{"id":2513,"date":"2019-12-28T16:46:05","date_gmt":"2019-12-28T08:46:05","guid":{"rendered":"http:\/\/www.sniper97.cn\/?p=2513"},"modified":"2019-12-28T16:46:05","modified_gmt":"2019-12-28T08:46:05","slug":"%e3%80%90%e7%ae%97%e6%b3%95%e5%88%86%e6%9e%90%e4%b8%8e%e8%ae%be%e8%ae%a1%e3%80%91%e5%8a%a8%e6%80%81%e8%a7%84%e5%88%92","status":"publish","type":"post","link":"http:\/\/www.sniper97.cn\/index.php\/note\/introduction_to_algorithms\/2513\/","title":{"rendered":"\u3010\u7b97\u6cd5\u5bfc\u8bba\u3011\u52a8\u6001\u89c4\u5212"},"content":{"rendered":"\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"274\" height=\"269\" src=\"\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-37.png\" alt=\"\" class=\"wp-image-2600\"\/><\/figure><\/div>\n\n\n<p>\u52a8\u6001\u89c4\u5212\uff08dynamic programming\uff09\u4e0e\u5206\u503c\u65b9\u6cd5\u76f8\u4f3c\uff0c\u90fd\u662f\u901a\u8fc7\u7ec4\u5408\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898\u3002<\/p>\n\n\n<p>\u5206\u6cbb\u65b9\u6cd5\u5c06\u95ee\u9898\u5212\u5206\u4e3a\u4e92\u4e0d\u76f8\u4ea4\u7684\u5b50\u95ee\u9898\uff0c\u9012\u5f52\u5730\u6c42\u89e3\u5b50\u95ee\u9898\uff0c\u518d\u5c06\u4ed6\u4eec\u7ec4\u5408\u8d77\u6765\uff0c\u6c42\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002<\/p>\n\n\n<p>\u800c\u52a8\u6001\u89c4\u5212\u5e94\u7528\u4e8e\u5b50\u95ee\u9898\u91cd\u53e0\u7684\u60c5\u51b5\uff0c\u5373\u4e0d\u540c\u7684\u5b50\u95ee\u9898\u5177\u6709\u516c\u5171\u7684\u5b50\u5b50\u95ee\u9898\u3002\u901a\u5e38\u7528\u6765\u6c42\u89e3<strong>\u6700\u4f18\u5316\u95ee\u9898<\/strong>\uff08optimization problem\uff09\u3002<\/p>\n\n\n<p>\u6211\u4eec\u901a\u5e38\u6309\u7167\u5982\u4e0b4\u4e2a\u6b65\u9aa4\u6765\u8bbe\u8ba1\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\uff1a<\/p>\n\n\n<ul><li>\u523b\u753b\u4e00\u4e2a\u6700\u4f18\u89e3\u7684\u7ed3\u6784\u7279\u6027<\/li><li>\u9012\u5f52\u5730\u5b9a\u4e49\u6700\u4f18\u89e3\u7684\u503c<\/li><li>\u8ba1\u7b97\u6700\u4f18\u89e3\u7684\u503c\uff0c\u901a\u5e38\u91c7\u7528\u81ea\u5e95\u5411\u4e0a\u7684\u65b9\u6cd5\u3002<\/li><li>\u5229\u7528\u8ba1\u7b97\u51fa\u7684\u4fe1\u606f\u6784\u9020\u4e00\u4e2a\u6700\u4f18\u89e3\u3002<\/li><\/ul>\n\n\n<h2 class=\"wp-block-heading\">1.\u94a2\u6761\u5207\u5272<\/h2>\n\n\n<p>\u672c\u4f8b\u4e3e\u4e86\u4e00\u4e2a\u4e00\u7ef4\u7684\u52a8\u6001\u89c4\u5212\u6817\u5b50\uff0c\u5982\u4e0b\uff0c\u4e0d\u540c\u957f\u5ea6\u7684\u94a2\u6761\u4ef7\u94b1\u4e0d\u4e00\u6837\uff0c\u73b0\u5728\u6c42\u957f\u5ea6\u4e3an\u7684\u94a2\u6761\u600e\u4e48\u5207\u83b7\u5229\u6700\u591a\u3002<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"607\" height=\"72\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-155.png\" alt=\"\" class=\"wp-image-2516\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-155.png 607w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-155-300x36.png 300w\" sizes=\"(max-width: 607px) 100vw, 607px\" \/><\/figure><\/div>\n\n\n<p>\u73b0\u5728\u6709\u4e24\u79cd\u601d\u8def\uff1a<\/p>\n\n\n<p>1.