{"id":1265,"date":"2019-09-26T20:44:58","date_gmt":"2019-09-26T12:44:58","guid":{"rendered":"https:\/\/mnihyc.com\/blog\/?p=1265"},"modified":"2020-02-09T02:14:10","modified_gmt":"2020-02-08T18:14:10","slug":"yzoj-p4621-csp-s-2019-%e5%9b%9b%e6%a0%a1%e8%81%94%e8%ae%ad-round-1%e7%94%9f%e6%97%a5%e6%82%96%e8%ae%ba","status":"publish","type":"post","link":"https:\/\/mnihyc.com\/blog\/archives\/1265","title":{"rendered":"YZOJ P4621 [CSP-S 2019 \u56db\u6821\u8054\u8bad Round 1]\u751f\u65e5\u6096\u8bba"},"content":{"rendered":"<h1 style=\"text-align: center;\">YZOJ P4621 [CSP-S 2019 \u56db\u6821\u8054\u8bad Round 1]\u751f\u65e5\u6096\u8bba<\/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: #ffcc00;\">\\(5.0\\)<\/span><\/p>\n<ul>\n<li>\n<h3><strong>\u9898\u76ee\u63cf\u8ff0<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u4e00\u5171\u6709 \\(N\\) \u79cd\u6570\u5b57\uff0c\u6240\u4ee5\u6309\u7167\u5305\u542b\u6570\u5b57\u7684\u96c6\u5408\u5206\u7c7b\uff0c\u53ef\u80fd\u6709 \\(2^N\\) \u7c7b\u4e0d\u540c\u7684\u96c6\u5408\u3002\u6bcf\u4e2a\u96c6\u5408\u5305\u542b\u4e00\u79cd\u7279\u5b9a\u7684\u6570\u5b57\u7684\u6982\u7387\u90fd\u662f \\(\\displaystyle\\frac{1}{2}\\) \u3002<\/p>\n<p>\u968f\u673a\u53d6\u5176\u4e2d \\(k\\) \u4e2a\u96c6\u5408\uff0c\u8bf7\u4f60\u5e2e\u5fd9\u7b97\u4e00\u4e0b\u5176\u4e2d\u5b58\u5728\u4e24\u4e2a\u96c6\u5408\u5b8c\u5168\u76f8\u540c\u7684\u6982\u7387\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u5165\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u4e00\u884c\u4e24\u4e2a\u6574\u6570 \\(N,k\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u51fa\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u4e24\u4e2a\u6b63\u6574\u6570\uff0c\u5206\u522b\u8868\u793a\u6982\u7387\uff08\u6700\u7b80\u5206\u6570\uff09\u7684\u5206\u5b50\u548c\u5206\u6bcd\u3002\u5206\u522b\u5bf9 \\(10^6+3\\) \u53d6\u6a21\u5427\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u6570\u636e\u89c4\u6a21\u4e0e\u7ea6\u5b9a<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\\(1 \\leq N \\leq 10^{18}\\)\uff0c\\(2 \\leq k \\leq 10^{18}\\) \u3002<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Source\uff1a <a href=\"https:\/\/codeforces.com\/contest\/711\/problem\/E\">CodeForces 711E ZS and The Birthday Paradox<\/a><\/p>\n<p><!--more--><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>\u82e5 \\(2^N&lt;K\\) \uff0c\u7b54\u6848\u4e3a \\(1\\) \u3002<\/p>\n<p>\u5426\u5219\u7b54\u6848\u4e3a \\(\\displaystyle 1- \\frac{P_{2^N}^{K}}{{2^N}^K}\\) \u3002<\/p>\n<p>\u53d1\u73b0\u8fd9\u4e24\u7684 \\(\\gcd\\) \u80af\u5b9a\u662f \\(2\\) \u7684\u67d0\u4e2a\u6b21\u5e42\uff0c\u5e76\u4e14\u5206\u6bcd\u4e2d\u56e0\u5b50 \\(2\\) \u7684\u4e2a\u6570\u80af\u5b9a\u6bd4\u5206\u5b50\u591a\u3002<\/p>\n<p>\u540c\u65f6\u7ea6\u6389 \\(2^N\\)\uff0c\u8981\u5728\u5206\u5b50 \\(\\displaystyle\\frac{P_{2^N}^{K}}{2^N} = (2^N-1)(2^N-2)\\cdots(2^N-K+1)\\) \u4e2d\u627e\u51fa\u56e0\u5b50 \\(2\\) \u7684\u4e2a\u6570\u3002<\/p>\n<p>\u6ce8\u610f\u5230 \\(\\left|-K+1\\right|\\) \u4e2d\u56e0\u5b50 \\(2\\) \u7684\u4e2a\u6570\u80af\u5b9a\u6bd4 \\(2^N\\) \u5c11\uff0c\u6240\u4ee5\u53ea\u9700\u8981\u627e\u51fa \\(1, 2, \\cdots, K-1\\) \u4e2d\u6bcf\u4e2a\u6570\u5404\u5305\u542b\u56e0\u5b50 \\(2\\) \u7684\u4e2a\u6570\u3002<\/p>\n<p>\u4e3a \\(\\displaystyle \\left\\lfloor\\frac{K-1}{2}\\right\\rfloor + \\left\\lfloor\\frac{K-1}{4}\\right\\rfloor + \\cdots + \\left\\lfloor\\frac{K-1}{2^i}\\right\\rfloor\\)\uff0c\u5176\u4e2d \\(i\\) \u4e3a\u6ee1\u8db3 \\(2^i \\leq K-1\\) \u6700\u5927\u7684\u6574\u6570\u3002<\/p>\n<p>\u8fd9\u6837 \\(\\gcd\\) \u548c\u5206\u6bcd\u5c31\u89e3\u51b3\u4e86\uff0c\u5269\u4e0b\u5206\u5b50\u8fd8\u6ca1\u7b97\u51fa\u6765\u3002<\/p>\n<p>\u6839\u636e\u62bd\u5c49\u539f\u7406\uff0c\u82e5 \\(K-1 \\geq MOD\\) \u5219\u8fde\u7eed\u7684 \\(K-1\\) \u4e2a\u6570\u4e58\u79ef\u4e00\u5b9a\u5305\u542b \\(MOD\\) \u7684\u500d\u6570\uff0c\u5373\u5206\u5b50\u4e3a \\(0\\) \uff1b<\/p>\n<p>\u5426\u5219\u56e0\u4e3a \\(MOD\\) \u6bd4\u8f83\u5c0f\uff0c\u66b4\u529b\u7b97\u5373\u53ef\u3002<\/p>\n<p>\u6700\u540e\u9006\u5143\u7b97\u4e00\u4e0b\u5373\u53ef\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 1000003\r\n\r\ninline int _pow(register int base,register long long b)\r\n{\r\n\tregister int ans=1;\r\n\twhile(b)\r\n\t{\r\n\t\tif(b&amp;1)\r\n\t\t\tans=(long long)ans*base%MOD;\r\n\t\tbase=(long long)base*base%MOD;\r\n\t\tb&gt;&gt;=1;\r\n\t}\r\n\treturn ans;\r\n}\r\n\r\nint main()\r\n{\r\n\tlong long N,K;\r\n\tscanf(\"%lld%lld\",&amp;N,&amp;K);\r\n\t\r\n\tif(N&lt;=60 &amp;&amp; ((long long)1&lt;&lt;N)&lt;K)\r\n\t{\r\n\t\tprintf(\"%d %d\\n\",1,1);\r\n\t\treturn 0;\r\n\t}\r\n\t\r\n\tlong long pw2=2,num=0;\r\n\tfor(;pw2&lt;=K-1;pw2*=2)\r\n\t\tnum+=(K-1)\/pw2;\r\n\tlong long ans=1;\r\n\tif(K-1&lt;MOD &amp;&amp; (pw2=_pow(2,N)))\r\n\t\tfor(register int i=1;i&lt;=K-1 &amp;&amp; ans;i++)\r\n\t\t\t(ans*=(pw2-i+MOD))%=MOD;\r\n\telse\r\n\t\tans=0;\r\n\t\r\n\tregister int invgcd=_pow(_pow(2,num),MOD-2);\r\n\tregister long long low=(long long)_pow(_pow(2,K-1),N)*invgcd%MOD;\r\n\tprintf(\"%lld %lld\\n\",(low-ans*invgcd%MOD+MOD)%MOD,low);\r\n\t\r\n\treturn 0;\r\n}<\/pre>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>YZOJ P4621 [CSP-S 2019 \u56db\u6821\u8054\u8bad Round 1]\u751f\u65e5\u6096\u8bba \u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0 &hellip; <a href=\"https:\/\/mnihyc.com\/blog\/archives\/1265\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">YZOJ P4621 [CSP-S 2019 \u56db\u6821\u8054\u8bad Round 1]\u751f\u65e5\u6096\u8bba<\/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":[36,26],"tags":[],"class_list":["post-1265","post","type-post","status-publish","format-standard","hentry","category-harda","category-number-theory"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>YZOJ P4621 [CSP-S 2019 \u56db\u6821\u8054\u8bad Round 1]\u751f\u65e5\u6096\u8bba - mnihyc&#039;s Blog<\/title>\n<meta name=\"description\" content=\"YZOJ P4621 \u751f\u65e5\u6096\u8bba \u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB \u96be\u5ea6\uff1a \u9898\u76ee\u63cf\u8ff0 \u4e00\u5171\u6709 (N) \u79cd\u6570\u5b57\uff0c\u6240\u4ee5\u6309\u7167\u5305\u542b\u6570\u5b57\u7684\u96c6\u5408\u5206\u7c7b\uff0c\u53ef\u80fd\u6709 (2^N) \u7c7b\u4e0d\u540c\u7684\u96c6\u5408\u3002\u6bcf\u4e2a\u96c6\u5408\u5305\u542b\u4e00\u79cd\u7279\u5b9a\u7684\u6570\u5b57\u7684\u6982\u7387\u90fd\u662f (displaystylefrac{1}{2}) \u3002\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cf.mnihyc.com\/blog\/archives\/1265\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"YZOJ P4621 [CSP-S 2019 \u56db\u6821\u8054\u8bad Round 1]\u751f\u65e5\u6096\u8bba - mnihyc&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"YZOJ P4621 \u751f\u65e5\u6096\u8bba \u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB \u96be\u5ea6\uff1a \u9898\u76ee\u63cf\u8ff0 \u4e00\u5171\u6709 (N) \u79cd\u6570\u5b57\uff0c\u6240\u4ee5\u6309\u7167\u5305\u542b\u6570\u5b57\u7684\u96c6\u5408\u5206\u7c7b\uff0c\u53ef\u80fd\u6709 (2^N) \u7c7b\u4e0d\u540c\u7684\u96c6\u5408\u3002\u6bcf\u4e2a\u96c6\u5408\u5305\u542b\u4e00\u79cd\u7279\u5b9a\u7684\u6570\u5b57\u7684\u6982\u7387\u90fd\u662f (displaystylefrac{1}{2}) \u3002\" \/>\n<meta property=\"og:url\" content=\"https:\/\/cf.mnihyc.com\/blog\/archives\/1265\" \/>\n<meta property=\"og:site_name\" content=\"mnihyc&#039;s Blog\" \/>\n<meta property=\"article:published_time\" content=\"2019-09-26T12:44:58+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-02-08T18:14:10+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\" 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