{"id":563,"date":"2026-04-07T16:54:10","date_gmt":"2026-04-07T08:54:10","guid":{"rendered":"https:\/\/www.xgnd.net\/?p=563"},"modified":"2026-04-07T17:33:42","modified_gmt":"2026-04-07T09:33:42","slug":"%e3%80%8c%e5%ad%a6%e4%b9%a0%e7%ac%94%e8%ae%b0%e3%80%8dlyndon-%e7%90%86%e8%ae%ba%ef%bc%88%e4%b8%80%ef%bc%89","status":"publish","type":"post","link":"https:\/\/www.xgnd.net\/index.php\/2026\/04\/07\/%e3%80%8c%e5%ad%a6%e4%b9%a0%e7%ac%94%e8%ae%b0%e3%80%8dlyndon-%e7%90%86%e8%ae%ba%ef%bc%88%e4%b8%80%ef%bc%89\/","title":{"rendered":"\u300c\u5b66\u4e60\u7b14\u8bb0\u300dLyndon \u7406\u8bba\uff08\u4e00\uff09"},"content":{"rendered":"<p>\u5b57\u7b26\u4e32\uff0c\u592a\u6df1\u523b\u4e86\u3002<\/p>\n<p>Lyndon \u7406\u8bba\uff0c\u592a\u795e\u5947\u4e86\u3002<\/p>\n<hr \/>\n<p>\u672c\u6587\u8be6\u7ec6\u4ecb\u7ecd\u4e86 Lyndon \u7406\u8bba\u7684\u76f8\u5173\u5185\u5bb9\uff0c\u5e76\u4e14\u51e0\u4e4e\u6240\u6709\u5b9a\u7406\u548c\u63a8\u8bba\u90fd\u7ed9\u51fa\u4e86\u76f8\u5173\u8bc1\u660e\uff0c\u540c\u65f6\u90e8\u5206\u7b97\u6cd5\u7ed9\u4e86\u51e0\u9053\u914d\u5957\u4f8b\u9898\u3002<\/p>\n<p>\u5e0c\u671b\u8bfb\u8005\u9605\u8bfb\u540e\u80fd\u5bf9 Lyndon \u7406\u8bba\u6709\u6df1\u523b\u7684\u7406\u89e3\u3002<\/p>\n<p>\u5168\u7bc7\u592a\u957f\u4e86\uff0c\u6545\u5206\u5f00\u6765\u53d1\u3002<\/p>\n<h2>0. \u521d\u6b65\u7684\u5b9a\u4e49 &amp; \u7ea6\u5b9a<\/h2>\n<ul>\n<li>\n<p>\u957f\u5ea6\uff1a\u8bb0\u4e3a $|s|$\u3002<\/p>\n<\/li>\n<li>\n<p>\u5b57\u7b26\uff1a\u7528 $s[i]$ \u8868\u793a\u4e32 $s$ \u7684\u7b2c $i$ \u4e2a\u5b57\u7b26\u3002$s$ \u4e2d\u7684\u5b57\u7b26\u6807\u53f7\u4e3a $1,2,\\ldots,n$\u3002\u5355\u4e2a\u5b57\u7b26\u4e5f\u88ab\u89c6\u4f5c\u5b57\u7b26\u4e32\u3002<\/p>\n<\/li>\n<li>\n<p>\u5b57\u7b26\u96c6\uff1a\u8bb0\u4e3a $\\Sigma_s$\u3002<\/p>\n<\/li>\n<li>\n<p>\u5b57\u7b26\u4e32\u7684\u6bd4\u8f83\uff1a\u82e5 $s$ \u7684\u5b57\u5178\u5e8f\u6bd4 $t$ \u5c0f\uff0c\u8bb0\u4e3a $s&lt;t$\u3002\u7c7b\u4f3c\u5730\u6709 $\\leq,&gt;,\\geq$\u3002<\/p>\n<\/li>\n<li>\n<p>\u5b50\u4e32\uff1a$s[l\\dots r]$ \u8868\u793a\u5c06 $s[l],s[l+1],\\ldots,s[r]$ \u987a\u6b21\u8fde\u63a5\u800c\u6210\u7684\u5b57\u7b26\u4e32\u3002<\/p>\n<\/li>\n<li>\n<p>\u524d\u7f00 &amp; \u540e\u7f00\uff1a\u524d\u7f00\u4e3a\u5b57\u7b26\u4e32\u5f00\u5934\u7684\u5b50\u4e32\uff0c\u5373 $s[1\\dots i]$\uff1b\u540e\u7f00\u4e3a\u5b57\u7b26\u4e32\u7ed3\u5c3e\u7684\u5b50\u4e32\uff0c\u5373 $s[i\\dots n]$<\/p>\n<p>\u8bb0 $\\mathrm{pre}(s)$ \u4e3a $s$ \u7684\u524d\u7f00\u96c6\u5408\uff0c$\\mathrm{suf}(s)$ \u4e3a $s$ \u7684\u540e\u7f00\u96c6\u5408\u3002<\/p>\n<\/li>\n<li>\n<p>\u771f\u524d\u7f00 &amp; \u771f\u540e\u7f00\uff1a\u79f0\u4e00\u4e2a\u524d\u7f00\/\u540e\u7f00\u4e3a\u771f\u524d\u7f00\/\u771f\u540e\u7f00\uff0c\u5f53\u4e14\u4ec5\u5f53\u5b83\u4e0e\u539f\u4e32 $s$ \u4e0d\u76f8\u7b49\u3002<\/p>\n<\/li>\n<li>\n<p>\u5b57\u7b26\u4e32\u7684\u62fc\u63a5\uff1a\u8bb0 $st$\uff0c$s \\cdot t$ \u4e3a\u5b57\u7b26\u4e32 $s$\uff0c$t$ \u987a\u6b21\u8fde\u63a5\u800c\u5f97\u7684\u4e32\u3002<\/p>\n<\/li>\n<li>\n<p>\u5b57\u7b26\u4e32\u7684\u5e42\uff1a\u5b9a\u4e49 $s^k = \\overbrace{ss\\ldots s}^{\\text{\u5171 } k \\text{ \u4e2a}}$\uff0c\u5373 $k$ \u4e2a $s$ \u987a\u6b21\u8fde\u63a5\u800c\u5f97\u7684\u4e32\u3002<\/p>\n<\/li>\n<\/ul>\n<h2>1. Lyndon \u4e32\uff08Lyndon Word\uff09<\/h2>\n<h3>1.1 \u5b9a\u4e49<\/h3>\n<ul>\n<li><strong>Lyndon \u4e32\uff08Lyndon Word\uff09<\/strong>\uff1a\u5bf9\u4e8e\u5b57\u7b26\u4e32 $s$\uff0c\u5982\u679c $s$ \u7684\u5b57\u5178\u5e8f\u4e25\u683c\u5c0f\u4e8e $s$ \u7684\u6240\u6709\u540e\u7f00\u7684\u5b57\u5178\u5e8f\uff0c\u6211\u4eec\u79f0 $s$ \u662f Lyndon \u4e32\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u5b57\u7b26\u4e32\uff0c\u82e5\u5176\u672c\u8eab\u5c31\u662f\u5176\u6700\u5c0f\u540e\u7f00\uff0c\u5219\u79f0\u5b83\u4e3a Lyndon \u4e32\u3002<\/li>\n<li><strong>Lyndon \u524d\u7f00\uff08Lyndon Prefix\uff09<\/strong>\uff1a\u5982\u679c\u5b57\u7b26\u4e32 $s$ \u7684\u4e00\u4e2a\u524d\u7f00 $s[1 \\dots i]$ \u662f\u4e00\u4e2a Lyndon \u4e32\uff0c\u90a3\u4e48\u5b83\u5c31\u662f\u4e00\u4e2a Lyndon \u524d\u7f00\u3002<\/li>\n<li><strong>Lyndon \u540e\u7f00\uff08Lyndon Suffix\uff09<\/strong>\uff1a\u5982\u679c\u5b57\u7b26\u4e32 $s$ \u7684\u4e00\u4e2a\u540e\u7f00 $s[i \\dots n]$ \u662f\u4e00\u4e2a Lyndon \u4e32\uff0c\u90a3\u4e48\u5b83\u5c31\u662f\u4e00\u4e2a Lyndon \u540e\u7f00\u3002<\/li>\n<\/ul>\n<h3>1.2 \u6027\u8d28<\/h3>\n<ul>\n<li><strong>\u5f15\u7406 1.2.1\uff1a\u82e5\u5b57\u7b26\u4e32 $a$ \u548c $b$ \u4e92\u4e0d\u4e3a\u5bf9\u65b9\u7684\u524d\u7f00\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f $c_1, c_2$\uff0c$ac_1 &lt; bc_2 \\Leftrightarrow a &lt; b$\u3002<\/strong><\/li>\n<\/ul>\n<p>\u5145\u5206\u6027\uff1a\u56e0\u4e3a $a$ \u4e0d\u662f $b$ \u7684\u771f\u524d\u7f00\uff0c\u90a3\u4e48\u6bd4\u8f83\u5b8c\u524d $min{|a|,|b|}$ \u4f4d\u5c31\u53ef\u4ee5\u786e\u5b9a $a&lt;b$\u3002<\/p>\n<p>\u5fc5\u8981\u6027\uff1a\u663e\u7136\u3002<\/p>\n<ul>\n<li><strong>\u5f15\u7406 1.2.2\uff1a\u82e5\u5b57\u7b26\u4e32 $a$ \u548c $b$ \u957f\u5ea6\u76f8\u540c\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f $c_1, c_2$\uff0c$ac_1 &lt; bc_2 \\implies a \\le b$\u3002<\/strong><\/li>\n<\/ul>\n<p>\u5982\u679c $a&gt;b$\uff0c\u663e\u7136 $ac_1 &gt; bc_2$\uff0c\u77db\u76fe\u3002<\/p>\n<p>\u6545 $a \\le b$\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 1.2.1\uff1aLyndon \u4e32\u7684\u5b57\u5178\u5e8f\u4e25\u683c\u5c0f\u4e8e\u5176\u6240\u6709\u7684\u975e\u7a7a\u771f\u540e\u7f00\u3002<\/strong><\/li>\n<\/ul>\n<p>\u6839\u636e\u5b9a\u4e49\u53ef\u5f97\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 1.2.2\uff1a\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2a\u957f\u5ea6\u5927\u4e8e 1 \u7684 Lyndon \u4e32 $s$\uff0c\u5176\u4e0d\u5b58\u5728 Border\uff08\u5373\u4e0d\u5b58\u5728\u65e2\u662f\u771f\u524d\u7f00\u53c8\u662f\u771f\u540e\u7f00\u7684\u5b50\u4e32\uff09\u3002<\/strong><\/li>\n<\/ul>\n<p>\u5982\u679c $s$ \u6709\u4e00\u4e2a Border $t$\uff0c\u90a3\u4e48 $t$ \u662f $s$ \u7684\u771f\u540e\u7f00\uff0c\u6839\u636e\u5b9a\u4e49 $s &lt; t$\u3002\u4f46 $t$ \u4e5f\u662f $s$ \u7684\u524d\u7f00\uff0c\u524d\u7f00\u7684\u5b57\u5178\u5e8f\u4e0d\u53ef\u80fd\u5927\u4e8e\u539f\u4e32\uff0c\u4ea7\u751f\u77db\u76fe\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 1.2.3\uff1a\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2a Lyndon \u4e32 $s=ab$\uff0c\u6709 $a&lt;b$\uff08$a,b$ \u5747\u975e\u7a7a\uff09\u3002<\/strong><\/li>\n<\/ul>\n<p>\u7531\u5b9a\u4e49\u6709 $ab&lt;b$\uff0c\u4e14\u663e\u7136\u6709 $a&lt;ab$\uff0c\u6240\u4ee5 $a&lt;b$\u3002<\/p>\n<ul>\n<li><strong>\u63a8\u8bba 1.2.3.1\uff1a\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2a Lyndon \u4e32 $s$\uff0c\u5176\u771f\u524d\u7f00\u548c\u771f\u540e\u7f00\u5206\u522b\u4e3a $a$ \u548c $b$\uff0c\u6709 $a&lt;b$\uff08$a,b$ \u5747\u975e\u7a7a\uff09\u3002<\/strong><\/li>\n<\/ul>\n<p>\u7531\u5b9a\u4e49\u6709 $s&lt;b$\uff0c\u4e14\u663e\u7136\u6709 $a&lt;s$\uff0c\u6240\u4ee5 $a&lt;b$\u3002<\/p>\n<ul>\n<li><strong>\u63a8\u8bba 1.2.3.2\uff1a\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2a Lyndon \u4e32 $s$ \u7684\u524d\u7f00 $t$\uff08\u53ef\u4ee5\u6709 $t$ \u548c $s$ \u76f8\u7b49\uff09\uff0c\u5176\u771f\u540e\u7f00\u7684\u5b57\u5178\u5e8f\u59cb\u7ec8<\/strong>\u5927\u4e8e\u6216\u7b49\u4e8e<strong>\u4e0e\u5176\u957f\u5ea6\u76f8\u540c\u7684\u771f\u524d\u7f00\u3002\u5f62\u5f0f\u5316\u5730\u8bb2\uff0c$\\forall i\\in[2, |t|]$\uff0c\u90fd\u6ee1\u8db3\uff1a<\/strong><br \/>\n$$t[i \\dots |t|] \\ge t[1 \\dots |t|-i+1]$$<\/li>\n<\/ul>\n<p>\u4ee4 $s=tv$\u3002<\/p>\n<p>\u6839\u636e\u5b9a\u4e49\uff0c\u6709\uff1a$$t[i \\dots |t|]v &gt; tv$$<\/p>\n<p>\u6211\u4eec\u8003\u8651\u524d $|t|-i+1$ \u4e2a\u5b57\u7b26\uff0c\u5206\u522b\u662f $t[i \\dots |t|]$ \u548c $t[1 \\dots |t|-i+1]$\u3002<\/p>\n<p>\u6839\u636e\u5f15\u7406 1.2.2\uff0c\u6709\uff1a$$t[i \\dots |t|]\\ge t[1 \\dots |t|-i+1]$$<\/p>\n<p>\u8bc1\u6bd5\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 1.2.4\uff1a\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2a\u4e32 $a$ \u548c\u4efb\u610f\u4e00\u4e2a Lyndon \u4e32 $b$ \uff0c\u6709 $a&lt;b \\Leftrightarrow ab&lt;b$<\/strong><\/li>\n<\/ul>\n<p>\u60c5\u51b5\u4e00\uff1a\u82e5 $a$ \u4e0d\u662f $b$ \u7684\u771f\u524d\u7f00<\/p>\n<p>\u5145\u5206\u6027\uff1a\u663e\u7136\u6210\u7acb\u3002<\/p>\n<p>\u5fc5\u8981\u6027\uff1a\u82e5 $|a|\\geq |b|$\uff0c\u663e\u7136\u6709 $a&lt;b$\uff1b\u82e5 $|a|&lt;|b|$\uff0c$b$ \u4e0d\u53ef\u80fd\u662f $a$ \u7684\u771f\u524d\u7f00\uff0c\u6839\u636e\u5f15\u7406 1.2.1\uff0c\u53ef\u5f97 $a&lt;b$\u3002<\/p>\n<p>\u60c5\u51b5\u4e8c\uff1a\u82e5 $a$ \u662f $b$ \u7684\u771f\u524d\u7f00<\/p>\n<p>\u5145\u5206\u6027\uff1a\u4ee4 $b=at$\uff0c\u5373\u8bc1 $b&lt;t$\uff0c\u5176\u4e2d $t$ \u4e3a $b$ \u7684\u540e\u7f00\u3002\u6839\u636e Lyndon \u4e32\u7684\u5b9a\u4e49\uff0c\u663e\u7136\u6709 $b&lt;t$\u3002<\/p>\n<p>\u5fc5\u8981\u6027\uff1a\u663e\u7136\u6709 $a&lt;b$ \uff0c\u5fc5\u8981\u6027\u6210\u7acb\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 1.2.5\uff1a$s$ \u662f Lyndon \u4e32\u5f53\u4e14\u4ec5\u5f53 $s=ab$\uff0c$a&lt;b$ \u4e14 $a,b$ \u5747\u4e3a Lyndon \u4e32\uff08$a,b$ \u5747\u975e\u7a7a\uff0c$|s|\\geq2$\uff09\u3002<\/strong><\/li>\n<\/ul>\n<p>\u5145\u5206\u6027\uff1a$a &lt; b$ \u4e14 $a, b$ \u5747\u4e3a Lyndon \u4e32 $\\Rightarrow$ $ab$ \u662f Lyndon \u4e32\u3002<\/p>\n<p>\u6211\u4eec\u8981\u8bc1\u660e $ab$ \u662f Lyndon \u4e32\uff0c\u5373\u8bc1\u660e\u5176\u4e25\u683c\u5c0f\u4e8e\u5b83\u7684\u6240\u6709\u975e\u7a7a\u771f\u540e\u7f00\u3002<\/p>\n<p>$ab$ \u7684\u771f\u540e\u7f00\u53ef\u4ee5\u5206\u4e3a $suf(a)b, b, suf(b)$ \u3002<\/p>\n<p>\u7531\u5b9a\u4e49\u6709 $a &lt; suf(a)$\uff0c\u663e\u7136\u6709\uff1a<\/p>\n<p>$$ab &lt; suf(a)b$$<\/p>\n<p>\u7531\u5b9a\u7406 1.2.4 \u6709 $ab &lt; b$\uff0c\u53c8 $b &lt; suf(b)$\uff0c\u6545\uff1a<\/p>\n<p>$$ab &lt; b &lt; suf(b)$$<\/p>\n<p>\u7efc\u4e0a\uff0c$ab$ \u4e25\u683c\u5c0f\u4e8e\u5176\u6240\u6709\u771f\u540e\u7f00\uff0c\u6545 $ab$ \u662f Lyndon \u4e32\uff0c\u8bc1\u6bd5\u3002<\/p>\n<p>\u5fc5\u8981\u6027\uff1a$s$ \u662f Lyndon \u4e32 $\\Rightarrow$ \u5b58\u5728 $i$ \u4f7f\u5f97 $s[1\\dots i-1]&lt;s[i\\dots n]$ \u4e14 $s[1\\dots i-1],s[i\\dots n]$ \u5747\u4e3a Lyndon \u4e32\u3002<\/p>\n<p>\u4ee4 $b$ \u4e3a $s$ \u7684\u6700\u5c0f\u771f\u540e\u7f00\uff0c$s=ab$\u3002\uff08\u8fd9\u91cc\u7684 $a,b$ \u5373\u662f\u547d\u9898\u4e2d\u7684 $s[1\\dots i-1],s[i\\dots n]$\uff09<\/p>\n<p>\u7531\u5b9a\u7406 1.2.3\uff0c\u6709 $a&lt;b$\u3002<\/p>\n<p>\u56e0\u4e3a\u4e0d\u5b58\u5728\u6bd4 $b$ \u66f4\u5c0f\u7684\u771f\u540e\u7f00\uff0c\u6545 $b$ \u4e3a Lyndon \u4e32\u3002<\/p>\n<p>\u8003\u8651\u540e\u7f00 $suf(a)b$\uff0c\u6709 $suf(a)b&gt;b$\uff0c\u53c8\u6839\u636e\u5b9a\u4e49\u6709 $b&gt;s=ab$\uff0c\u5219 $$suf(a)b&gt;ab$$<\/p>\n<p>\u82e5 $suf(a)$ \u4e0d\u662f $a$ \u7684\u771f\u524d\u7f00\uff0c\u56e0\u4e3a $a$ \u4e0d\u53ef\u80fd\u662f $suf(a)$ \u7684\u771f\u524d\u7f00\uff0c\u7531\u5f15\u7406 