\u5047\u5982\u73b0\u5728\u53c8\u4e00\u4e2a\u94a2\u7ba1\uff0c\u6211\u4eec\u5207\u5272\u95ee\u9898\u4e4b\u540e\u5c31\u53d8\u6210\u4e86\u4e24\u4e2a\u94a2\u7ba1\u7684\u5207\u5272\u95ee\u9898\uff0c\u6211\u4eec\u901a\u8fc7\u4e24\u4e2a\u76f8\u5173\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u5e76\u5728\u6240\u6709\u53ef\u80fd\u7684\u4e24\u6bb5\u65b9\u6848\u4e2d\u627e\u51fa\u6700\u5927\u503c\uff0c\u6784\u6210\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u6211\u4eec\u79f0\u94a2\u6761\u5207\u5272\u95ee\u9898\u6ee1\u8db3<strong>\u6700\u4f18\u5b50\u7ed3\u6784<\/strong>\uff08optimal substructure\uff09\u3002<\/p>\n\n\n<p>2.\u94a2\u6761\u5207\u5272\u8fd8\u5b58\u5728\u4e00\u79cd\u76f8\u4f3c\u4f46\u662f\u66f4\u4e3a\u7b80\u5355\u7684\u9012\u5f52\u6c42\u89e3\u6cd5\uff0c\u6211\u4eec\u5c06\u94a2\u6761\u4ece\u5de6\u8fb9\u5207\u5272ixa\u4e00\u4e2a\u957f\u5ea6\u4e3ai\u7684\u4e00\u6bb5\uff0c\u5bf9\u53f3\u8fb9\u5269\u4e0b\u7684n-i\u957f\u5ea6\u8fdb\u884c\u9012\u5f52\u6c42\u89e3\uff08\u5207\u5272\uff09\uff0c\u5bf9\u5de6\u8fb9\u5219\u4e0d\u518d\u8fdb\u884c\u5207\u5272\u3002\u5373\u73b0\u5728\u7684\u95ee\u9898\u4e3a\u5de6\u8fb9\u7684\u4ef7\u683c\u548c\u53f3\u8fb9\u5207\u5272\u5f97\u5230\u7684\u6700\u4f18\u89e3\uff0c\u4ece\u5bfb\u627e\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3\u53d8\u6210\u4e86\u4e00\u4e2a\u3002<\/p>\n\n\n<p>\u90a3\u4e48\u6211\u4eec\u5148\u6765\u5b9e\u73b0\u4e00\u4e0b\u7b2c\u4e8c\u79cd\u8f83\u7b80\u5355\u7684\u65b9\u6cd5\uff1a<\/p>\n\n\n<pre class=\"wp-block-preformatted\">def CUT_OD(self, length, cost):<br \/>    <em>\"\"\"<br \/><\/em><em>    <\/em><em>\u94a2\u7ba1\u5207\u5272<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> length:<\/em><em>\u94a2\u7ba1\u957f\u5ea6<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> cost:<\/em><em>\u6bcf\u4e2a\u957f\u5ea6\u7684\u4ef7\u683c\uff08\u7d22\u5f15\u5373\u957f\u5ea6\uff09<br \/><\/em><em>    <\/em><strong><em>:return<\/em><\/strong><em>:<br \/><\/em><em>    \"\"\"<br \/><\/em><em>    <\/em># \u5904\u7406\u94a2\u7ba1\u957f\u5ea6\u4e3a0\u7684\u60c5\u51b5<br \/>    if length == 0:<br \/>        return 0<br \/>    # \u521d\u59cb\u5316\u7ed3\u679c\uff08\u6700\u4f18\u89e3\uff09<br \/>    q = -1<br \/>    for i in range(1, length+1):<br \/>        # \u6c42\u89e3\u6700\u4f18\u89e3<br \/>        q = max(q, self.CUT_OD(length - i, cost) + cost[i])<br \/>    return q<\/pre>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"242\" height=\"33\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-157.png\" alt=\"\" class=\"wp-image-2518\"\/><\/figure><\/div>\n\n\n<p>\u4f46\u662f\u8fd9\u79cd\u8fd0\u884c\u6548\u7387\u5341\u5206\u4f4e\uff0c\u56e0\u4e3a\u8fdb\u884c\u4e86\u8fc7\u591a\u7684\u91cd\u590d\u8fd0\u7b97\uff0c\u5f53length\u4e0a\u5347\u65f6\uff0c\u8fd0\u884c\u65f6\u95f4\u6307\u6570\u7ea7\u4e0a\u5347\uff0c\u4e0b\u56fe\u662f\u5f53\u957f\u5ea6\u4e3a4\u65f6\u7684\u8c03\u7528\u6811\uff0c\u8fd0\u884c\u65f6\u95f4\u4e3an\u7684\u6307\u6570\u51fd\u6570\u3002<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"387\" height=\"205\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-156.png\" alt=\"\" class=\"wp-image-2517\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-156.png 