{"id":801,"date":"2019-01-10T20:03:40","date_gmt":"2019-01-10T12:03:40","guid":{"rendered":"https:\/\/mnihyc.tk\/blog\/?p=801"},"modified":"2020-02-09T03:10:16","modified_gmt":"2020-02-08T19:10:16","slug":"yzoj-p3129-%e6%a0%a1%e5%86%85%e8%ae%ad%e7%bb%8320170627%e8%b7%b3%e6%a0%bc%e5%ad%90","status":"publish","type":"post","link":"https:\/\/0.mnihyc.com\/blog\/archives\/801","title":{"rendered":"YZOJ P3129 [\u6821\u5185\u8bad\u7ec320170627]\u8df3\u683c\u5b50"},"content":{"rendered":"<h1 style=\"text-align: center;\">YZOJ P3129 [\u6821\u5185\u8bad\u7ec320170627]\u8df3\u683c\u5b50<\/h1>\n<p style=\"text-align: center;\">\u65f6\u95f4\u9650\u5236\uff1a1000MS \u5185\u5b58\u9650\u5236\uff1a524288KB<\/p>\n<p style=\"text-align: center;\">\u96be\u5ea6\uff1a<span style=\"color: #993300;\">\\(7.0\\)<\/span><\/p>\n<ul>\n<li>\n<h3><strong>\u9898\u76ee\u63cf\u8ff0<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u5728\u4e00\u4e2a\u82b1\u56ed\u91cc\u6709\u4e00\u6392 \\(n\\) \u4e2a\u683c\u5b50\uff0c\u8fd9\u4e9b\u683c\u5b50\u7f16\u53f7\u4e3a \\(1\\) \u5230 \\(n\\) \u3002\u888b\u9f20\u559c\u6b22\u5728\u82b1\u56ed\u7684\u683c\u5b50\u4e0a\u8df3\u8dc3\u3002<\/p>\n<p>\u888b\u9f20\u7684\u8df3\u8dc3\u65b9\u5f0f\u662f\u8fd9\u6837\u7684\uff1a\u4e00\u5f00\u59cb\uff0c\u888b\u9f20\u4f4d\u4e8e \\(s\\) \u53f7\u683c\u5b50\u4e0a\uff0c\u63a5\u7740\u5b83\u4f1a\u9009\u62e9\u4e00\u4e2a\u5c1a\u672a\u7ecf\u8fc7\u7684\u683c\u5b50\u4e0a\u8df3\u8dc3\uff0c\u7ecf\u8fc7 \\(n-1\\) \u6b21\u8df3\u8dc3\u540e\uff0c\u6240\u6709\u683c\u5b50\u90fd\u88ab\u7ecf\u8fc7\u4e86\u4e00\u6b21\uff0c\u888b\u9f20\u7684\u8df3\u8dc3\u5c06\u7ed3\u675f\u3002\u888b\u9f20\u603b\u662f\u4f1a\u5728 \\(t\\) \u53f7\u683c\u5b50\u7ed3\u675f\u8df3\u8dc3\u3002<\/p>\n<p>\u7531\u4e8e\u888b\u9f20\u4e2d\u4e86\u75c5\u6bd2\uff0c\u6240\u4ee5\u5982\u679c\u4e0a\u4e00\u6b65\u5f80\u524d\u8df3\uff0c\u90a3\u4e48\u8fd9\u4e00\u6b65\u5fc5\u987b\u5f80\u540e\u8df3\uff1b\u53cd\u4e4b\uff0c\u82e5\u4e0a\u4e00\u6b65\u5f80\u540e\u8df3\uff0c\u90a3\u4e48\u8fd9\u4e00\u6b65\u5f80\u524d\u8df3\u3002<\/p>\n<p>\u8bf7\u95ee\u888b\u9f20\u6709\u591a\u5c11\u79cd\u8df3\u8dc3\u7684\u65b9\u6848\u5462\uff1f<\/p>\n<p>\u5f62\u5f0f\u5316\u5730\uff0c\u4f60\u9700\u8981\u6c42\u51fa\u6709\u591a\u5c11\u4e2a \\(1\\) \u5230 \\(n\\) \u7684\u6392\u5217 \\(p_1, p_2, \\cdots, p_n\\)\uff0c\u6ee1\u8db3\uff1a<\/p>\n<p style=\"padding-left: 30px;\">1\uff0c \\(p_1=s \uff0cp_n=t\\)\uff1b<\/p>\n<p style=\"padding-left: 30px;\">2\uff0c \u5bf9\u4efb\u610f \\(2 \\leq i \\leq n-1\\)\uff0c\u6216\u8005 \\((p_i-1&lt;p_i) \\land (p_i+1&lt;p_i)\\) \uff0c\u6216\u8005 \\((p_i-1&gt;p_i) \\land (p_i+1&gt;p_i)\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u5165\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u8f93\u5165\u5171\u4e00\u884c\uff0c\u5305\u542b\u4e09\u4e2a\u6b63\u6574\u6570 \\(n, s, t\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u51fa\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u8f93\u51fa\u5171\u4e00\u884c\uff0c\u5305\u542b\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a\u8df3\u8dc3\u7684\u65b9\u6848\u6570\u5bf9 \\(1,000,000,007\\) \u53d6\u6a21\u7684\u7ed3\u679c\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8f93\u5165\u3010\u5305\u542b\u4e24\u7ec4\u6837\u4f8b\u3011<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"\">\u3010\u6837\u4f8b1\u3011\r\n4 2 3\r\n\r\n\u3010\u6837\u4f8b2\u3011\r\n26 1 26<\/pre>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8f93\u51fa<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"\">\u3010\u6837\u4f8b1\u3011 \r\n2\r\n\r\n\u3010\u6837\u4f8b2\u3011 \r\n955348527<\/pre>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b 1 \u8bf4\u660e<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u4ece \\(2\\) \u53f7\u683c\u5b50\u5230 \\(3\\) \u53f7\u683c\u5b50\u7684\u8df3\u8dc3\u65b9\u6848\u6709\u4e24\u79cd\uff0c\u4e00\u79cd\u662f \\(2,1,4,3\\)\uff0c\u4e00\u79cd\u662f \\(2,4,1,3\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u6570\u636e\u89c4\u6a21\u4e0e\u7ea6\u5b9a<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-803\" src=\"https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/3129_Constant.png\" alt=\"\" width=\"602\" height=\"658\" srcset=\"https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/3129_Constant.png 602w, https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/3129_Constant-274x300.png 274w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><!--more--><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>\u8003\u573a\u4e0a\u75af\u72c2\u6253\u8868\u627e\u89c4\u5f8b\uff0c\u7136\u540e\u8fd8\u6ca1\u627e\u51fa\u6765\uff0c\u8c03\u8bd5\u4ee3\u7801\u6ca1\u5220\uff0c\u6240\u4ee5\u5c31\u7206\u4e86\u96f6\u3002<\/p>\n<p>\u542c\u8bf4\u672c\u9898\u6709\u6570\u5b66\u89e3\uff0c\u4f46\u662f\u6211\u592a\u83dc\u6211\u4e0d\u4f1a\u6211\u60f3\u4e0d\u51fa\u6765\uff0c<a href=\"https:\/\/en.wikipedia.org\/wiki\/Alternating_permutation\">\u8fd9\u91cc\u662fWiki \u7684\u8bf4\u660e<\/a>\u3002<\/p>\n<p>&nbsp;<\/p>\n<p>\u672c\u9898\u8fd0\u7528 <strong>\u63a5\u5934DP<\/strong> \u7684\u65b9\u6cd5\u7edf\u8ba1\u65b9\u6848\u6570\uff08\u63a5\u5934DP\uff1f\uff1f\uff1f\u6211\u4e5f\u662f\u7b2c\u4e00\u6b21\u542c\u8bf4\uff09\uff0c\u5373\u5728DP\u7684\u65f6\u5019\u8003\u8651\u5f53\u524d\u683c\u5b50\u548c\u5b83\u7684\u4e0a\u4e00\u4e2a\u53ca\u4e0b\u4e00\u4e2a\u683c\u5b50\uff08\u82e5\u5b58\u5728\uff09\u7684\u8fde\u63a5\u60c5\u51b5\u3002<\/p>\n<p>\u8bb0 <code><span style=\"color: #ff0000;\">o-<\/span><\/code> \u8868\u793a \u4ece\u8fd9\u4e2a\u683c\u5b50<strong>\u51fa\u53d1\u5411\u53f3<\/strong> \u6216\u8005 <strong>\u4ece\u53f3\u8fb9<\/strong>\u5230\u8fbe\u8fd9\u4e2a\u683c\u5b50<strong>\u7ed3\u675f<\/strong>\uff0c<code><span style=\"color: #ff0000;\">-o<\/span><\/code> \u8868\u793a \u4ece\u8fd9\u4e2a\u683c\u5b50<strong>\u51fa\u53d1\u5411\u5de6<\/strong> \u6216\u8005 <strong>\u4ece\u5de6\u8fb9<\/strong>\u5230\u8fbe\u8fd9\u4e2a\u683c\u5b50<strong>\u7ed3\u675f<\/strong>\uff1b<code><span style=\"color: #ff0000;\">&gt;o<\/span><\/code> \u8868\u793a \u4ece<strong>\u5de6\u8fb9\u8fdb\u5165<\/strong>\u8fd9\u4e2a\u683c\u5b50\u518d\u4ece<strong>\u5de6\u8fb9\u51fa\u53bb<\/strong>\uff0c<code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u8868\u793a \u4ece<strong>\u53f3\u8fb9\u8fdb\u5165<\/strong>\u8fd9\u4e2a\u683c\u5b50\u518d\u4ece<strong>\u53f3\u8fb9\u51fa\u53bb<\/strong>\u3002\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u9898\u76ee\u7684\u7279\u6b8a\u9650\u5236\uff0c\u6240\u4ee5\u53ea\u6709\u8fd9\u56db\u79cd\u60c5\u51b5\uff09<\/p>\n<p>\u90a3\u4e48\u5bf9\u4e8e\u7b2c\u4e00\u4e2a\u683c\u5b50\uff0c\u56e0\u4e3a\u5b83\u5de6\u8fb9\u5df2\u7ecf\u6ca1\u6709\u683c\u5b50\uff0c\u6240\u4ee5\u5b83\u7684\u63a5\u5934\u53ea\u6709\u4e24\u79cd\u60c5\u51b5\u3002<\/p>\n<p style=\"padding-left: 30px;\">1\uff0c \u82e5 \\((s = 1) \\lor (t = 1)\\) \uff0c\u90a3\u4e48\u5b83\u53ea\u53ef\u80fd\u662f <code><span style=\"color: #ff0000;\">o-<\/span><\/code> \u8fd9\u79cd\u60c5\u51b5\u3002<\/p>\n<p style=\"padding-left: 30px;\">2\uff0c \u82e5 \\((s \\ne 1) \\land (t \\ne 1)\\) \uff0c\u90a3\u4e48\u5b83\u53ea\u53ef\u80fd\u662f <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u8fd9\u79cd\u60c5\u51b5\u3002<\/p>\n<p>\u540c\u6837\uff0c\u5bf9\u4e8e\u7b2c \\(n\\) \u4e2a\u683c\u5b50\uff0c\u5b83\u53f3\u8fb9\u5df2\u7ecf\u6ca1\u6709\u683c\u5b50\uff0c\u6240\u4ee5\u5b83\u7684\u63a5\u5934\u4e5f\u53ea\u6709\u4e24\u79cd\u60c5\u51b5\u3002<\/p>\n<p style=\"padding-left: 30px;\">1\uff0c \u82e5 \\((s = n) \\lor (t = n)\\) \uff0c\u90a3\u4e48\u5b83\u53ea\u53ef\u80fd\u662f <code><span style=\"color: #ff0000;\">-o<\/span><\/code> \u8fd9\u79cd\u60c5\u51b5\u3002<\/p>\n<p style=\"padding-left: 30px;\">2\uff0c \u82e5 \\((s \\ne n) \\land (t \\ne n)\\) \uff0c\u90a3\u4e48\u5b83\u53ea\u53ef\u80fd\u662f <code><span style=\"color: #ff0000;\">&gt;o<\/span><\/code> \u8fd9\u79cd\u60c5\u51b5\u3002<\/p>\n<p>\u90a3\u4e48\u5bf9\u4e8e\u5176\u4ed6\u7684\u975e \\(1\\) \u975e \\(n\\) \u7684\u683c\u5b50 \\(i\\)\uff0c\u5b83\u7684\u63a5\u5934\u4e5f\u4f1a\u6709\u4e24\u79cd\u60c5\u51b5\u3002<\/p>\n<p style=\"padding-left: 30px;\">1\uff0c\u82e5 \\((i = s) \\lor (i = t)\\)\uff0c\u90a3\u4e48\u5b83\u53ef\u80fd\u4e3a <code><span style=\"color: #ff0000;\">o-<\/span><\/code> \u6216 <code><span style=\"color: #ff0000;\">-o<\/span><\/code> \u7684\u5176\u4e2d\u4e00\u79cd\u60c5\u51b5\u3002<\/p>\n<p