{"id":1223,"date":"2019-07-02T16:58:54","date_gmt":"2019-07-02T08:58:54","guid":{"rendered":"https:\/\/0.mnihyc.com\/blog\/?p=1223"},"modified":"2020-02-09T02:22:52","modified_gmt":"2020-02-08T18:22:52","slug":"yzoj-p3451-sdoi2013%e9%9a%8f%e6%9c%ba%e6%95%b0%e7%94%9f%e6%88%90%e5%99%a8%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"https:\/\/0.mnihyc.com\/blog\/archives\/1223","title":{"rendered":"YZOJ P3451 [SDOI2013]\u968f\u673a\u6570\u751f\u6210\u5668\u95ee\u9898"},"content":{"rendered":"<h1 style=\"text-align: center;\">YZOJ P3451 [SDOI2013]\u968f\u673a\u6570\u751f\u6210\u5668\u95ee\u9898<\/h1>\n<p style=\"text-align: center;\">\u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a262144KB<\/p>\n<p style=\"text-align: center;\">\u96be\u5ea6\uff1a<span style=\"color: #ff6600;\">\\(6.0\\)<\/span><\/p>\n<ul>\n<li>\n<h3><strong>\u9898\u76ee\u63cf\u8ff0<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u6709\u4e00\u4e2a\u968f\u673a\u6570\u751f\u6210\u5668\uff0c\u53ef\u4ee5\u751f\u6210 \\(0\\) ~ \\(p-1\\) \u4e4b\u95f4\u7684\u4f2a\u968f\u673a\u6574\u6570\u3002\u5728\u751f\u6210\u4e4b\u524d\uff0c\u9700\u8981\u8bbe\u5b9a\u4e00\u4e2a\u968f\u673a\u6570\u79cd\u5b50 \\(k\\) \uff08\\(0 \\leq k &lt; p\\)\uff09\uff0c\u5219\u751f\u6210\u5668\u7b2c \\(1\\) \u6b21\u751f\u6210\u7684\u6574\u6570\u4e3a \\(k\\)\u3002\u8be5\u968f\u673a\u6570\u751f\u6210\u5668\u6709 \\(2\\) \u4e2a\u53c2\u6570 \\(a, b\\)\uff0c\u5982\u679c\u7b2c \\(i\\) \u6b21\u751f\u6210\u7684\u6574\u6570\u4e3a \\(x\\)\uff0c\u5219\u7b2c \\(i+1\\) \u6b21\u751f\u6210\u7684\u6574\u6570\u4e3a \\((ax+b) \\bmod p\\) \u3002<\/p>\n<p>\u73b0\u7ed9\u5b9a\u6574\u6570 \\(t\\) \uff0c\u6211\u4eec\u7684\u95ee\u9898\u662f\uff0c\u8be5\u968f\u673a\u6570\u751f\u6210\u5668\u81f3\u5c11\u9700\u8981\u751f\u6210\u591a\u5c11\u4e2a\u6570\uff0c\u624d\u80fd\u751f\u6210\u5f97\u5230 \\(t\\)\uff1f<\/p>\n<p>\u5bf9\u4e8e\u7ed9\u5b9a\u7684 \\(p,a,b,k,t\\) \uff0c\u8bf7\u8ba1\u7b97\u8981\u4f7f\u968f\u673a\u6570\u751f\u6210\u5668\u751f\u6210\u6574\u6570 \\(t\\)\uff0c\u6240\u9700\u751f\u6210\u6570\u7684\u6b21\u6570\u7684\u6700\u5c0f\u503c\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u5165\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u591a\u7ec4\u6570\u636e\u3002\u7b2c\u4e00\u884c\u4e00\u4e2a\u6574\u6570 \\(T\\) \u8868\u793a\u6570\u636e\u7ec4\u6570\uff0c\\(T \\leq 50\\) \u3002<\/p>\n<p>\u63a5\u4e0b\u6765 \\(T\\) \u884c\uff0c\u6bcf\u884c\u4e94\u4e2a\u6574\u6570 \\(p,a,b,k,t\\) \u8868\u793a\u4e00\u7ec4\u6570\u636e\uff0c\u5176\u4e2d \\(0 \\leq a,b,k,t &lt; p\\) \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u51fa\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u5bf9\u4e8e\u6bcf\u7ec4\u6570\u636e\uff0c\u5c06\u968f\u673a\u6570\u751f\u6210\u5668\u7684\u6700\u5c0f\u751f\u6210\u6b21\u6570\u8f93\u51fa\u5230\u6587\u4ef6\u4e2d\u3002<\/p>\n<p>\u5982\u679c\u8be5\u968f\u673a\u6570\u751f\u6210\u5668\u65e0\u6cd5\u751f\u6210\u6574\u6570 \\(t\\)\uff0c\u8f93\u51fa \\(-1\\) \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 \">3\r\n7 1 1 3 3\r\n7 2 2 2 0\r\n7 2 2 2 1<\/pre>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8f93\u51fa<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"lang:default decode:true \">1\r\n3\r\n-1<\/pre>\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\\(p \\leq 100\\) \u3002<\/p>\n<p>\u5bf9\u4e8e\u53e6\u5916 \\(30\\%\\) \u7684\u6570\u636e\uff0c\\(a=1\\) \u3002<\/p>\n<p>\u5bf9\u4e8e\u53e6\u5916 \\(30\\%\\) \u7684\u6570\u636e\uff0c\\(b=0\\) \u3002<\/p>\n<p>\u5bf9\u4e8e \\(100\\%\\) \u7684\u6570\u636e\uff0c\\(p \\leq 10^9\\) \u4e14 \\(p\\) \u662f\u8d28\u6570\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><del>\u7a0d\u5fae\u63a8\u4e00\u4e0b\u5f0f\u5b50\u5c31\u4f1a\u53d1\u73b0\u8fd9\u662f\u6a21\u677f\u9898<\/del><\/p>\n<p>\u53ef\u4ee5\u628a\u6bcf\u4e00\u9879\u90fd\u5c55\u5f00\uff0c\u627e\u627e\u89c4\u5f8b\uff0c\u6216\u8005\u76f4\u63a5\u7b49\u6bd4\u6570\u5217\u6c42\u548c\uff1a<\/p>\n<p>\u7b2c \\(n\\) \u9879\u4e3a\uff1a\\(a^{n-1}k-(a^{n-2}+a^{n-3}+&#8230;+a+1)b \\bmod p\\)<\/p>\n<p>\u5373\uff1a\\(a^{n-1}k-\\frac{a^{n-1}-1}{a-1} b \\bmod p\\)<\/p>\n<p>\u4ee4\u5176\u540c\u4f59\u4e8e \\(t\\) \uff0c\u5219\u6709\uff1a<\/p>\n<p>\\(a^{n-1}k-\\frac{a^{n-1}-1}{a-1} b\\equiv t\\pmod p\\)<\/p>\n<p>\u5373\uff1a\\(a^{n-1}\\equiv \\frac{t-\\frac{b}{a-1}}{k-\\frac{b}{a-1}}\\pmod p\\)<\/p>\n<p>\u53d1\u73b0\u53f3\u8fb9\u662f\u5e38\u6570\uff0c\u90a3\u5c31\u662f\u4e00\u4e2a BSGS \u4e86\u3002<\/p>\n<p>BSGS \u4e00\u79cd\u6bd4\u8f83\u597d\u7684\u5199\u6cd5\uff1a<\/p>\n<p>\u6709 \\(a^n \\equiv t (\\bmod p)\\) \uff0c\u6c42 \\(n\\) \u6700\u5c0f\u6b63\u6574\u6570\u89e3\u3002<\/p>\n<p>\u53ef\u4ee5\u53d1\u73b0 \\(n &lt; p\\) \uff0c\u6240\u4ee5\u53ef\u4ee5\u8bbe \\(n=Ax-B\\)\uff0c\u5176\u4e2d \\(x=\\lfloor \\sqrt p \\rfloor, 1 \\leq A \\leq x, 0 \\leq B&lt;x\\) \u3002<\/p>\n<p>\u6240\u4ee5 \\(a^{Ax} \\equiv t+a^B (\\bmod p)\\) \uff0c\u7528\u4e00\u4e2a <code>hash_table<\/code> \u5b58\u503c\u5373\u53ef\u3002<\/p>\n<p>\u8fd8\u6709\u672c\u9898\u8981\u7279\u522b\u6ce8\u610f \\(a=0\\)\uff0c\\(a=1\\) \u7684\u70b9\uff0c\u7279\u5224\u4e0d\u6e05\u695a\u7684\u8bdd\u5c31\u3002\u3002\u3002\u3002\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#include &lt;ext\/pb_ds\/assoc_container.hpp&gt;\r\n#include &lt;ext\/pb_ds\/hash_policy.hpp&gt;\r\n\r\ninline int _gcd(register int a,register int b)\r\n{\r\n\tregister int r=a%b;\r\n\twhile(r)\r\n\t\ta=b,b=r,r=a%b;\r\n\treturn