387w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-156-300x159.png 300w\" sizes=\"(max-width: 387px) 100vw, 387px\" \/><\/figure><\/div>\n\n\n<p>\u4e8e\u662f\u6211\u4eec\u8bd5\u56fe<strong>\u4f7f\u7528\u4e00\u4e2a\u66f4\u9ad8\u6548\u7387\u7684\u52a8\u6001\u89c4\u5212\u7b97\u6cd5<\/strong>\u6765\u6c42\u89e3\u6700\u4f18\u94a2\u6761\u5207\u5272\u95ee\u9898\uff1a<\/p>\n\n\n<p>\u7b2c\u4e00\u79cd\u65b9\u6cd5\uff1a\u65e2\u7136\u6211\u4eec\u628a\u597d\u591a\u65f6\u95f4\u90fd\u7528\u6765\u505a\u91cd\u590d\u6027\u7684\u5de5\u4f5c\u4e86\uff0c\u90a3\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4\uff08\u94fe\u8868\uff09\u6765\u8bb0\u5f55\u90a3\u4e9b\u6211\u4eec\u5df2\u7ecf\u8bb0\u5f55\u8fc7\u7684\u7ed3\u679c\uff0c\u8fd9\u6837\u4e0d\u5c31\u5feb\u4e86\u4e48\uff0c\u6211\u4eec\u79f0\u8fd9\u4e2a\u65b9\u6cd5\u53eb<strong>\u5e26\u5907\u5fd8\u7684\u81ea\u9876\u5411\u4e0b\u6cd5<\/strong>\uff08top-down with memoization\uff09\u3002<\/p>\n\n\n<p>\u7b2c\u4e8c\u79cd\u65b9\u6cd5\uff1a\u8fd9\u79cd\u65b9\u6cd5\u4e00\u822c\u9700\u8981\u6070\u5f53\u5b9a\u4e49\u5b50\u95ee\u9898\u201c\u89c4\u6a21\u201d\u7684\u6982\u5ff5\uff0c\u4f7f\u5f97\u4efb\u4f55\u5b50\u95ee\u9898\u7684\u6c42\u89e3\u90fd\u53ea\u4f9d\u8d56\u4e8e\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\u6c42\u89e3\u3002\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5c06\u5b50\u95ee\u9898\u6309\u89c4\u6a21\u6392\u5e8f\uff0c\u6309\u7531\u5c0f\u5230\u5927\u4f9d\u6b21\u6c42\u89e3\u3002\u6211\u4eec\u79f0\u8fd9\u79cd\u65b9\u6cd5\u53eb\u81ea\u5e95\u5411\u4e0a\u6cd5\uff08bottom-up method\uff09\u3002<\/p>\n\n\n<p>\u4f46\u662f\u4e24\u79cd\u7b97\u6cd5\u7684\u6e10\u8fdb\u65f6\u95f4\u76f8\u540c\u3002<\/p>\n\n\n<p>\u4e0b\u9762\u662f\u52a0\u5165\u4e86\u5907\u5fd8\u673a\u5236\u7684\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\uff1a<\/p>\n\n\n<pre class=\"wp-block-preformatted\">def memoized_CUT_ROD(self, length, cost):<br \/>    <em>\"\"\"<br \/><\/em><em>    <\/em><em>\u94a2\u7ba1\u5207\u5272<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> length:<\/em><em>\u94a2\u7ba1\u957f\u5ea6<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> cost:<\/em><em>\u6bcf\u4e2a\u957f\u5ea6\u7684\u4ef7\u683c\uff08\u7d22\u5f15\u5373\u957f\u5ea6\uff09<br \/><\/em><em>    <\/em><strong><em>:return<\/em><\/strong><em>:<br \/><\/em><em>    \"\"\"<br \/><\/em><em>    <\/em># \u521d\u59cb\u5316\u5207\u5272\u4ef7\u683c\u6570\u7ec4\uff0c\u5e76\u76f4\u63a5\u5b9a\u4e49\u957f\u5ea60\u83b7\u76ca0\uff08\u4ec5\u7b2c\u4e00\u6b21\u521d\u59cb\u5316\uff09<br \/>    if self.value is None:<br \/>        self.value = [-1 for _ in range(length + 1)]<br \/>        self.value[0] = 0<br \/>    # \u5982\u679c\u8ba1\u7b97\u8fc7\u5f53\u524d\u957f\u5ea6\u7684\u6700\u4f18\u89e3\uff0c\u76f4\u63a5\u8fd4\u56de<br \/>    if self.value[length] &gt;= 