u4e0e\u4efb\u4f55\u5148\u524d\u5217\u7684\u4e3b\u5143\u76f8\u649e. \u8fd9\u91cc\u7684&#8221;\u5df2\u7ea6\u5316&#8221; \u662f\u6307\u4e3b\u5143\u5df2\u6ee1\u8db3\u914d\u5bf9\u7b97\u6cd5\u7684\u8981\u6c42, \u4e0d\u8981\u6c42\u628a\u4e3b\u5143\u4e0a\u65b9\u7684\u975e\u96f6\u9879\u4e5f\u5168\u90e8\u6d88\u53bb.<\/p>\n<p><strong>16. \u8bc1\u660e\u8865\u8bb0: A.6\u7684\u5206\u9636\u6bb5\u8fc1\u79fb<\/strong><\/p>\n<p>\u4ee4$B$\u4e3a\u6240\u9009\u5207\u5272\u7684\u76ee\u6807\u51fa\u751f\u80de, $D$\u4e3a\u6240\u9009\u586b\u5145\u7684\u76ee\u6807\u6b7b\u4ea1\u80de. \u5019\u9009\u5df2\u7ecf\u9650\u5b9a\u6765\u81ea\u6b21\u8981\u914d\u5bf9, \u6240\u4ee5$B$\u4e0d\u4f1a\u6df7\u5165\u4e3b\u8981\u914d\u5bf9\u7684\u76ee\u6807\u51fa\u751f\u80de. A.6\u6309$C \\setminus B$, $F \\setminus D$, $B$, $D$\u7684\u6b21\u5e8f\u8bc1\u660e\u8fc1\u79fb\u6548\u679c.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/03-migration.png\" alt=\"\" width=\"2200\" height=\"980\" class=\"aligncenter size-full wp-image-5510\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/03-migration.png 2200w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/03-migration-300x134.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/03-migration-768x342.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/03-migration-1536x684.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/03-migration-2048x912.png 2048w\" sizes=\"(max-width: 2200px) 100vw, 2200px\" \/><\/p>\n<p>$\\\\$ \u751f\u6210\u96c6\u5408\u7684\u8f85\u52a9\u90e8\u5206\u53ef\u4ee5\u6309\u5220\u9664\u6b21\u5e8f\u7ec4\u7ec7\u6210\u7b80\u5355\u5bf9; \u7b2c$i$\u5bf9\u53ea\u9700\u5728\u524d$i &#8211; 1$\u5bf9\u79fb\u9664\u540e\u4e3a\u7b80\u5355\u5bf9. \u524d\u4e24\u6bb5\u5206\u522b\u5728\u5f62\u72b6\u548c\u80cc\u666f\u5e94\u7528A.5, \u4fdd\u7559\u539f\u6709\u6d3b\u8dc3\u914d\u5bf9\u4e0e\u4ee3\u8868\u94fe. \u6b64\u65f6\u76ee\u6807\u80de\u5b64\u7acb, \u624d\u80fd\u9010\u4e2a\u5e94\u7528A.3\u8fc1\u79fb$B$\u4e0e$D$.<br \/>\n$\\\\$ \u8fc1\u79fb$B$\u4e4b\u540e, \u8fd8\u8981\u786e\u8ba4$D$\u4ecd\u7531\u6b21\u8981\u914d\u5bf9\u7684\u6b7b\u4ea1\u80de\u7ec4\u6210, \u4e14\u5176\u5bf9\u5076\u4fdd\u6301\u5b64\u7acb. \u5bf9\u6bcf\u6b21\u8fc1\u79fb\u7684$b \\in B$, A.4(2) \u8bf4\u6b21\u8981\u6b7b\u4ea1\u80de\u96c6\u5408\u53ea\u4f1a\u5c11\u4e00\u4e2a\u80de$e$, \u4e14$e$\u6ee1\u8db3\u4e24\u79cd\u6761\u4ef6\u4e4b\u4e00: $e$\u662f$b$\u7684\u4e0a\u80de, \u6216$\\widehat e$\u5728\u8fc1\u79fb\u540e\u7684\u80cc\u666f\u4e2d\u4e0d\u5b64\u7acb. \u8981\u8bc1\u660e$d \\in D$\u4e0d\u4f1a\u9000\u51fa, \u5fc5\u987b\u628a\u8fd9\u4e24\u79cd\u53ef\u80fd\u90fd\u6392\u9664.