{"id":897,"date":"2019-02-03T00:17:50","date_gmt":"2019-02-02T16:17:50","guid":{"rendered":"https:\/\/0.mnihyc.com\/blog\/?p=897"},"modified":"2020-02-09T03:05:20","modified_gmt":"2020-02-08T19:05:20","slug":"yzoj-p2021-apio2014sequence","status":"publish","type":"post","link":"https:\/\/0.mnihyc.com\/blog\/archives\/897","title":{"rendered":"YZOJ P2021 [APIO2014]sequence"},"content":{"rendered":"<h1 style=\"text-align: center;\">YZOJ P2021 [APIO2014]sequence<\/h1>\n<p style=\"text-align: center;\">\u65f6\u95f4\u9650\u5236\uff1a2000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB<\/p>\n<p style=\"text-align: center;\">\u96be\u5ea6\uff1a <span style=\"color: #ff9900;\">\\(5.7\\)<\/span><\/p>\n<ul>\n<li>\n<h3><strong>\u9898\u76ee\u63cf\u8ff0<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(N\\) \u7684\u975e\u8d1f\u6574\u6570\u5e8f\u5217 \\(a\\) \uff0c\u73b0\u8981\u628a\u5b83\u5206\u5272\u6210 \\(k+1\\) \u4e2a\u8fde\u7eed\u975e\u7a7a\u7684\u5b50\u5e8f\u5217\u3002<\/p>\n<p>\u6bcf\u6b21\u5206\u5272\u53ef\u4ee5\u9009\u62e9\u4e00\u6bb5\u5269\u4f59\u957f\u5ea6\u8d85\u8fc7 \\(1\\) \u7684\u5e8f\u5217\uff0c\u5e76\u5728\u5176\u4e2d\u9009\u62e9\u4e00\u4e2a\u4f4d\u7f6e\uff0c\u628a\u5b83\u5206\u5272\u6210\u4e24\u4e2a\u8fde\u7eed\u975e\u7a7a\u7684\u5b50\u5e8f\u5217\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u5f97\u5230\u4e00\u4e9b \\(Bonus\\)\uff0c\u4e3a\u5206\u5272\u51fa\u7684\u4e24\u4e2a\u65b0\u5e8f\u5217\u4e2d<strong>\u5143\u7d20\u548c<\/strong>\u7684<strong>\u4e58\u79ef<\/strong>\u3002<\/p>\n<p>\u73b0\u8981\u627e\u5230\u4e00\u79cd\u65b9\u6848\uff0c\u4f7f\u5f97\u7ecf\u8fc7 \\(k\\) \u6b21\u5206\u5272\u540e\uff0c\u80fd\u5f97\u5230\u7684 \\(Bonus\\) \u6700\u591a\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u5165\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u7b2c\u4e00\u884c\u5305\u542b\u4e24\u4e2a\u6574\u6570 \\(n,\u00a0 k\\) \uff1b<\/p>\n<p>\u7b2c\u4e8c\u884c\u5305\u542b \\(n\\) \u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u5e8f\u5217 \\(a\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u51fa\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u7b2c\u4e00\u884c\u5305\u542b\u4e00\u4e2a\u6574\u6570\uff0c\u4e3a\u6700\u5927 \\(Bonus\\) \u3002<\/p>\n<p>\u7b2c\u4e8c\u884c\u5305\u542b \\(k\\) \u4e2a \\(1\\) \u5230 \\(n-1\\) \u7684\u6574\u6570\uff0c\u8868\u793a\u6700\u4f18\u65b9\u6848\u3002\u5176\u4e2d\u7b2c \\(i\\) \u4e2a\u6570 \\(x_i\\) \u8868\u793a\u5728\u8be5\u65b9\u6848\u4e2d\uff0c\u8fdb\u884c\u7b2c \\(i\\) \u6b21\u64cd\u4f5c\u65f6\uff0c\u5e94\u8be5\u9009\u62e9\u7b2c \\(x_i\\) \u4e2a\u6570\u540e\u9762\u7684\u4f4d\u7f6e\uff0c\u5e76\u5c06\u8fd9\u4e2a\u6570\u6240\u5728\u7684\u5e8f\u5217\u5206\u6210\u4e24\u90e8\u5206\u3002<\/p>\n<p>\u5982\u679c\u6709\u591a\u4e2a\u6700\u4f18\u65b9\u6848\uff0c\u8f93\u51fa\u5176\u4e2d\u4efb\u610f\u4e00\u79cd\u5373\u53ef\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8f93\u5165<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"\">7 3\r\n4 1 3 4 0 2 