0:<br \/>        print('return', length)<br \/>        return self.value[length]<br \/>    print('compute', length)<br \/>    for i in range(1, length + 1):<br \/>        self.value[length] = max(self.value[length], self.memoized_CUT_ROD(length - i, cost) + cost[i])<br \/>    return self.value[length]<\/pre>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"454\" height=\"32\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-158.png\" alt=\"\" class=\"wp-image-2520\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-158.png 454w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-158-300x21.png 300w\" sizes=\"(max-width: 454px) 100vw, 454px\" \/><\/figure><\/div>\n\n\n<p>\u81ea\u5e95\u5411\u4e0a\u7684\u7248\u672c\uff1a<\/p>\n\n\n<pre class=\"wp-block-preformatted\">def bottom_up_CUT_ROD(self, length, cost):<br \/>    <em>\"\"\"<br \/><\/em><em>    <\/em><em>\u81ea\u5e95\u5411\u4e0a<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> length:<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> cost:<br \/><\/em><em>    <\/em><strong><em>:return<\/em><\/strong><em>:<br \/><\/em><em>    \"\"\"<br \/><\/em><em>    <\/em># \u521d\u59cb\u5316\u5207\u5272\u4ef7\u683c\u6570\u7ec4\uff0c\u5e76\u76f4\u63a5\u5b9a\u4e49\u957f\u5ea60\u83b7\u76ca0\uff08\u4ec5\u7b2c\u4e00\u6b21\u521d\u59cb\u5316\uff09<br \/>    if self.value is None:<br \/>        self.value = [-1 for _ in range(length + 1)]<br \/>        self.value[0] = 0<br \/>    for i in range(1, length + 1):<br \/>        for j in range(1, i + 1):<br \/>            self.value[length] = max(self.value[length], self.bottom_up_CUT_ROD(i - j, cost) + cost[j])<br \/>    return self.value[length]<\/pre>\n\n\n<p>\u4e24\u79cd\u6c42\u89e3\u65b9\u6cd5\u590d\u6742\u5ea6\u5747\u4e3aO(n^2)\u3002<\/p>\n\n\n<h2 class=\"wp-block-heading\">2.\u77e9\u9635\u94fe\u4e58\u6cd5<\/h2>\n\n\n<p>\u4e3e\u4f8b\u4e8c\u7ef4\u52a8\u6001\u89c4\u5212\uff1a\u5229\u7528\u77e9\u9635\u7684\u4e58\u6cd5\u7ed3\u5408\u5f8b\u5bfb\u627e\u77e9\u9635\u76f8\u4e58\u4e2d\u8ba1\u7b97\u6b21\u6570\u6700\u5c11\u7684\u76f8\u4e58\u7ec4\u5408\u3002<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"182\" height=\"151\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-159.png\" alt=\"\" class=\"wp-image-2523\"\/><\/figure><\/div>\n\n\n<p><strong>\u6b65\u9aa4\u4e00\uff1a\u6700\u4f18\u62ec\u53f7\u5316\u65b9\u6848\u7684\u7ed3\u6784\u7279\u5f81<\/strong><\/p>\n\n\n<p>\u52a8\u6001\u89c4\u5212\u7684\u7b2c\u4e00\u6b65\u5c31\u662f\u5bfb\u627e\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u7136\u540e\u5229\u7528\u8fd9\u79cd\u5b50\u7ed3\u6784\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u9020\u51fa\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002<\/p>\n\n\n<p>\u672c\u4f8b\u4e2d\u7684\u7ed3\u6784\u7279\u5f81\u5c31\u662f\uff0c\u5047\u8bbe\u6211\u4eec\u5728A_{1~j}\u4ee5A_k\u4e3a\u5206\u5272\u70b9\uff0c\u90a3\u4e48A_{1~k}\u548cA_{k+1~j}\u5206\u522b\u4f7f\u7528\u6700\u4f18\u5206\u5272\u4ee3\u4ef7\u6765\u8ba1\u7b97 