style=\"padding-left: 30px;\">2\uff0c\u82e5 \\((i \\ne s) \\land (i \\ne t)\\)\uff0c\u90a3\u4e48\u5b83\u53ef\u80fd\u4e3a <code><span style=\"color: #ff0000;\">&gt;o<\/span><\/code> \u6216 <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u7684\u5176\u4e2d\u4e00\u79cd\u60c5\u51b5\u3002<\/p>\n<p>\u4f8b\u5982\u6837\u4f8b \\(1\\) \u5bf9\u5e94 <code><span style=\"color: #ff0000;\">o&lt; -o o- &gt;o<\/span><\/code> \u548c <code><span style=\"color: #ff0000;\">o&lt; o- -o &gt;o<\/span><\/code> \u8fd9\u4e24\u79cd\u8fde\u63a5\u60c5\u51b5\u3002<\/p>\n<p>\u6ce8\u610f\u5230\uff0c\u8fde\u63a5\u7684\u65b9\u5411\u53ef\u80fd\u76f8\u540c\uff0c\u4f46\u662f\u8fde\u63a5\u7684\u987a\u5e8f\u53ef\u80fd\u4e0d\u540c\uff0c\u5c31\u5982\u540c\u4e0a\u9762\u7684\u7b2c \\(1\\) \u4e2a\u683c\u5b50\uff0c\u6240\u4ee5\u72b6\u6001\u53ea\u6709\u4e00\u7ef4\u662f\u4e0d\u591f\u7684\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u8bbe \\(f_{i, j}\\) \u4e3a\u8868\u793a\u5f53\u524d\u6309\u987a\u5e8f\u505a\u5230<strong>\u7b2c \\(i\\) \u4e2a<\/strong>\u683c\u5b50\uff0c<strong>\u6b64\u65f6<\/strong>\u5269\u4f59\u7684<strong>\u672a\u914d\u5bf9\u7684<\/strong> <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u5f62\u8fde\u63a5\u8fd8\u6709 \\(j\\) \u4e2a\u7684\u65b9\u6848\u6570\u3002<\/p>\n<p>\u8003\u8651\u521d\u59cb\u503c\uff0c\u5176\u5b9e\u5df2\u7ecf\u5728\u4e0a\u9762\u6709\u6240\u6d89\u53ca\uff0c\u5373 \\((s = 1) \\lor (t = 1)\\) \u65f6 \\(f_{1, 0}=1\\) \uff0c\\((s \\ne 1) \\land (t \\ne 1)\\) \u65f6 \\(f_{1, 1}=1\\) \u3002<\/p>\n<p>\u8003\u8651\u8f6c\u79fb\uff0c\u5047\u8bbe\u5f53\u524d\u505a\u5230\u7b2c \\(i\\) \u4e2a\u683c\u5b50\uff0c\u540c\u6837\u6709\u4e24\u79cd\u60c5\u51b5\u3002<\/p>\n<p style=\"padding-left: 30px;\">1\uff0c\\((i = s) \\lor (i = t)\\) \uff0c\u53ef\u80fd\u4e3a <code><span style=\"color: #ff0000;\">o-<\/span><\/code> \u6216 <code><span style=\"color: #ff0000;\">-o<\/span><\/code> \u3002\u82e5\u662f <code><span style=\"color: #ff0000;\">o-<\/span><\/code> \uff0c\u5219\u5b83\u5bf9\u672a\u914d\u5bf9\u7684 <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u5f62\u8fde\u63a5\u7684\u4e2a\u6570<strong>\u6ca1\u6709\u5f71\u54cd<\/strong>\uff1b\u82e5\u662f <code><span style=\"color: #ff0000;\">-o<\/span><\/code> \uff0c\u5219\u5b83\u9700\u8981\u9009\u62e9\u524d\u9762\u7684\u4e00\u4e2a<strong>\u672a\u914d\u5bf9\u7684<\/strong> <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u8fdb\u884c\u63a5\u5934<strong>\u914d\u5bf9<\/strong>\u3002\u5373 \\(f_{i, j}=f_{i-1, j}+\\left( j+1 \\right) \\cdot f_{i-1, j+1}\\) \u3002<\/p>\n<p style=\"padding-left: 30px;\">2\uff0c\\((i \\ne s) \\land (i \\ne t)\\)\uff0c\u53ef\u80fd\u4e3a <code><span style=\"color: #ff0000;\">&gt;o<\/span><\/code><code><\/code> \u6216 <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u3002\u82e5\u662f <code><span style=\"color: #ff0000;\">&gt;o<\/span><\/code>\uff0c\u5219\u5b83\u53ef\u4ee5\u9009\u62e9\u4e4b\u524d<strong>\u4efb\u610f\u4e24\u4e2a<\/strong>\u672a\u914d\u5bf9\u7684 <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u5e76\u628a\u5b83\u4eec\u8fde\u5728\u4e00\u8d77\uff0c\u6216\u8005\u662f\u8fde\u63a5\u4ece \\(s\\) \u6216 \\(t\\) \u51fa\u6765\u7684<strong>\u4e00\u6761\u94fe<\/strong>\uff08\u6ce8\u610f\uff0c<strong>\u4ec5\u5f53 \\(i&gt;s\\) \u6216 \\(i&gt;t\\) \u65f6<\/strong>\u624d\u80fd\u8fdb\u884c\u8fde\u63a5\uff09\uff1b\u82e5\u662f <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code>\uff0c\u5219\u5b83\u4ec5\u4ec5\u53ea\u662f<strong>\u589e\u52a0<\/strong>\u4e86\u4e00\u4e2a\u672a\u914d\u5bf9\u7684 <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u3002\u5373 \\(f_{i, j}=f_{i-1, j-1}+j(j+1) \\cdot f_{i-1, j+1} + [i&gt;s]f_{i-1, j+1}+[i&gt;t]f_{i-1, j+1}\\) \uff0c\u5316\u7b80\u5f97 \\(f_{i, j}=f_{i-1, j-1}+(j+1)(j+[i&gt;s]+[i&gt;t]) \\cdot f_{i-1, j+1}\\) \u3002<\/p>\n<p>\u7531\u4e8e\u5168\u90e8\u505a\u5b8c\u4e00\u5b9a<strong>\u4e0d\u4f1a\u5269\u4e0b<\/strong>\u672a\u914d\u5bf9\u7684 <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u5f62\u8fde\u63a5\uff0c\u800c\u4e0d\u8bba\u662f\u54ea\u79cd\u60c5\u51b5\u4e00\u5b9a\u4e0d\u6539\u53d8\u672a\u914d\u5bf9\u7684 <code><span style=\"color: #ff0000;\">o&lt;<\/span><\/code> \u5f62\u8fde\u63a5\u7684\u6570\u91cf\uff0c\u7b54\u6848\u5373\u4e3a \\(f_{n, 0}=f_{n-1, 0}\\) \u3002<\/p>\n<p>\u8fd9\u6837\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002<\/p>\n<p>&nbsp;<\/p>\n<pre class=\"lang:default decode:true \">#include &lt;cstdio&gt;\r\n#include &lt;cstdlib&gt;\r\n#include &lt;cstring&gt;\r\n#include &lt;climits&gt;\r\n\r\n#define MOD 1000000007\r\n\r\nint f[2050][2550];\r\n\r\nint main()\r\n{\r\n\tint N,S,T;scanf(\"%d%d%d\",&amp;N,&amp;S,&amp;T);\r\n\t\r\n\tif(S==1 || T==1)\r\n\t\tf[1][0]=1;\r\n\telse\r\n\t\tf[1][1]=1;\r\n\tfor(register int i=2;i&lt;=N-1;i++)\r\n\t\tfor(register int j=0;j&lt;=i &amp;&amp; j&lt;=N-i;j++)\r\n\t\t\tif(S==i || T==i)\r\n\t\t\t\tf[i][j]=((long long)f[i-1][j]+(long long)(j+1)*f[i-1][j+1])%MOD;\r\n\t\t\telse\r\n\t\t\t\tf[i][j]=((long long)f[i-1][j-1]+(long long)(j+1)*f[i-1][j+1]*(j+(i&gt;S)+(i&gt;T)))%MOD;\r\n\t\r\n\tprintf(\"%d\\n\",f[N-1][0]);\r\n}<\/pre>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>YZOJ P3129 [\u6821\u5185\u8bad\u7ec320170627]\u8df3\u683c\u5b50 \u65f6\u95f4\u9650\u5236\uff1a1000MS \u5185\u5b58\u9650\u5236\uff1a524288KB  &hellip; <a href=\"https:\/\/0.mnihyc.com\/blog\/archives\/801\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">YZOJ P3129 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