<br \/>\n$\\\\$ \u7b2c\u4e00\u79cd\u53ef\u80fd\u7531\u5207\u5272\u4e0e\u586b\u5145\u4e92\u4e0d\u76f8\u4ea4\u6392\u9664: $b \\in C$, $d \\in F$, \u6240\u4ee5$d$\u4e0d\u53ef\u80fd\u662f$b$\u7684\u4e0a\u80de. \u7b2c\u4e8c\u79cd\u53ef\u80fd\u8981\u5148\u68c0\u67e5\u80cc\u666f\u7684\u53d8\u5316: \u5220\u9664\u5f62\u72b6\u4e2d\u7684$b$, \u5c31\u662f\u5728\u5bf9\u5076\u80cc\u666f\u91cc\u52a0\u5165$\\widehat b$. \u539f\u6765\u5b64\u7acb\u7684$\\widehat d$\u53ea\u6709\u5728\u65b0\u589e\u80de$\\widehat b$\u6210\u4e3a\u5176\u4e0a\u80de\u65f6\u624d\u53ef\u80fd\u5931\u53bb\u5b64\u7acb\u6027; \u5bf9\u5076\u53cd\u8f6c\u9762\u5173\u7cfb, \u8fd9\u53c8\u7b49\u4ef7\u4e8e$d$\u662f$b$\u7684\u4e0a\u80de, \u5df2\u88ab\u6392\u9664. \u56e0\u6b64$d$\u65e2\u4e0d\u662f\u7b2c\u4e00\u7c7b$e$, \u4e5f\u4e0d\u662f\u7b2c\u4e8c\u7c7b$e$, \u4ecd\u662f\u6b21\u8981\u6b7b\u4ea1\u80de. \u6bcf\u8fc1\u79fb\u4e00\u4e2a$b$\u90fd\u80fd\u91cd\u590d\u8fd9\u4e2a\u68c0\u67e5, \u6240\u4ee5\u7ed3\u8bba\u5bf9\u6574\u4e2a$B$\u6210\u7acb; \u4f46\u5b83\u4e0d\u4fdd\u8bc1$d$\u7684\u51fa\u751f\u642d\u6863\u59cb\u7ec8\u4e0d\u53d8.<br \/>\n$\\\\$ \u5206\u9636\u6bb5\u8fc1\u79fb\u6539\u53d8\u4e86$C$, $F$\u5185\u90e8\u987a\u5e8f. \u6062\u590d\u539f\u987a\u5e8f\u6240\u9700\u7684\u76f8\u90bb\u4ea4\u6362\u53ef\u4ee5\u9010\u4e2a\u6784\u9020: \u5728\u67d0\u4e00\u7ec4\u5185, \u627e\u51fa\u5f53\u524d\u987a\u5e8f\u4e0e\u76ee\u6807\u987a\u5e8f\u7b2c\u4e00\u6b21\u4e0d\u540c\u7684\u4f4d\u7f6e, \u628a\u76ee\u6807\u4f4d\u7f6e\u5e94\u6709\u7684\u80de$x$\u4ece\u540e\u65b9\u9010\u6b65\u5411\u524d\u4ea4\u6362. \u8bbe$x$\u5373\u5c06\u8de8\u8fc7\u7684\u80de\u4e3a$y$. \u5f53\u524d\u987a\u5e8f\u4e2d$y$\u5728$x$\u4e4b\u524d, \u76ee\u6807\u987a\u5e8f\u4e2d$x$\u5728$y$\u4e4b\u524d. \u82e5$y$\u662f$x$\u7684\u9762, \u76ee\u6807\u987a\u5e8f\u5c31\u4e0d\u5408\u6cd5; \u82e5$x$\u662f$y$\u7684\u9762, \u5f53\u524d\u987a\u5e8f\u5c31\u4e0d\u5408\u6cd5. \u4e24\u4e2a\u987a\u5e8f\u5747\u5408\u6cd5, \u6240\u4ee5\u4e8c\u8005\u5fc5\u65e0\u9762\u5173\u7cfb, \u53ef\u4ee5\u4ea4\u6362. \u6bcf\u5b8c\u6210\u4e00\u8f6e\u5c31\u591a\u56fa\u5b9a\u4e00\u4e2a\u4f4d\u7f6e, \u6709\u9650\u6b21\u540e\u4fbf\u6062\u590d\u8be5\u7ec4\u7684\u539f\u76f8\u5bf9\u987a\u5e8f, \u4e14\u4e0d\u5fc5\u8de8\u8d8a\u7ec4\u5916\u7684\u80de.