A_{1~j} \u7684\u6700\u5c0f\u4ee3\u4ef7\u3002<\/p>\n\n\n<p><strong>\u6b65\u9aa4\u4e8c\uff1a\u4e00\u4e2a\u9012\u5f52\u6c42\u89e3\u65b9\u6848<\/strong><\/p>\n\n\n<p>\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u77e9\u9635\uff0c\u5bf9\u4e8em[i,j]\u7684\u5b9a\u4e49\u5982\u4e0b\uff1a\u77e9\u9635A_{ i,j }\u6240\u9700\u6807\u91cf\u4e58\u6cd5\u6b21\u6570\u7684\u6700\u5c0f\u503c\u3002<\/p>\n\n\n<p>\u90a3\u4e48\u5bf9\u4e8e\u5728 A_k \u5206\u5272\u7684\u77e9\u9635\u5e8f\u5217 A_{i~j} \uff0c\u6211\u4eec\u6709:<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"373\" height=\"31\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-161.png\" alt=\"\" class=\"wp-image-2525\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-161.png 373w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-161-300x25.png 300w\" sizes=\"(max-width: 373px) 100vw, 373px\" \/><\/figure><\/div>\n\n\n<p>\u5176\u4e2dp\u4e3a\u4e24\u4e2a\u77e9\u9635\u7684\u884c\u5217\u53c2\u6570\uff0c\u56e0\u6b64\uff1a<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"552\" height=\"58\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-162.png\" alt=\"\" class=\"wp-image-2526\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-162.png 552w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-162-300x32.png 300w\" sizes=\"(max-width: 552px) 100vw, 552px\" \/><\/figure><\/div>\n\n\n<p><strong>\u6b65\u9aa4\u4e09\uff1a\u8ba1\u7b97\u6700\u4f18\u4ee3\u4ef7<\/strong><\/p>\n\n\n<p>\u5373\u4f7f\u7528\u4ee3\u7801\u8ba1\u7b97\u6700\u4f18\u89e3\uff1a<\/p>\n\n\n<pre class=\"wp-block-preformatted\">def matrix_chain_order(self, p):<br \/>    <em>\"\"\"<br \/><\/em><em><br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> p:<br \/><\/em><em>    <\/em><strong><em>:return<\/em><\/strong><em>:<br \/><\/em><em>    \"\"\"<br \/><\/em><em>    <\/em>if self.value is None:<br \/>        self.value = [[float('inf') for _ in range(len(p))] for _ in range(len(p))]<br \/>        self.cut_loc = [[-1 for _ in range(len(p))] for _ in range(len(p))]<br \/><br \/>    p_length = len(p)<br \/>    # \u521d\u59cb\u5316\u5bf9\u89d2\u7ebf\u5143\u7d20<br \/>    for i in range(p_length):<br \/>        self.value[i][i] = 0<br \/>    # \u957f\u5ea6\u4e3an\u7684 \uff08\u5148\u8ba1\u7b97\u4e24\u4e24\u7684\u62ec\u53f7\u7684\u3001\u4e09\u4e09\u62ec\u53f7\u7684\u3002\u3002\u3002\uff09<br \/>    for l in range(2, p_length + 1):<br \/>        # \u5bfb\u627elength-n+1 \u4e2a\u957f\u5ea6\u4e3an\u7684\u6392\u5e8f<br \/>        for i in range(0, p_length - l + 1):<br \/>            j = i + l - 1<br \/>            for k in range(i, j):<br \/>                if self.value[i][j] &gt; self.value[i][k] + self.value[k + 1][j] + p[i][0] * p[k][1] * p[j][1]:<br \/>                    self.value[i][j] = self.value[i][k] + self.value[k + 1][j] + p[i][0] * p[k][1] * p[j][1]<br \/>                    self.cut_loc[i][j] = k<\/pre>\n\n\n<p>\u8f93\u5165\u4e3a\u6570\u7ec4p = [[30, 35], [35, 15], [15, 5], [5, 10], [10, 20], [20, 