3<\/pre>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8f93\u51fa<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"\">108\r\n1 3 5<\/pre>\n<ul>\n<li>\n<h3><strong>\u6570\u636e\u89c4\u6a21\u4e0e\u7ea6\u5b9a<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u6570\u636e\u6ee1\u8db3 \\(2 \\leq n \\leq 100000\\)\uff0c\\(1 \\leq k \\leq min\\{n-1,200\\}\\) \u3002<\/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>\u5957\u8def\u9898\uff0c\u8bbe \\(f_{i, k}\\) \u4e3a\u5272\u5230\u7b2c \\(i\\) \u4e2a\u6570\u4e14\u5df2\u7ecf\u5272\u4e86 \\(k\\) \u5200\u65f6 \\(Bonus\\) \u7684\u6700\u5927\u503c\uff0c\u8ba1 \\(sum\\) \u4e3a \\(a\\) \u7684\u524d\u7f00\u548c\u3002<\/p>\n<p>\u6240\u4ee5 <span style=\"font-size: 20px;\">\\(f_{i, k}=\\min\\limits_{j=1}^{i-1}\\{f_{j, k-1}+sum_j(sum_i-sum_j)\\}\\)<\/span> \u3002<\/p>\n<p>\u8fd9\u6837\u5c31\u662f \\(O(kN^2)\\) \u65e0\u6cd5\u901a\u8fc7\u672c\u9898\u3002<\/p>\n<p>\u4e0d\u96be\u53d1\u73b0\u5176\u5b9e\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, k} &amp;=f_{j, k-1}+sum_j(sum_i-sum_j) \\\\ \\underbrace{f_{i, k}}_{b_i} &amp;=\\underbrace{f_{j, k-1}-sum^2_j}_{y_j}+\\underbrace{sum_i}_{k_i} \\cdot \\underbrace{sum_j}_{-x_j} \\end{align}\\)<\/span><\/p>\n<p>\u8fd9\u91cc\u7684 \\(k_i\\) \u548c \\(x_j\\) \u90fd\u662f<strong>\u5355\u8c03<\/strong>\u7684\uff0c\u76f4\u63a5\u4e0a\u5355\u8c03\u961f\u5217 \\(O(kN)\\) \u5373\u53ef\u3002<\/p>\n<p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\\(x_j\\) \u662f<strong>\u9012\u51cf<\/strong>\u7684\uff0c\u6240\u4ee5\u5224\u65ad\u7684\u65f6\u5019\u7b26\u53f7\u8981\u53cd\u8fc7\u6765\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#ifdef ONLINE_JUDGE\r\n\tchar __B[1&lt;&lt;15],*__S=__B,*__T=__B;\r\n\t#define getchar() (__S==__T&amp;&amp;(__T=(__S=__B)+fread(__B,1,1&lt;&lt;15,stdin),__S==__T)?EOF:*__S++)\r\n#endif\r\n\r\ntypedef long long LLONG;\r\n\r\ninline int getnum()\r\n{\r\n\tregister char c=getchar();\r\n\twhile(!(c&gt;='0' &amp;&amp; c&lt;='9'))\r\n\t\tc=getchar();\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 a;\r\n}\r\n\r\nint sum[105050];\r\nLLONG f[105050][2];\r\nint path[105050][205],stk[205];\r\n\r\nstruct _point\r\n{\r\n\tLLONG x,y;int wh;\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[105050];\r\nint front,rear;\r\n\r\nint main()\r\n{\r\n\tregister int N=getnum(),K=getnum(),oK=K;\r\n\tfor(register int i=1;i&lt;=N;i++)\r\n\t\tsum[i]=sum[i-1]+getnum();\r\n\t\r\n\t\/*for(register int k=1;k&lt;=K;k++)\r\n\t\tfor(register int i=k;i&lt;=N;i++)\r\n\t\t\tfor(register int j=1;j&lt;i;j++)\r\n\t\t\t{\r\n\t\t\t\tULLONG tf=f[j][k-1]+sum[j]*(sum[i]-sum[j]);\r\n\t\t\t\tif(tf&gt;f[i][k])\r\n\t\t\t\t\tf[i][k]=tf,path[i][k]=j;\r\n\t\t\t}*\/\r\n\t\r\n\tfor(register int k=1,x=0;k&lt;=K;k++,x^=1)\r\n\t{\r\n\t\tfront=rear=0;\r\n\t\tfor(register int i=k;i&lt;=N;i++)\r\n\t\t{\r\n\t\t\t_point ts=(_point){1,sum[i],0};\r\n\t\t\twhile(front+1&lt;rear &amp;&amp; (list[front+1]-list[front])*ts&lt;=0)\r\n\t\t\t\tfront++;\r\n\t\t\tif(front&lt;rear)\r\n\t\t\t\tf[i][x]=list[front].y-(LLONG)list[front].x*sum[i],path[i][k]=list[front].wh;\r\n\t\t\t\r\n\t\t\t_point