<br \/>\n$\\\\$ \u4e0a\u8ff0\u6784\u9020\u53ea\u8bc1\u660e\u987a\u5e8f\u5408\u6cd5, \u914d\u5bf9\u4e0d\u53d8\u8fd8\u9700\u53e6\u4e00\u6761\u4f9d\u636e: A.4\u4e0eA.5\u7684\u8fc1\u79fb\u8bba\u8bc1\u4fdd\u8bc1\u6b64\u65f6$C$, $F$\u4e2d\u7684\u80de\u53ea\u6d89\u53ca\u975e\u6d3b\u8dc3\u914d\u5bf9. \u5728\u5404\u7ec4\u5185\u90e8\u5e94\u7528\u5c40\u90e8\u4ea4\u6362\u89c4\u5219, \u624d\u80fd\u6392\u9664\u5bf9\u6d3b\u8dc3\u914d\u5bf9\u7684\u5f71\u54cd. \u65f6\u95f4\u503c\u8fd8\u987b\u6cbf\u6062\u590d\u540e\u7684\u987a\u5e8f\u4fdd\u6301\u4e0d\u51cf, \u4e0d\u80fd\u53ea\u6362\u987a\u5e8f\u5374\u4efb\u610f\u4fdd\u7559\u4e00\u7ec4\u4e0b\u964d\u7684\u65f6\u95f4\u503c. \u5b9e\u9645\u5b9e\u73b0\u53ef\u4ee5\u8df3\u8fc7\u8fd9\u6761\u8bc1\u660e\u8def\u5f84, \u76f4\u63a5\u6267\u884c\u7b2c7\u8282\u7684\u7a33\u5b9a\u91cd\u6392.<\/p>\n<p><strong>17. \u7591\u4f3c\u7b14\u8bef\u4e0e\u5f15\u7528\u8303\u56f4<\/strong><\/p>\n<p>\u4ee5\u4e0b\u4e3a\u5bf9\u7167\u5f53\u524d\u8bba\u6587\u7248\u672c\u6240\u5f97\u7684\u9605\u8bfb\u5224\u65ad, \u4e0d\u4ee3\u8868\u4f5c\u8005\u53d1\u5e03\u7684\u6b63\u5f0f\u52d8\u8bef.<br \/>\n$\\\\$ $\\cdot$ A.3\u4e2d\u63cf\u8ff0\u51fa\u751f\u80de\u4ea4\u6362\u5e76\u6539\u53d8\u914d\u5bf9\u65f6\u5f15\u7528Case 2.1.2, \u5e94\u4e3aCase 1.1.2. \u524d\u8005\u662f\u6b7b\u4ea1\u80de\u4ea4\u6362\u89c4\u5219.<br \/>\n$\\\\$ $\\cdot$ A.6\u8fc1\u79fb$C \\setminus B$\u4e0e$F \\setminus D$\u65f6\u4e24\u6b21\u5f15\u7528A.3\u6765\u4fdd\u7559\u5168\u90e8\u6d3b\u8dc3\u914d\u5bf9\u53ca\u5176\u4ee3\u8868\u94fe, \u4e0e\u524d\u63d0\u548c\u7ed3\u8bba\u5bf9\u5e94\u7684\u5e94\u662fA.5. \u9700\u8981\u4fee\u6b63\u5f15\u7406\u7f16\u53f7, \u800c\u975e\u628a\u7ed3\u8bba\u964d\u4e3a\u53ea\u4fdd\u7559\u4e3b\u8981\u914d\u5bf9.<br \/>\n$\\\\$ $\\cdot$ A.5\u524d\u5bfc\u6587\u5b57\u8bf4\u4fdd\u7559\u6240\u6709\u914d\u5bf9, \u6b63\u5f0f\u9648\u8ff0\u9650\u5b9a\u4e3a\u539f\u6709\u6d3b\u8dc3\u914d\u5bf9, \u5e94\u4ee5\u540e\u8005\u4e3a\u51c6.<br \/>\n$\\\\$ $\\cdot$ A.5\u8fc1\u79fb$\\sigma$\u65f6\u5f15\u7528Case 2.1.1\u4e0e2.1.2\u6b63\u786e, \u56e0\u4e3a\u5b83\u6b64\u65f6\u662f\u6b7b\u4ea1\u80de. \u5b64\u7acb$\\phi$\u4e0d\u80fd\u6210\u4e3a\u975e\u6d3b\u8dc3\u914d\u5bf9\u51fa\u751f\u80de\u7684\u8bba\u8bc1\u5e94\u5f15\u7528A.2(2), \u4e5f\u4e0d\u9700\u6539\u6210A.2(1).