25]]\uff0c\u4ee3\u8868\u77e9\u96350\u4e3a30*35\uff0c\u77e9\u96351\u4e3a35*15&#8230;&#8230;<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"336\" height=\"331\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-163.png\" alt=\"\" class=\"wp-image-2527\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-163.png 336w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-163-300x296.png 300w\" sizes=\"(max-width: 336px) 100vw, 336px\" \/><\/figure><\/div>\n\n\n<p>\u53ef\u4ee5\u770b\u51fa\uff0c\u5f53\u6211\u4eec\u8bd5\u56fe\u77e5\u9053A_{0~5}\u7684\u6700\u4f18\u89e3\u65f6\uff0c\u9700\u8981\u505a15125\uff08[0][5]\uff09\u6b21\u4e58\u6cd5\uff0c\u800c\u9700\u8981\u5728idx2\u5904\u5206\u5272\uff0c\u5373\u3002 A_{0~2}\u548c A_{3~5}\uff0c\u800c A_{0~2}\u56fe\u4e2d\u4e3a0\uff0c\u5219\u7ee7\u7eed\u5206\u4e3a  A_{0}\u548c A_{1~2} \uff1b A_{3~5}\u4e3a4\uff0c\u5373\u53c8\u5206\u5272\u4e3a A_{4~5} \u548c A_{6}\u4ee5\u6b64\u8ba1\u7b97\u51fa\u6700\u4f18<\/p>\n\n\n<p>((A_0(A_1*A_2))((A_4*A_5)A_6))\uff09\u3002    <\/p>\n\n\n<p><strong>\u6b65\u9aa4\u56db\uff1a\u6784\u9020\u6700\u4f18\u89e3<\/strong><\/p>\n\n\n<p>\u8f93\u51fa\u7ed3\u679c\uff1a<\/p>\n\n\n<pre class=\"wp-block-preformatted\">def print_optimal_parens(self, i, j):<br \/>    <em>\"\"\"<br \/><\/em><em>    <\/em><em>\u6253\u5370\u62ec\u53f7\u5206\u5e03<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> i: from<br \/><\/em><em>    <\/em><strong><em>:param<\/em><\/strong><em> j: to<br \/><\/em><em>    <\/em><strong><em>:return<\/em><\/strong><em>:<br \/><\/em><em>    \"\"\"<br \/><\/em><em>    <\/em>if i == j:<br \/>        print('A_', i, ' ', end='')<br \/>    else:<br \/>        print('(', end='')<br \/>        self.print_optimal_parens(i, self.cut_loc[i][j])<br \/>        self.print_optimal_parens(self.cut_loc[i][j] + 1, j)<br \/>        print(')', end='')<\/pre>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"436\" height=\"34\" src=\"\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-164.png\" alt=\"\" class=\"wp-image-2529\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-164.png 436w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/11\/\u56fe\u7247-164-300x23.png 300w\" sizes=\"(max-width: 436px) 100vw, 436px\" \/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\">3.\u52a8\u6001\u89c4\u5212\u539f\u7406<\/h2>\n\n\n<h3 class=\"wp-block-heading\">1.\u6700\u4f18\u5b50\u7ed3\u6784<\/h3>\n\n\n<p>\u52a8\u6001\u89c4\u5212\u7684\u601d\u60f3\u5c31\u662f\u627e\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u4f46\u662f\u540c\u65f6\u9700\u8981\u6ce8\u610f\uff0c\u5e76\u4e0d\u662f\u6240\u6709\u7684\u95ee\u9898\u90fd\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u6bd4\u5982\u4e0b\u56fe\uff1a<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"178\" height=\"116\" src=\"\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-32.png\" alt=\"\" class=\"wp-image-2587\"\/><\/figure><\/div>\n\n\n<p>\u5982\u679c\u6211\u4eec\u627e\u6700\u77ed\u8ddd\u79bb\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u5355\u6b65\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u4f46\u662f\u5982\u679c\u6211\u4eec\u662f\u627e\u6700\u957f\u8ddd\u79bb\uff0c\u90a3\u4e48\u6211\u4eec\u6c38\u8fdc\u4e5f\u627e\u4e0d\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff08\u56e0\u4e3a\u6c38\u8fdc\u4f1a\u6709\u66f4\u957f\u7684\uff09\u3002<\/p>\n\n\n<h3 class=\"wp-block-heading\">2.