{"id":913,"date":"2019-02-03T01:21:45","date_gmt":"2019-02-02T17:21:45","guid":{"rendered":"https:\/\/mnihyc.com\/blog\/?p=913"},"modified":"2020-02-09T03:04:54","modified_gmt":"2020-02-08T19:04:54","slug":"sdoi2012%e4%bb%bb%e5%8a%a1%e5%ae%89%e6%8e%92","status":"publish","type":"post","link":"https:\/\/mnihyc.com\/blog\/archives\/913","title":{"rendered":"[SDOI2012]\u4efb\u52a1\u5b89\u6392"},"content":{"rendered":"<h1 style=\"text-align: center;\">[SDOI2012]\u4efb\u52a1\u5b89\u6392<\/h1>\n<p style=\"text-align: center;\">\u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB<\/p>\n<p style=\"text-align: center;\">\u96be\u5ea6\uff1a <span style=\"color: #ff6600;\">\\(7.1\\)<\/span><\/p>\n<ul>\n<li>\n<h3><strong>\u9898\u76ee\u63cf\u8ff0<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u6309\u987a\u5e8f\u7ed9\u5b9a \\(N\\) \u4e2a\u5b50\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u7528\u65f6 \\(t_i\\) \uff0c\u8d39\u7528\u7cfb\u6570 \\(f_i\\)\u3002<\/p>\n<p>\u53ef\u4ee5\u628a<strong>\u8fde\u7eed<\/strong>\u7684\u82e5\u5e72\u4e2a\uff08\u6216\u4e00\u4e2a\uff09\u5b50\u4efb\u52a1\u5408\u6210\u4e3a\u4e00\u4e2a\u5927\u4efb\u52a1\uff0c\u5927\u4efb\u52a1\u7684\u7528\u65f6\u548c\u8d39\u7528\u7cfb\u6570\u5206\u522b\u4e3a\u5176\u4e2d\u6bcf\u4e2a\u5b50\u4efb\u52a1\u7684\u7528\u65f6\u4e4b<strong>\u548c<\/strong>\u4e0e\u8d39\u7528\u7cfb\u6570\u4e4b<strong>\u548c<\/strong>\u3002\u5f00\u59cb\u4e00\u4e2a\u5927\u4efb\u52a1\u4e4b\u524d\u9700\u8981\u51c6\u5907\u65f6\u95f4 \\(S\\)\uff0c\u4e00\u4e2a\u5927\u4efb\u52a1\u7684\u8d39\u7528\u4e3a\u4ed6\u7684 \u5b8c\u6210\u65f6\u523b \\(\\times\\) \u8d39\u7528\u7cfb\u6570 \uff0c\u95ee\u6309\u987a\u5e8f\u6267\u884c\u5b8c\u6240\u6709\u7684\u5927\u4efb\u52a1\u540e\uff0c\u6700\u5c0f\u7684\u8d39\u7528\u548c\u4e3a\u591a\u5c11\u3002<\/p>\n<p>\u5f62\u5f0f\u5316\u7684\uff0c\u5c06 \\(N\\) \u4e2a\u4efb\u52a1\u5212\u5206\u6210\u82e5\u5e72\u4e2a\u5757\uff0c\u6bcf\u4e00\u7ec4\u4efb\u52a1 \\(M_i\\) \u82b1\u8d39\u4ee3\u4ef7 \\((T+\\sum{t_j}+S) \\times \\sum{f_j}\\)\uff0c\\(j \\in M_i\\)\uff0c\\(T\\) \u4e3a\u6267\u884c\u5230\u8fd9\u4e2a\u4efb\u52a1\u4e4b\u524d\u82b1\u8d39\u7684\u65f6\u95f4\uff0c\u6c42\u6267\u884c\u5b8c\u6240\u6709\u4efb\u52a1\u7684\u6700\u5c0f\u4ee3\u4ef7\u548c\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u5165\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u7b2c\u4e00\u884c\u4e00\u4e2a\u6574\u6570 \\(N\\) \uff1b<\/p>\n<p>\u800c\u540e \\(N\\) \u884c\u4e2d\uff0c\u7b2c \\(i\\) \u884c\u5305\u542b\u4e00\u4e2a<strong>\u53ef\u80fd\u4e3a\u8d1f\u7684<\/strong>\u6574\u6570 \\(t_i\\) \u548c\u4e00\u4e2a\u975e\u8d1f\u6574\u6570 \\(f_i\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u51fa\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a\u6700\u5c0f\u7684\u4ee3\u4ef7\u548c\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8f93\u5165<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"lang:default decode:true\">5 1\r\n1 3\r\n3 2\r\n4 3\r\n2 3\r\n1 4<\/pre>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8f93\u51fa<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"\">153<\/pre>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8bf4\u660e<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u5982\u679c\u5206\u7ec4\u65b9\u6848\u662f \\(\\{1,2\\}\\)\u3001\\(\\{3\\}\\)\u3001\\(\\{4,5\\}\\)\uff0c\u5219\u5b8c\u6210\u65f6\u95f4\u5206\u522b\u4e3a \\(\\{5,5,10,14,14\\}\\)\uff0c\u8d39\u7528 \\(C=\\{15,10,30,42,56\\}\\)\uff0c\u603b\u8d39\u7528\u5c31\u662f \\(153\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u6570\u636e\u89c4\u6a21\u4e0e\u7ea6\u5b9a<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u5bf9\u4e8e \\(20\\%\\) \u7684\u6570\u636e\uff0c\\(1 \\leq N \\leq 1000\\) \uff1b<\/p>\n<p>\u5bf9\u4e8e\u53e6\u5916 \\(40\\%\\) \u7684\u6570\u636e\uff0c \\(1 \\leq N \\leq 300000\\) \uff1b<\/p>\n<p>\u5bf9\u4e8e\u524d \\(60\\%\\) \u7684\u6570\u636e\uff0c\\(0 \\leq t_i \\leq 2^8\\) \uff1b<\/p>\n<p>\u5bf9\u4e8e\u540e \\(40\\%\\) \u7684\u6570\u636e\uff0c\\(1 \\leq N \\leq 100000\\)\uff0c<span style=\"font-size: 20px;\"><strong>\\(-2^8 \\leq t_i \\leq 2^8\\)<\/strong><\/span> \uff1b<\/p>\n<p>\u5bf9\u4e8e \\(100\\%\\) \u7684\u6570\u636e\uff0c \\(0 \\leq S, f_i \\leq 2^8\\) \u3002<\/p>\n<p><del>\u6211\u4e5f\u4e0d\u77e5\u9053\u4e3a\u4ec0\u4e48\u65f6\u95f4\u53ef\u4ee5\u4e3a\u8d1f\uff0c\u4e0d\u8981\u95ee\u6211\u3002<\/del><\/p>\n<p>\u6570\u636e\u6765\u6e90\u4e8e <a href=\"https:\/\/lydsy.com\/JudgeOnline\/wttl\/wttl.php?pid=2726\">BZOJ 2726<\/a> \u3002<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>\u5df2\u642c\u81f3 YZOJ P4253<\/p>\n<p><!--more--><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>\u8bbe \\(f_i\\) \u4e3a\u5206\u5272\u524d \\(i\\) \u4e2a\u4efb\u52a1\u7684\u6700\u5c0f\u8d39\u7528\u548c\u3002\u8ba1 \\(sumt\\) \u4e3a\u7528\u65f6\u7684\u524d\u7f00\u548c\uff0c\\(sumf\\) \u4e3a\u8d39\u7528\u7cfb\u6570\u7684\u524d\u7f00\u548c\u3002<\/p>\n<p>\u5982\u679c\u4e0d\u8003\u8651\u51c6\u5907\u65f6\u95f4 \\(S\\) \u7684\u8bdd\uff0c\u53ef\u4ee5\u5f88\u8f7b\u677e\u7684\u5199\u51fa\u8f6c\u79fb\u65b9\u7a0b <span style=\"font-size: 20px;\">\\(f_i = \\min\\limits_{j=0}^{i-1}\\{f_j+{sumt}_i({sumf}_i-{sumf}_j)\\}\\)<\/span> \u3002<\/p>\n<p>\u4f46\u662f\u8981\u8003\u8651\u51c6\u5907\u65f6\u95f4 \\(S\\) \u7684\u8bdd\uff0c\u7531\u4e8e\u6539\u53d8\u524d\u9762\u4efb\u52a1\u7684\u65f6\u95f4\u4f1a\u5f71\u54cd\u5230\u540e\u9762\u4efb\u52a1\u7684\u8d39\u7528\uff0c\u6240\u4ee5\u8fd9\u4e2a DP \u5c31\u53d8\u6210\u6709\u540e\u6548\u6027\u4e86\u3002\uff08\u5f53\u7136\uff0c\u5012\u8fc7\u6765 DP \u4e5f\u662f\u4e00\u79cd\u529e\u6cd5\u3002\uff09<\/p>\n<p>\u6ce8\u610f\u5230\u8fd9\u4e2a\u5f71\u54cd\u662f<strong>\u53ef\u4ee5\u8ba1\u7b97<\/strong>\u7684\u3002\u5982\u679c\u5728 \\(j\\) \u8fd9\u4e2a\u70b9\u8fdb\u884c\u5206\u5272\uff0c\u90a3\u4e48\u8fd9\u4e2a\u70b9\u4e4b\u540e\u7684\u6240\u6709\u4efb\u52a1\u7684\u5b8c\u6210\u65f6\u95f4\u90fd\u8981\u591a\u52a0\u4e0a \\(S\\) \uff0c\u5373\u4efb\u52a1 \\(k\\) \uff08\\(k &gt; j\\)\uff09 \u7684\u8d39\u7528\u591a\u4e0a \\(S \\times f_k\\) \uff08\\(f\\) \u8868\u793a\u8d39\u7528\u7cfb\u6570\uff09 \uff0c\u603b\u8d39\u7528\u5c31\u591a\u4e0a\u4e86 \\(S(sumf_N-sumf_j)\\) \u3002<\/p>\n<p>\u5728 DP \u7684\u65f6\u5019\u53ef\u4ee5\u628a\u8fd9\u4e2a\u5f71\u54cd\u5148\u52a0\u5230 \\(f_i\\) \u91cc\u9762\u6765\uff0c\u56e0\u4e3a\u53cd\u6b63\u90fd\u662f\u8981\u52a0\u4e0a\u53bb\u7684\uff0c\u52a0\u7684\u5148\u540e\u987a\u5e8f\u6ca1\u6709\u5173\u7cfb\u3002\u6240\u4ee5\u8f6c\u79fb\u65b9\u7a0b\u5c31\u53d8\u4e3a <span style=\"font-size: 20px;\">\\(f_i = \\min\\limits_{j=0}^{i-1}\\{f_j+{sumt}_i({sumf}_i-{sumf}_j)+S(sumf_N-sumf_j)\\}\\)<\/span> \u3002<\/p>\n<p>\u8fd9\u6837\u5c31\u5f97\u5230\u4e86\u4e00\u4e2a \\(O(N^2)\\) \u7684 DP \u80fd\u62ff\u5230 \\(60pts\\) \u3002<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"color: black;\">\u4e0d\u96be\u53d1\u73b0\u5176\u5b9e<\/span>\u8fd9\u662f\u4e00\u4e2a<strong>\u659c\u7387\u4f18\u5316<\/strong>\u7684\u7ecf\u5178\u6a21\u578b\u3002\u5bf9\u8fd9\u4e2a\u8f6c\u79fb\u65b9\u7a0b\u8fdb\u884c\u4e00\u4e9b\u5904\u7406<\/p>\n<p><span style=\"font-size: 20px;\">\\(\\begin{align} f_i &amp;= f_j+{sumt}_i({sumf}_i-{sumf}_j)+S(sumf_N-sumf_j) \\\\ \\underbrace{f_i}_{b_i} &amp;= \\underbrace{f_j+S(sumf_N-sumf_j)}_{y_j}-\\underbrace{sumt_i}_{k_i} \\cdot \\underbrace{sumf_j}_{x_j}+\\underbrace{sumt_i \\cdot sumf_i}_{const_i} \\end{align}\\)<\/span><\/p>\n<p>\u5176\u4e2d \\(x_j\\) \u662f<strong>\u5355\u8c03\u7684<\/strong>\uff0c\u4f46\u662f<strong>\u659c\u7387<\/strong> \\(k_i\\) \u5374<strong>\u4e0d\u662f\u5355\u8c03\u7684<\/strong>\uff08\u56e0\u4e3a\u53ef\u80fd\u51fa\u73b0 \\(t\\) \u4e3a\u8d1f\u6570\uff0c\u6240\u4ee5 \\(sumt\\) \u4e0d\u5177\u6709\u5355\u8c03\u6027\uff09\u3002<\/p>\n<p>\\(x_j\\) \u662f\u5355\u8c03\u7684\uff0c\u610f\u5473\u7740\u4ecd\u7136\u53ef\u4ee5\u6309\u7167\u8001\u529e\u6cd5\u52a0\u5165\u65b0\u7684\u70b9\u7ef4\u62a4\u4e0b\u51f8\u58f3\u3002\u4f46\u662f \\(k_i\\) \u4e0d\u662f\u5355\u8c03\u7684\uff0c\u8fd9\u5c31\u610f\u5473\u7740\u6ca1\u6709\u529e\u6cd5\u7ef4\u62a4\u5355\u8c03\u961f\u5217\uff0c\u4f7f\u961f\u5934\u5143\u7d20\u6c38\u8fdc\u662f\u6700\u5c0f\u7684\u3002<\/p>\n<p>\u6240\u4ee5\u6700\u7b80\u5355\u7684\u529e\u6cd5\u5c31\u662f\u4e0d\u8981\u5f39\u51fa\u961f\u5934\u3002\u56e0\u4e3a\u7ef4\u62a4\u7684\u4e0b\u51f8\u58f3\u4e0a\u6240\u6709\u70b9\u7684\u659c\u7387\u90fd\u662f<strong>\u5355\u8c03\u9012\u589e<\/strong>\u7684\uff0c\u6240\u4ee5\u53ef\u4ee5\u5728\u4e0a\u9762<strong>\u4e8c\u5206\u659c\u7387<\/strong>\uff0c\u627e\u5230<strong>\u6bd4\u5f53\u524d\u659c\u7387\u5927<\/strong>\u7684\u76f4\u7ebf\u4e2d<strong>\u659c\u7387<\/strong><strong>\u6700\u5c0f<\/strong>\u7684\u76f4\u7ebf\uff0c\u7528\u5b83\u7684<strong>\u5de6\u7aef\u70b9<\/strong>\u6765\u8d21\u732e\u7b54\u6848\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-926\" src=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2019\/02\/13.png\" alt=\"\" width=\"733\" height=\"443\" srcset=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2019\/02\/13.png 733w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2019\/02\/13-300x181.png 300w\" sizes=\"auto, (max-width: 733px) 100vw, 733px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>\u8fd9\u6837\u5c31\u53ef\u4ee5\u6709 \\(O(NlogN)\\) \u7684\u590d\u6742\u5ea6\u8db3\u4ee5\u901a\u8fc7\u672c\u9898\u4e86\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\ntypedef long long LLONG;\r\n\r\ninline int getnum()\r\n{\r\n\tregister int sgn=1;\r\n\tregister char c=getchar();\r\n\t(c=='-' ? sgn=-1 : 0);\r\n\twhile(!(c&gt;='0' &amp;&amp; c&lt;='9'))\r\n\t\tc=getchar(),(c=='-' ? sgn=-1 : 0);\r\n\tregister int a=0;\r\n\twhile(c&gt;='0' &amp;&amp; c&lt;='9')\r\n\t{\r\n\t\ta*=10;a+=c-'0';\r\n\t\tc=getchar();\r\n\t}\r\n\treturn sgn*a;\r\n}\r\n\r\nint st[305050],sf[305050];\r\nLLONG f[305050];\r\n\r\nstruct _point\r\n{\r\n\tLLONG x,y;\r\n\tLLONG operator * (const _point&amp;o)const\r\n\t{\r\n\t\treturn (x*o.y-y*o.x);\r\n\t}\r\n\t_point operator - (const _point&amp;o)const\r\n\t{\r\n\t\treturn (_point){x-o.x,y-o.y};\r\n\t}\r\n}list[305050];\r\nint front,rear;\r\n\r\nint main()\r\n{\r\n\tregister int N=getnum(),S=getnum();\r\n\tfor(register int i=1;i&lt;=N;i++)\r\n\t\tst[i]=st[i-1]+getnum(),sf[i]=sf[i-1]+getnum();\r\n\t\r\n\t\/*for(register int i=1;i&lt;=N;i++)\r\n\t{\r\n\t\tf[i]=INT_MAX;\r\n\t\tfor(register int j=0;j&lt;i;j++)\r\n\t\t{\r\n\t\t\tregister int ts=f[j]+st[i]*(sf[i]-sf[j])+S*(sf[N]-sf[j]);\r\n\t\t\tif(ts&lt;f[i])\r\n\t\t\t\tf[i]=ts;\r\n\t\t}\r\n\t}*\/\r\n\t\r\n\tlist[rear++]=(_point){sf[0],f[0]+(LLONG)S*(sf[N]-sf[0])};\r\n\tfor(register int i=1;i&lt;=N;i++)\r\n\t{\r\n\t\t_point ts=(_point){1,st[i]};\r\n\t\tregister int l=front,r=rear-1,ans=-1;\r\n\t\tif(l==r)\r\n\t\t\tans=l;\r\n\t\telse\r\n\t\t\twhile(l&lt;=r)\r\n\t\t\t{\r\n\t\t\t\tregister int mid=(l+r)&gt;&gt;1;\r\n\t\t\t\tif((list[mid+1]-list[mid])*ts&lt;=0)\r\n\t\t\t\t\tr=mid-1,ans=mid;\r\n\t\t\t\telse\r\n\t\t\t\t\tl=mid+1;\r\n\t\t\t}\r\n\t\tif(ans&gt;=0)\r\n\t\t\tf[i]=list[ans].y-(LLONG)st[i]*list[ans].x+(LLONG)st[i]*sf[i];\r\n\t\telse\r\n\t\t\tf[i]=LLONG_MAX&gt;&gt;2;\r\n\t\t\r\n\t\t_point tn=(_point){sf[i],f[i]+(LLONG)S*(sf[N]-sf[i])};\r\n\t\twhile(front+1&lt;rear &amp;&amp; (list[rear-1]-list[rear-2])*(tn-list[rear-1])&lt;=0)\r\n\t\t\trear--;\r\n\t\tlist[rear++]=tn;\r\n\t}\r\n\t\r\n\t\r\n\tprintf(\"%lld\\n\",f[N]);\r\n\t\r\n\treturn 0;\r\n}<\/pre>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[SDOI2012]\u4efb\u52a1\u5b89\u6392 \u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB \u96be\u5ea6\uff1a \u9898\u76ee\u63cf\u8ff0  &hellip; <a href=\"https:\/\/mnihyc.com\/blog\/archives\/913\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">[SDOI2012]\u4efb\u52a1\u5b89\u6392<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[37,63,30,43],"tags":[],"class_list":["post-913","post","type-post","status-publish","format-standard","hentry","category-crazya","category-bindivide","category-otherdp","category-slopedp"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>[SDOI2012]\u4efb\u52a1\u5b89\u6392 - mnihyc&#039;s Blog<\/title>\n<meta name=\"description\" content=\"\u4efb\u52a1\u5b89\u6392 \u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB \u96be\u5ea6\uff1a \u9898\u76ee\u63cf\u8ff0 \u6309\u987a\u5e8f\u7ed9\u5b9a (N) \u4e2a\u5b50\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u7528\u65f6 (t_i) 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