<\/p>\n<p><strong>18. UE\u5b9e\u73b0\u4f8b\u5b50<\/strong><\/p>\n<p>\u51fa\u4e8eLicense\u9650\u5236, \u65e0\u6cd5\u653e\u51fa\u6e90\u7801.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_original_view.png\" alt=\"\" width=\"1395\" height=\"843\" class=\"aligncenter size-full wp-image-5524\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_original_view.png 1395w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_original_view-300x181.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_original_view-768x464.png 768w\" sizes=\"(max-width: 1395px) 100vw, 1395px\" \/><center><em>\u7b80\u5316\u524d($\\beta_0 = 1$, $\\beta_1 = 1$, $\\beta_2 = 0$)<\/em><\/center><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_after_topology_view.png\" alt=\"\" width=\"1395\" height=\"843\" class=\"aligncenter size-full wp-image-5525\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_after_topology_view.png 1395w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_after_topology_view-300x181.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/09\/prescribed_topology_three_state_preview_after_topology_view-768x464.png 768w\" sizes=\"(max-width: 1395px) 100vw, 1395px\" \/><center><em>\u7b80\u5316\u540e($\\beta_0 = 1$, $\\beta_1 = 0$, $\\beta_2 = 0$)<\/em><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u5728\u901bSiggraph 2026\u7684\u6587\u7ae0\u65f6\u6dd8\u5230\u4e86\u4e00\u7bc7\u5c06\u6301\u7eed\u540c\u8c03\u4e0e\u56fe\u5f62\u5b66\u7ed3\u5408\u5f97\u6781\u4e3a\u5de7\u5999\u7684\u6587\u7ae0Rong L, Ju &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2026\/09\/23\/interpretation_paper_persistence_guided_prescribed_topological_simplification\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u300aPersistence-guided Prescribed Topological Simplification\u300b\u8bba\u6587\u89e3\u8bfb<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15,24,5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/5494"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=5494"}],"version-history":[{"count":34,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/5494\/revisions"}],"predecessor-version":[{"id":5531,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/5494\/revisions\/5531"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=5494"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=5494"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=5494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}