\u91cd\u53e0\u5b50\u95ee\u9898<\/h3>\n\n\n<p>\u9002\u7528\u4e8e\u52a8\u6001\u89c4\u5212\u65b9\u6cd5\u6c42\u89e3\u7684\u6700\u4f18\u5316\u95ee\u9898\u5e94\u8be5\u5177\u5907\u7684\u7b2c\u4e8c\u4e2a\u6027\u8d28\u662f\u5b50\u95ee\u9898\u7a7a\u95f4\u5fc5\u987b\u8db3\u591f\u5c0f\uff0c\u5e76\u4e14\u6709\u53cd\u590d\u6c42\u89e3\u7684\u76f8\u540c\u5b50\u95ee\u9898\uff08\u5982\u524d\u9762\u94a2\u6761\u95ee\u9898\u5bf9\u4e8e\u540c\u4e00\u533a\u95f4\u5229\u6da6\u7684\u53cd\u590d\u6c42\u89e3\uff09\uff0c\u6211\u4eec\u5c31\u79f0\u8fd9\u4e2a\u6700\u4f18\u5316\u95ee\u9898\u5177\u6709<strong>\u91cd\u53e0\u5b50\u95ee\u9898<\/strong>\u3002<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"251\" src=\"\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-33.png\" alt=\"\" class=\"wp-image-2589\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-33.png 640w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-33-300x118.png 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><\/figure><\/div>\n\n\n<p>\u5982\u4e0a\u56fe\uff0c\u5728\u94a2\u6761\u5229\u76ca\u8ba1\u7b97\u65f6\uff0c\u91cd\u590d\u8ba1\u7b97\u4e86\u591a\u7ec4\u76f8\u540c\u89e3\uff0c\u5982\u679c\u4f7f\u7528\u52a8\u5f52\u8bb0\u5f55\u8fd9\u4e9b\u503c\u5c31\u4e0d\u9700\u8981\u8fdb\u884c\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u51cf\u5c11\u8fd0\u884c\u65f6\u95f4\u3002<\/p>\n\n\n<h3 class=\"wp-block-heading\">3.\u91cd\u6784\u6700\u4f18\u89e3<\/h3>\n\n\n<p>\u4e0d\u4ec5\u8bb0\u5f55\u5c40\u90e8\u6700\u4f18\u7ed3\u679c\uff0c\u8fd8\u8981\u8bb0\u5f55\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u4ee5\u964d\u4f4e\u4e0b\u4e00\u6b65\u9700\u8981\u8ba1\u7b97\u51fa\u5168\u5c40\u6700\u4f18\u89e3\u65f6\u7684\u4ee3\u4ef7\u3002<\/p>\n\n\n<h2 class=\"wp-block-heading\">4.\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217<\/h2>\n\n\n<p>\u5b9e\u9645\u4e0a\u5c31\u662fKMP\u7b97\u6cd5\u3002\u5728\u8fd9\u91cc\u4e0d\u505a\u8bf4\u660e\uff0c\u4e0a\u4e00\u4e2a\u89c6\u9891\u3002<\/p>\n\n\n<p><iframe src=\"\/\/player.bilibili.com\/player.html?aid=22409335&amp;cid=37123410&amp;page=1\" scrolling=\"no\" border=\"0\" framespacing=\"0\" allowfullscreen=\"true\" frameborder=\"no\"> <\/iframe><\/p>\n\n\n<h2 class=\"wp-block-heading\">5.