tn=(_point){-sum[i],f[i][x^1]-(LLONG)sum[i]*sum[i],i};\r\n\t\t\twhile(front+1&lt;rear &amp;&amp; (list[rear-1]-list[rear-2])*(tn-list[rear-1])&lt;=0)\r\n\t\t\t\trear--;\r\n\t\t\tlist[rear++]=tn;\r\n\t\t}\r\n\t}\r\n\t\r\n\tprintf(\"%lld\\n\",f[N][(K+1)&amp;1]);\r\n\twhile(K)\r\n\t\tN=path[N][K],stk[K]=N,K--;\r\n\tfor(register int i=oK;i&gt;=1;i--)\r\n\t\tprintf(\"%d \",stk[i]);\r\n\tprintf(\"\\n\");\r\n\t\r\n\treturn 0;\r\n}<\/pre>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>YZOJ P2021 [APIO2014]sequence \u65f6\u95f4\u9650\u5236\uff1a2000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131 &hellip; <a href=\"https:\/\/0.mnihyc.com\/blog\/archives\/897\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">YZOJ P2021 [APIO2014]sequence<\/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":[36,43],"tags":[],"class_list":["post-897","post","type-post","status-publish","format-standard","hentry","category-harda","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>YZOJ P2021 [APIO2014]sequence - mnihyc&#039;s Blog<\/title>\n<meta name=\"description\" content=\"YZOJ P2021 sequence \u65f6\u95f4\u9650\u5236\uff1a2000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB \u96be\u5ea6\uff1a \u9898\u76ee\u63cf\u8ff0 \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a (N) \u7684\u975e\u8d1f\u6574\u6570\u5e8f\u5217 (a) \uff0c\u73b0\u8981\u628a\u5b83\u5206\u5272\u6210 (k+1) \u4e2a\u8fde\u7eed\u975e\u7a7a\u7684\u5b50\u5e8f\u5217\u3002 \u6bcf\u6b21\u5206\u5272\u53ef\u4ee5\u9009\u62e9\u4e00\u6bb5\u5269\u4f59\u957f\u5ea6\u8d85\u8fc7 (1)\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/0self.mnihyc.com\/blog\/archives\/897\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"YZOJ P2021 [APIO2014]sequence - mnihyc&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"YZOJ P2021 sequence \u65f6\u95f4\u9650\u5236\uff1a2000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB \u96be\u5ea6\uff1a \u9898\u76ee\u63cf\u8ff0 \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a (N) \u7684\u975e\u8d1f\u6574\u6570\u5e8f\u5217 (a) \uff0c\u73b0\u8981\u628a\u5b83\u5206\u5272\u6210 (k+1) \u4e2a\u8fde\u7eed\u975e\u7a7a\u7684\u5b50\u5e8f\u5217\u3002 \u6bcf\u6b21\u5206\u5272\u53ef\u4ee5\u9009\u62e9\u4e00\u6bb5\u5269\u4f59\u957f\u5ea6\u8d85\u8fc7 (1)\" \/>\n<meta property=\"og:url\" content=\"https:\/\/0self.mnihyc.com\/blog\/archives\/897\" \/>\n<meta property=\"og:site_name\" content=\"mnihyc&#039;s Blog\" \/>\n<meta property=\"article:published_time\" content=\"2019-02-02T16:17:50+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-02-08T19:05:20+00:00\" \/>\n<meta name=\"author\" content=\"mnihyc\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@mnihyc\" \/>\n<meta name=\"twitter:site\" content=\"@mnihyc\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005\" \/>\n\t<meta name=\"twitter:data1\" content=\"mnihyc\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 \u5206\" \/>\n<script type=\"application\/ld+json\" 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