b;\r\n}\r\n\r\ninline int _pow(register int base,register int b,const int&amp;MOD)\r\n{\r\n\tbase%=MOD;\r\n\tif(!base)\r\n\t\treturn 0;\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\n__gnu_pbds::gp_hash_table&lt;int,int&gt; hash;\r\n\r\nint main()\r\n{\r\n\tint _T;scanf(\"%d\",&amp;_T);\r\n\twhile(_T--)\r\n\t{\r\n\t\tint P,A,B,K,T;scanf(\"%d%d%d%d%d\",&amp;P,&amp;A,&amp;B,&amp;K,&amp;T);\r\n\t\t\r\n\t\tif(T==K)\r\n\t\t{\r\n\t\t\tprintf(\"%d\\n\",1);\r\n\t\t\tcontinue;\r\n\t\t}\r\n\t\tif(A==0)\r\n\t\t{\r\n\t\t\tprintf(\"%d\\n\",(B==T ? 2 : -1));\r\n\t\t\tcontinue;\r\n\t\t}\r\n\t\tif(A==1)\r\n\t\t{\r\n\t\t\tif(B==0)\r\n\t\t\t\tprintf(\"%d\\n\",-1);\r\n\t\t\telse\r\n\t\t\t\tprintf(\"%lld\\n\",(long long)((T-K)%P+P)%P*_pow(B,P-2,P)%P + 1);\r\n\t\t\tcontinue;\r\n\t\t}\r\n\t\tif(_gcd(A,P) != 1)\r\n\t\t{\r\n\t\t\tprintf(\"%d\\n\",-1);\r\n\t\t\tcontinue;\r\n\t\t}\r\n\t\t\r\n\t\thash.clear();\r\n\t\tregister int sp=__builtin_sqrt(P);\r\n\t\tif(P%sp != 0)\r\n\t\t\tsp++;\r\n\t\tregister int tv=(long long)B*_pow(A-1,P-2,P)%P;\r\n\t\tregister int rov=(long long)(T+tv)%P*_pow(K+tv,P-2,P)%P;\r\n\t\t\r\n\t\tfor(register int j=0;j&lt;sp;j++)\r\n\t\t\thash[(long long)rov*_pow(A,j,P)%P]=j+1;\r\n\t\tregister int ans=0;\r\n\t\tfor(register int i=1;i&lt;=sp;i++)\r\n\t\t\tif((ans=hash[_pow(A,i*sp,P)]))\r\n\t\t\t{\r\n\t\t\t\tprintf(\"%d\\n\",i*sp-(ans-1)+1);\r\n\t\t\t\tbreak;\r\n\t\t\t}\r\n\t\tif(ans&lt;=0)\r\n\t\t\tprintf(\"%d\\n\",-1);\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 P3451 [SDOI2013]\u968f\u673a\u6570\u751f\u6210\u5668\u95ee\u9898 \u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a262 &hellip; <a href=\"https:\/\/0.mnihyc.com\/blog\/archives\/1223\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">YZOJ P3451 [SDOI2013]\u968f\u673a\u6570\u751f\u6210\u5668\u95ee\u9898<\/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":[41,106],"tags":[],"class_list":["post-1223","post","type-post","status-publish","format-standard","hentry","category-proa","category-bsgs"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>YZOJ P3451 [SDOI2013]\u968f\u673a\u6570\u751f\u6210\u5668\u95ee\u9898 - mnihyc&#039;s Blog<\/title>\n<meta name=\"description\" content=\"YZOJ P3451 \u968f\u673a\u6570\u751f\u6210\u5668\u95ee\u9898 \u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a262144KB \u96be\u5ea6\uff1a \u9898\u76ee\u63cf\u8ff0 \u6709\u4e00\u4e2a\u968f\u673a\u6570\u751f\u6210\u5668\uff0c\u53ef\u4ee5\u751f\u6210 (0) ~ (p-1) \u4e4b\u95f4\u7684\u4f2a\u968f\u673a\u6574\u6570\u3002\u5728\u751f\u6210\u4e4b\u524d\uff0c\u9700\u8981\u8bbe\u5b9a\u4e00\u4e2a\u968f\u673a\u6570\u79cd\u5b50 (k) \uff08(0 leq k &lt; 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