\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811<\/h2>\n\n\n<p>\u5047\u5982\u6211\u4eec\u9700\u8981\u8fdb\u884c\u4e00\u4e2a\u6811\u67e5\u627e\u64cd\u4f5c\uff0c\u5982\u679c\u8981\u5feb\u901f\u7684\u8fdb\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u53ef\u4ee5\u4f7f\u7528\u5e73\u8861\u4e8c\u53c9\u6811\u6216\u8005\u7ea2\u9ed1\u6811\u6765\u8fdb\u884c\uff0c\u4f46\u662f\u5047\u5982\u5176\u4e2d\u67d0\u4e2a\u8282\u70b9\u9700\u8981\u9891\u7e41\u7684\u8fdb\u884c\u67e5\u8be2\u64cd\u4f5c\uff0c\u800c\u6070\u597d\u8fd9\u4e2a\u7ed3\u70b9\u53c8\u662f\u6811\u7684\u53f6\u5b50\u7ed3\u70b9\uff0c\u90a3\u4e48\u65e0\u7591\u4e5f\u662f\u9700\u8981\u5f88\u957f\u7684\u65f6\u95f4\u6765\u8fdb\u884c\u67e5\u627e\u64cd\u4f5c\u3002<\/p>\n\n\n<p>\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\u7684\u529e\u6cd5\u5c31\u662f<strong>\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811<\/strong>\u3002\u6211\u4eec\u5c06\u67e5\u8be2\u6b21\u6570\u8f83\u591a\u7684\u70b9\u5411\u6839\u9760\u62e2\uff0c\u5c06\u67e5\u8be2\u6b21\u6570\u4e0d\u90a3\u4e48\u591a\u7684\u70b9\u5411\u53f6\u5b50\u9760\u62e2\u3002<\/p>\n\n\n<p>\u6bd4\u5982\u5982\u4e0b\u7684\u4f8b\u5b50\uff0c\u5176\u4e2d\uff0cp\u8868\u793a\u8be5\u6570\u5b57\u51fa\u73b0\u7684\u6982\u7387\uff0cq\u8868\u793a\u4e0d\u662f\u8be5\u6570\u5b57\uff08\u662f\u8be5\u533a\u95f4\uff09\u51fa\u73b0\u7684\u6982\u7387\uff0c\u4e5f\u5c31\u662fp_i+q_i=1\uff1a<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"557\" height=\"100\" src=\"\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-34.png\" alt=\"\" class=\"wp-image-2594\" srcset=\"http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-34.png 557w, http:\/\/www.sniper97.cn\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-34-300x54.png 300w\" sizes=\"(max-width: 557px) 100vw, 557px\" \/><\/figure><\/div>\n\n\n<p>\u5bf9\u4e8e\u8fd9\u4e2a\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u671f\u671b\u641c\u7d22\u4ee3\u4ef7\u4e3a2.80\u3002<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"274\" height=\"164\" src=\"\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-35.png\" alt=\"\" class=\"wp-image-2595\"\/><\/figure><\/div>\n\n\n<p>\u800c\u5bf9\u4e8e\u4e0b\u9762\u8fd9\u4e2a\uff08\u6700\u4f18\uff09\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u671f\u671b\u641c\u7d22\u4ee3\u4ef7\u53ea\u67092.75\u3002<\/p>\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"266\" height=\"196\" src=\"\/wp-content\/uploads\/2019\/12\/\u56fe\u7247-36.png\" alt=\"\" class=\"wp-image-2596\"\/><\/figure><\/div>\n\n\n<p>\u5728\u751f\u6210\u8fd9\u6837\u4e00\u4e2a\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811\u65f6\u4e5f\u662f\u9075\u5faa\u52a8\u6001\u89c4\u5212\u7684\u601d\u60f3\uff0c\u5373\u6bcf\u4e00\u4e2a\u5b50\u6811\u90fd\u662f\u5f53\u524d\u7684\u6700\u4f18\u89e3\uff0c\u6211\u4eec\u4ece\u53f6\u5b50\u8282\u70b9\u9009\u53d6\u4e00\u4e2a\u6982\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u70b9\u4f5c\u4e3a\u6839\uff0c\u7136\u540e\u9010\u6b65\u5411\u4e0a\u6c42\u548c\u53d6\u6700\u4f4e\uff0c\u6700\u7ec8\u6784\u9020\u51fa\u4e00\u4e2a\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u52a8\u6001\u89c4\u5212\uff08dynamic programming\uff09\u4e0e\u5206\u503c\u65b9\u6cd5\u76f8\u4f3c\uff0c\u90fd\u662f\u901a\u8fc7\u7ec4\u5408\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898 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