{"id":70,"date":"2025-11-29T15:01:52","date_gmt":"2025-11-29T07:01:52","guid":{"rendered":"https:\/\/tblog.zeabur.app\/?p=70"},"modified":"2025-11-29T15:07:50","modified_gmt":"2025-11-29T07:07:50","slug":"%e9%a2%98%e8%a7%a3-htoj-lgp2252-%e3%80%90%e6%a8%a1%e6%9d%bf%e3%80%91%e5%a8%81%e4%bd%90%e5%a4%ab%e5%8d%9a%e5%bc%88-shoi2002-%e5%8f%96%e7%9f%b3%e5%ad%90%e6%b8%b8%e6%88%8f","status":"publish","type":"post","link":"https:\/\/tblog.zeabur.app\/?p=70","title":{"rendered":"\u9898\u89e3 HTOJ LGP2252 \u3010\u6a21\u677f\u3011\u5a01\u4f50\u592b\u535a\u5f08 \/ [SHOI2002] \u53d6\u77f3\u5b50\u6e38\u620f"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\">\u9898\u610f<\/h3>\n\n\n\n<p>\u7ecf\u5178\u7684 <strong>\u5a01\u4f50\u592b\u535a\u5f08\uff08Wythoff Game\uff09<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u6709\u4e24\u5806\u77f3\u5b50\uff0c\u6570\u91cf\u5206\u522b\u4e3a $a$ \u548c $b$<\/li>\n\n\n\n<li>\u6bcf\u6b21\u53ef\u4ee5\u8fdb\u884c\u4e24\u79cd\u64cd\u4f5c\uff1a<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u4ece\u4efb\u610f\u4e00\u5806\u53d6\u4efb\u610f\u591a\u77f3\u5b50<\/li>\n\n\n\n<li>\u4ece\u4e24\u5806\u4e2d\u53d6\u76f8\u540c\u6570\u91cf\u7684\u77f3\u5b50<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u6700\u540e\u628a\u77f3\u5b50\u5168\u90e8\u53d6\u5b8c\u7684\u4eba\u83b7\u80dc<\/li>\n<\/ul>\n\n\n\n<p>\u8981\u6c42\uff1a\u5224\u65ad\u5148\u624b\u5728\u521d\u59cb\u72b6\u6001\u4e0b\u662f\u5426\u80fd\u83b7\u80dc\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u89e3\u6790<\/h3>\n\n\n\n<p>\u5a01\u4f50\u592b\u535a\u5f08\u4e2d\uff0c\u6709\u4e24\u79cd\u5173\u952e\u72b6\u6001\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>P-position\uff08\u5fc5\u8d25\u70b9\uff09<\/strong>\uff1a\u5982\u679c\u5148\u624b\u5904\u4e8e\u6b64\u72b6\u6001\uff0c\u5fc5\u8d25<\/li>\n\n\n\n<li><strong>N-position\uff08\u5fc5\u80dc\u70b9\uff09<\/strong>\uff1a\u5982\u679c\u5148\u624b\u5904\u4e8e\u6b64\u72b6\u6001\uff0c\u5fc5\u80dc<\/li>\n<\/ul>\n\n\n\n<p>\u8bbe $m=\\max(a,b)$\uff0c$n=\\min(a,b)$\uff0c\u5219 $(n,m)$ \u662f P-position \u5f53\u4e14\u4ec5\u5f53\uff1a<\/p>\n\n\n\n<p>$\\displaystyle n=\\lfloor(m-n)\\cdot\\phi\\rfloor$<\/p>\n\n\n\n<p>\u5176\u4e2d $\\phi$ \u662f\u9ec4\u91d1\u6bd4\u4f8b\uff1a<\/p>\n\n\n\n<p>$\\displaystyle\\phi=\\frac{1+\\sqrt{5}}{2}\\approx 1.6180339887$<\/p>\n\n\n\n<p>\u6240\u6709\u5176\u4ed6\u72b6\u6001\u90fd\u662f N-position\uff08\u5148\u624b\u5fc5\u80dc\uff09\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5224\u5b9a<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u4ee4 $m=\\max(a,b)$\uff0c$n=\\min(a,b)$\uff1b<\/li>\n\n\n\n<li>\u8ba1\u7b97\u5dee\u503c $k=m-n$\uff1b<\/li>\n\n\n\n<li>\u82e5 $n=\\lfloor k\\cdot\\phi\\rfloor$\uff0c\u5219\u8f93\u51fa <code>0<\/code>\uff08\u5148\u624b\u5fc5\u8d25\uff09\uff1b<\/li>\n\n\n\n<li>\u5426\u5219\u8f93\u51fa <code>1<\/code>\uff08\u5148\u624b\u5fc5\u80dc\uff09\u3002<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">\u8bc1\u660e<\/h3>\n\n\n\n<p>\u5df2\u77e5\u6e38\u620f\u72b6\u6001 $(a,b),a\\le b$ <\/p>\n\n\n\n<p>\u5b9a\u4e49\uff1a$P$ \u4e3a\u5fc5\u8d25\u72b6\u6001\uff0c$N$ \u4e3a\u5fc5\u80dc\u72b6\u6001<\/p>\n\n\n\n<p>\u8bbe $P$ \u96c6\u5408 $(x_n,y_n),x_n&lt;y_n$\uff0c\u5dee\u503c $d_n = y_n-x_n$<\/p>\n\n\n\n<p>$\\because$ $P$ \u4e0d\u80fd\u4e92\u76f8\u5e72\u6270<\/p>\n\n\n\n<p>$\\therefore\\forall i\\ne j,x_i \\ne x_j,y_i \\ne y_j,d_i \\ne d_j$<\/p>\n\n\n\n<p>$\\because$ \u82e5 $\\alpha&gt;1$ \u65e0\u7406\u6570\uff0c$\\beta = \\frac{\\alpha}{\\alpha-1}$(<strong>Beatty \u5b9a\u7406<\/strong>)<\/p>\n\n\n\n<p>$\\displaystyle\\therefore{\\lfloor n\\alpha \\rfloor}_{n\\ge1} \\cup {\\lfloor n\\beta \\rfloor}_{n\\ge1} = \\mathbb{Z}^+,{\\lfloor n\\alpha \\rfloor} \\cap {\\lfloor n\\beta \\rfloor} = \\emptyset$<\/p>\n\n\n\n<p>\u4ee4 $\\phi = \\frac{1+\\sqrt{5}}{2}$<\/p>\n\n\n\n<p>$\\because$ \u9ec4\u91d1\u6bd4\u4f8b\u6ee1\u8db3 Beatty \u5b9a\u7406<\/p>\n\n\n\n<p>$\\therefore x_n = \\lfloor n\\phi \\rfloor,y_n = x_n+n$\uff0c\u5dee\u503c $y_n-x_n = n$<\/p>\n\n\n\n<p>$\\because$ $x_n$ \u5e8f\u5217\u548c $n$ \u6784\u6210 Beatty \u5e8f\u5217<\/p>\n\n\n\n<p>$\\therefore$ \u4e0d\u53ef\u80fd\u901a\u8fc7\u4e00\u6b65\u5230\u8fbe\u53e6\u4e00\u4e2a $P$<\/p>\n\n\n\n<p>\u8bbe $N(a,b),a&lt;b,k=b-a$<\/p>\n\n\n\n<p>$\\because$ \u5b58\u5728\u552f\u4e00 $n$\uff0c$x_n \\le a &lt; x_{n+1}$<\/p>\n\n\n\n<p>$\\therefore$ \u5b58\u5728\u4e00\u6b65 $(a,b)\\to(x_n,y_n)$<\/p>\n\n\n\n<p>\u8bbe $m = \\max(a,b),n = \\min(a,b),k = m-n$<br>$\\therefore(a,b) \\in P \\iff n = \\lfloor k \\phi \\rfloor$<\/p>\n\n\n\n<p>$\\because$ $P$ \u6784\u9020 $(x_n,y_n)=(\\lfloor n\\phi \\rfloor,\\lfloor n\\phi \\rfloor+n)$<\/p>\n\n\n\n<p>$\\because$ \u5dee\u503c $k=y_n-x_n=n$<\/p>\n\n\n\n<p>$\\therefore$ $x_n = \\lfloor k \\phi \\rfloor$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6807\u7a0b<\/h3>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\n#define int long long\nusing namespace std;\nsigned main()\n{\n    int a,b;\n    cin&gt;&gt;a&gt;&gt;b;\n    if(a&gt;b)swap(a,b);\n    int k=b-a;\n    int t=k*16180339887498948482\/10000000000000000000;\/\/\u907f\u514d\u7cbe\u5ea6\u8bef\u5dee\uff0c\u4f7f\u7528\u6574\u578b\n    cout&lt;&lt;(t==a?0:1)&lt;&lt;endl;\n    return 0;\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">\u540e\u8bb0<\/h3>\n\n\n\n<p>\u590d\u6742\u5ea6\uff1a$\\mathcal{O}(1)$\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u9898\u610f \u7ecf\u5178\u7684 \u5a01\u4f50\u592b\u535a\u5f08\uff08Wythoff Game\uff09\uff1a \u8981\u6c42\uff1a\u5224\u65ad\u5148\u624b\u5728\u521d\u59cb\u72b6\u6001\u4e0b\u662f\u5426\u80fd\u83b7\u80dc\u3002 \u89e3\u6790 \u5a01\u4f50\u592b\u535a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[3],"class_list":["post-70","post","type-post","status-publish","format-standard","hentry","category-oi","tag-oi"],"featured_image_src":null,"author_info":{"display_name":"CaelumRadish_c","author_link":"https:\/\/tblog.zeabur.app\/?author=1"},"_links":{"self":[{"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=\/wp\/v2\/posts\/70","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=70"}],"version-history":[{"count":8,"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=\/wp\/v2\/posts\/70\/revisions"}],"predecessor-version":[{"id":79,"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=\/wp\/v2\/posts\/70\/revisions\/79"}],"wp:attachment":[{"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=70"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=70"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tblog.zeabur.app\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}