1.2.1\uff0c\u53ef\u5f97 $suf(a)&gt;a$\uff0c\u6545 $a$ \u4e3a Lyndon \u4e32\u3002<\/p>\n<p>\u82e5 $suf(a)$ \u662f $a$ \u7684\u771f\u524d\u7f00\uff0c\u5373 $pre(a)=suf(a)$\uff0c\u5e26\u5165\u4e0a\u5f0f\u5f97 $pre(a)b&gt;ab$\uff0c\u628a\u76f8\u540c\u7684\u524d\u7f00\u62ff\u6389\u540e\uff0c\u6709 $b&gt;suf'(a)b$\uff08\u6ce8\u610f\u8fd9\u91cc $suf'(b)$ \u4e0d\u4e00\u5b9a\u548c $suf(a)$ \u76f8\u7b49\uff09\uff0c\u4e0e $b$ \u662f $s$ \u7684\u6700\u5c0f\u771f\u540e\u7f00\u77db\u76fe\uff0c\u6545 $suf(a)$ \u4e0d\u80fd\u662f $a$ \u7684\u771f\u524d\u7f00\u3002<\/p>\n<p>\u7efc\u4e0a\uff0c$a$ \u4e3a Lyndon \u4e32\uff0c\u8bc1\u6bd5\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 1.2.6\uff1a\u82e5\u5b57\u7b26\u4e32 $v$ \u548c\u5b57\u7b26 $c$ \u6ee1\u8db3 $vc$ \u662f\u67d0\u4e2a Lyndon \u4e32 $s$ \u7684\u524d\u7f00\uff0c\u5219\u5bf9\u4e8e\u5b57\u7b26 $d &gt; c$ \u6709 $vd$ \u662f Lyndon \u4e32\u3002<\/strong><\/li>\n<\/ul>\n<p>\u8bbe $s=vct$\uff0c\u7531\u5b9a\u4e49\u5f97 $v[i\\dots |v|]ct&gt;vct$\u3002<\/p>\n<p>\u6211\u4eec\u8981\u8bc1\u660e $vd$ \u662f Lyndon \u4e32\uff0c\u5373\u8bc1\u660e\u5176\u4e25\u683c\u5c0f\u4e8e\u5b83\u7684\u6240\u6709\u975e\u7a7a\u771f\u540e\u7f00\u3002\u771f\u540e\u7f00\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\uff1a<\/p>\n<p>\u60c5\u51b5\u4e00\uff1a\u957f\u5ea6\u5927\u4e8e 1 \u7684\u771f\u540e\u7f00\uff0c\u5f62\u5982 $v[i\\dots |v|] d \\quad (i \\in [2, |v|])$\u3002<\/p>\n<p>\u6bd4\u8f83\u4e24\u8fb9\u524d $|v|-i+2$ \u4e2a\u5b57\u7b26\uff08\u5373\u53ea\u6bd4\u5230 $c$ \u7684\u4f4d\u7f6e\uff09\uff0c\u7531\u5f15\u7406 1.2.2\uff0c\u5fc5\u7136\u6709\uff1a<\/p>\n<p>$$v[i\\dots |v|] c \\ge v[1 \\dots |v|-i+2]$$<\/p>\n<p>\u53c8 $d &gt; c$\uff0c\u6709\uff1a<\/p>\n<p>$$v[i\\dots |v|] d &gt; v[1 \\dots |v|-i+2]$$<\/p>\n<p>\u56e0\u6b64\uff0c\u5728\u5b57\u5178\u5e8f\u6bd4\u8f83\u4e2d\uff0c$v[i\\dots |v|]  d$ \u5728\u524d\u7f00\u9636\u6bb5\u5c31\u5df2\u7ecf\u4e25\u683c\u5927\u4e8e $vd$ \u7684\u524d\u7f00\uff0c\u5219\u6709\uff1a<\/p>\n<p>$$v[i\\dots |v|] d&gt;vd$$<\/p>\n<p>\u60c5\u51b5\u4e8c\uff1a\u957f\u5ea6\u4e3a 1 \u7684\u771f\u540e\u7f00\uff0c\u5373 $d$\u3002<\/p>\n<p>\u56e0\u4e3a $s$ \u662f Lyndon \u4e32\uff0c\u663e\u7136\u5176\u9996\u5b57\u7b26 $v[1]$ \u5fc5\u7136\u662f $s$ \u4e2d\u5b57\u5178\u5e8f\u6700\u5c0f\u7684\u5b57\u7b26\uff08\u6216\u4e4b\u4e00\uff09\uff0c\u6545\u5fc5\u6709\uff1a<\/p>\n<p>$$c \\ge v[1]$$<\/p>\n<p>\u7ed3\u5408\u5df2\u77e5\u6761\u4ef6 $d &gt; c$\uff0c\u53ef\u5f97\uff1a<\/p>\n<p>$$d &gt; v[1]$$<\/p>\n<p>\u6545\uff1a<\/p>\n<p>$$d&gt;vd$$<\/p>\n<p>\u7efc\u4e0a\uff0c$vd$ \u4e25\u683c\u5c0f\u4e8e\u5176\u6240\u6709\u771f\u540e\u7f00\uff0c\u6545 $vd$ \u662f Lyndon \u4e32\u3002\u8bc1\u6bd5\u3002<\/p>\n<h2>2. Lyndon \u5206\u89e3\uff08CFL \u5206\u89e3\uff09<\/h2>\n<h3>2.1 \u5b9a\u4e49<\/h3>\n<ul>\n<li>Lyndon \u5206\u89e3\uff1a\u5b57\u7b26\u4e32 $s$ \u7684 Lyndon \u5206\u89e3\u8bb0\u4e3a $s=w_1w_2\\dots w_k$\uff0c\u5176\u4e2d\u6240\u6709 $w_i$ \u4e3a Lyndon \u4e32\uff0c\u5e76\u4e14\u4ed6\u4eec\u7684\u5b57\u5178\u5e8f\u6309\u7167\u975e\u4e25\u683c\u9012\u51cf\u6392\u5e8f\uff0c\u5373 $w_1\\geq w_2\\geq\\dots\\geq w_k$\u3002\u53ef\u4ee5\u53d1\u73b0\uff0c\u8fd9\u6837\u7684\u5206\u89e3\u5b58\u5728\u4e14\u552f\u4e00\u3002<\/li>\n<\/ul>\n<h3>2.2 \u6027\u8d28<\/h3>\n<p>\u5728 Lyndon \u5206\u89e3 $s = w_1 w_2 \\dots w_k$ \u4e2d<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 2.3.1\uff1a\u4efb\u4f55\u5b57\u7b26\u4e32\u7684 Lyndon \u5206\u89e3\u90fd\u662f\u552f\u4e00\u7684\u3002<\/strong><\/li>\n<\/ul>\n<p>\u5047\u8bbe\u5b58\u5728\u4e24\u79cd\u5206\u89e3\u65b9\u5f0f\uff0c\u8bc1\u660e\u77db\u76fe\u5373\u53ef\u3002<\/p>\n<p><del>\u8fd9\u91cc\u5c31\u5077\u61d2\u4e0d\u8bc1\u4e86\u3002<\/del><\/p>\n<ul>\n<li><strong>\u5b9a\u7406 2.3.2\uff1a\u5982\u679c\u622a\u53d6\u5206\u89e3\u4e2d\u7684\u8fde\u7eed\u4e00\u6bb5 $w<em>i w<\/em>{i+1} \\dots w_j$\uff08\u5176\u4e2d $1 \\le i \\le j \\le k$\uff09\uff0c\u8fd9\u6bb5\u5b50\u4e32\u7684 Lyndon \u5206\u89e3\u6070\u597d\u5c31\u662f $w<em>i, w<\/em>{i+1}, \\dots, w_j$ \u672c\u8eab\u3002<\/strong><\/li>\n<\/ul>\n<p>\u663e\u7136 $w<em>i, w<\/em>{i+1}, \\dots, w_j$ \u7b26\u5408 Lyndon \u5206\u89e3\u7684\u5b9a\u4e49\uff0c\u5e76\u4e14\u7531\u4e8e Lyndon \u5206\u89e3\u7684\u552f\u4e00\u6027\uff0c\u6545 $w<em>i w<\/em>{i+1} \\dots w_j$ \u7684 Lyndon \u5206\u89e3\u6070\u597d\u5c31\u662f $w<em>i, w<\/em>{i+1}, \\dots, w_j$ \u672c\u8eab<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 2.3.3\uff1aLyndon \u5206\u89e3\u4e2d\u7684\u6700\u540e\u4e00\u4e2a Lyndon \u56e0\u5b50 $w_k$ \u5fc5\u7136\u662f\u6574\u4e2a\u5b57\u7b26\u4e32 $s$ \u7684\u5b57\u5178\u5e8f\u4e25\u683c\u6700\u5c0f\u7684\u540e\u7f00\u3002<\/strong><\/li>\n<\/ul>\n<p>\u6211\u4eec\u8981\u8bc1\u660e\u5bf9\u4e8e $s$ \u7684\u4efb\u610f\u975e\u7a7a\u540e\u7f00 $t$\uff0c\u90fd\u6709 $w_k \\le t$\u3002<\/p>\n<p>$s$ \u7684\u540e\u7f00\u53ef\u4ee5\u5206\u4e3a\u4e24\u7c7b\uff1a<\/p>\n<p>\u60c5\u51b5\u4e00\uff1a $t$ \u662f $w_k$ \u7684\u4e00\u4e2a\u975e\u7a7a\u540e\u7f00\u3002<\/p>\n<p>\u6839\u636e\u5b9a\u4e49\uff0c$w_k \\le t$ \u663e\u7136\u6210\u7acb\u3002<\/p>\n<p>\u60c5\u51b5\u4e8c\uff1a$t$ \u8de8\u8d8a\u4e86\u591a\u4e2a\u56e0\u5b50<\/p>\n<p>\u8bbe\u540e\u7f00 $t = w<em>i' w<\/em>{i+1} \\dots w_k$\uff0c\u5176\u4e2d $w_i'$ \u662f $w_i$ \u7684\u67d0\u4e2a\u540e\u7f00\uff08\u53ef\u4ee5\u662f $w_i$ \u672c\u8eab\uff09\uff0c\u4e14 $i &lt; k$\u3002<\/p>\n<p>\u5047\u8bbe\u6b64\u65f6\u6070\u6709 $w_i'=w_i$\u3002<\/p>\n<p>\u5982\u679c $w_i &gt; w_k$\uff0c\u6545 $t=w<em>i w<\/em>{i+1} \\dots w_k &gt; w_k$\u3002<\/p>\n<p>\u5982\u679c $w_i = w_k$\uff0c\u7531\u4e8e $w_1\\geq w_2\\geq\\dots\\geq w_k$\uff0c\u663e\u7136\u4e2d\u95f4\u6240\u6709\u7684\u56e0\u5b50\u4e5f\u5fc5\u987b\u76f8\u7b49\uff0c\u5373 $w<em>m = w<\/em>{m+1} = \\dots = w_k$\u3002\u6b64\u65f6 $t = w_k^r$\uff08\u5176\u4e2d $r$ \u662f\u56e0\u5b50\u7684\u4e2a\u6570\uff09\uff0c\u663e\u7136 $t=w_k^r &gt; w_k$\u3002<\/p>\n<p>\u5373\u4f7f $w_i'$ \u662f $w_i$ \u7684\u771f\u540e\u7f00\uff0c\u6839\u636e Lyndon \u4e32\u5b9a\u4e49 $w_i' &gt; w_i$\uff0c\u540c\u6837\u53ef\u4ee5\u63a8\u5bfc\u51fa $t=w<em>i' w<\/em>{i+1} \\dots w_k &gt; w_k$\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 2.3.4\uff1aLyndon \u5206\u89e3\u4e2d\u7684\u7b2c\u4e00\u4e2a Lyndon \u56e0\u5b50 $w_1$ \u662f $s$ \u7684\u6700\u957f Lyndon \u524d\u7f00\uff1b\u6700\u540e\u4e00\u4e2a\u56e0\u5b50 $w_k$ \u662f $s$ \u7684\u6700\u957f Lyndon \u540e\u7f00\u3002<\/strong><\/li>\n<\/ul>\n<p>\u6211\u4eec\u8981\u8bc1\u660e\uff1a\u5bf9\u4e8e\u4efb\u4f55\u957f\u5ea6\u5927\u4e8e $|w_1|$ \u7684\u524d\u7f00 $p = s[1 \\dots j]$\uff08\u5373 $j &gt; |w_1|$\uff09\uff0c$p$ \u4e00\u5b9a\u4e0d\u662f Lyndon \u4e32\u3002<\/p>\n<p>\u8bbe $p$ \u8de8\u8d8a\u4e86\u524d $r$ \u4e2a Lyndon \u56e0\u5b50\uff08$r \\ge 2$\uff09\uff0c\u5373 $p = w_1 w<em>2 \\dots w<\/em>{r-1} w_r'$\uff0c\u5176\u4e2d $w_r'$ \u662f $w_r$ \u7684\u4e00\u4e2a\u975e\u7a7a\u524d\u7f00\uff08\u53ef\u4ee5\u662f $w_r$ \u672c\u8eab\uff09\u3002<\/p>\n<p>\u6839\u636e Lyndon \u5206\u89e3\u7684\u6027\u8d28\uff0c$w_1 \\ge w_2$\uff0c\u6211\u4eec\u53ef\u4ee5\u5206\u7c7b\u8ba8\u8bba\u3002<\/p>\n<p>\u60c5\u51b5\u4e00\uff1a$w_1 &gt; w_2$<\/p>\n<p>\u5f80\u540e\u9762\u52a0\u4e00\u4e9b\u4e1c\u897f\u663e\u7136\u4e0d\u6539\u53d8\u4e0d\u7b49\u53f7\u65b9\u5411\uff0c\u4e8e\u662f\u6709 $w_1 w_2 \\dots &gt; w_2 \\dots$\u3002<\/p>\n<p>\u7531\u4e8e $p$ \u62e5\u6709\u4e00\u4e2a\u771f\u540e\u7f00 $suf(p) = w_2 \\dots w_r'$\uff0c\u663e\u7136 $p &gt; suf(p)$\u3002<\/p>\n<p>\u6839\u636e Lyndon \u4e32\u7684\u5b9a\u4e49\uff0c$p$ \u4e0d\u662f Lyndon \u4e32\u3002<\/p>\n<p>\u60c5\u51b5\u4e8c\uff1a$w_1 = w_2$<\/p>\n<p>\u6709 $p = w_1 w<em>2 \\dots w<\/em>{r-1} w_r' = w_1 w<em>1 \\dots w<\/em>{r-1} w_r'$\u3002<\/p>\n<p>\u90a3\u4e48 $p$ \u7684\u4e00\u4e2a\u771f\u540e\u7f00\u5c31\u662f $suf(p)=w_2 \\dots w_r'=w_1 \\dots w_r'$\u3002<\/p>\n<p>\u4e0d\u96be\u53d1\u73b0\u524d\u7f00\u90fd\u6709 $w_1$\uff0c\u6b64\u65f6\u53ea\u9700\u6bd4\u8f83 $w_1w<em>3 \\dots w<\/em>{r-1} w_r'$ \u548c $w_3 \\dots w_r'$\u3002<\/p>\n<p>\u82e5 $w_1 &gt; w_3$\uff0c\u663e\u7136\u6709 $p &gt; suf(p)$\uff0c$p$ \u4e0d\u662f Lyndon \u4e32\u3002<\/p>\n<p>\u82e5 $w_1 = w_3$\uff0c\u91cd\u590d\u4e0a\u8ff0\u6b65\u9aa4\uff0c\u540e\u9762\u4ee5\u6b64\u7c7b\u63a8\u3002<\/p>\n<p>\u5982\u679c\u4e00\u76f4\u76f8\u7b49\uff0c\u6700\u540e\u4f1a\u5269\u4e0b $w_1 w_r'$ \u548c $w_r'$\uff0c\u7531\u4e8e $w_1\\ge w_r \\ge w_r'$\uff0c\u90a3\u4e48 $w_1 w_r'&gt;w_r'$\uff0c\u4e8e\u662f\u6709 $p &gt; suf(p)$\uff0c$p$ \u4e0d\u662f Lyndon \u4e32\u3002<\/p>\n<p>\u7efc\u4e0a\uff0c\u59cb\u7ec8\u6709 $p &gt; suf(p)$\uff0c\u6545 $p$ \u4e0d\u662f Lyndon \u4e32\uff0c\u8bc1\u6bd5\u3002<\/p>\n<p>\u540e\u7f00\u7684\u8bc1\u660e\u548c\u524d\u7f00\u7c7b\u4f3c\uff0c\u8282\u7701\u7a7a\u95f4\uff0c\u8fd9\u91cc\u5c31\u4e0d\u8bc1\u4e86\u3002<\/p>\n<ul>\n<li><strong>\u5b9a\u7406 2.3.5\uff1a\u540e\u7f00 $s[i \\dots n]$ \u7684\u7b2c\u4e00\u4e2a Lyndon \u56e0\u5b50\u7684\u672b\u5c3e\uff0c\u6070\u597d\u5728\u201c\u7b2c\u4e00\u4e2a\u5b57\u5178\u5e8f\u4e25\u683c\u5c0f\u4e8e $s[i \\dots n]$ \u7684\u540e\u7f00\u201d\u51fa\u73b0\u7684\u4f4d\u7f6e\u4e4b\u524d\u3002<\/strong><\/li>\n<\/ul>\n<p>\u4ee4 $t=s[i \\dots n]$\u3002<\/p>\n<p>\u8bbe $t$ \u7684\u552f\u4e00 Lyndon \u5206\u89e3\u4e3a $t = w_1 w_2 \\dots w_k$\uff0c\u4e14 $w_1 \\ge w_2 \\ge \\dots \\ge w_k$\u3002<\/p>\n<p>\u6211\u4eec\u9996\u5148\u8bc1\u660e\uff1a\u5bf9\u4e8e\u6240\u6709\u8d77\u59cb\u4e8e $w_1$ \u5185\u90e8\u7684\u4f4d\u7f6e $j$\uff08\u5373 $1 &lt; j \\le |w_1|$\uff09\uff0c\u4ee5\u5b83\u5f00\u5934\u7684\u540e\u7f00 $t[j\\dots n]$ \u4e00\u5b9a\u4e25\u683c\u5927\u4e8e $t$\uff0c\u5373 $t[j\\dots n]&gt;t$\u3002<\/p>\n<p>\u663e\u7136 $t[j\\dots n] = w_1[j \\dots |w_1|] w_2 \\dots w_k$\u3002<\/p>\n<p>\u6211\u4eec\u8981\u5c06\u5b83\u4e0e\u539f\u4e32 $t = w_1 w_2 \\dots w_k$ \u8fdb\u884c\u6bd4\u8f83\u3002<\/p>\n<p>\u6839\u636e Lyndon \u4e32\u7684\u5b9a\u4e49\uff0c$w_1$ \u4e25\u683c\u5c0f\u4e8e\u5b83\u7684\u6240\u6709\u771f\u540e\u7f00\u3002\u56e0\u6b64\uff1a<\/p>\n<p>$$w_1 &lt; w_1[j \\dots |w_1|]$$<\/p>\n<p>\u6839\u636e\u5b9a\u7406 1.2.2\uff0c$w_1[j \\dots |w_1|]$ \u4e0d\u662f $w_1$ \u7684\u524d\u7f00\uff0c\u53c8\u56e0\u4e3a $w_1$ \u663e\u7136\u4e0d\u53ef\u80fd\u662f $w_1[j \\dots |w_1|]$ \u7684\u524d\u7f00\uff0c\u6839\u636e\u5f15\u7406 1.2.1\uff0c\u6709\uff1a<\/p>\n<p>$$w_1 w_2 \\dots w_k &lt; w_1[j \\dots |w_1|] w_2 \\dots w_k$$<\/p>\n<p>\u5373 $t&lt;t[j\\dots n]$\u3002<\/p>\n<p>\u63a5\u4e0b\u6765\u6211\u4eec\u9700\u8981\u8bc1\u660e\uff1a\u4ee5 $|w_1| + 1$ \u5f00\u5934\u7684\u540e\u7f00\uff0c\u4e00\u5b9a\u4e25\u683c\u5c0f\u4e8e $t$\uff0c\u5373 $t[|w_1| + 1\\dots n]&lt;t$\u3002<\/p>\n<p>\u663e\u7136 $t[|w_1| + 1\\dots n] = w_2 w_3 \\dots w_k$\u3002<\/p>\n<p>\u6211\u4eec\u53ea\u9700\u8bc1\u660e $w_2 w_3 \\dots w_k&lt;w_1 w_2 \\dots w_k$\u3002<\/p>\n<p>\u8003\u8651\u5230 $w_1 \\ge w_2 \\ge \\dots \\ge w_k$\uff0c\u5047\u8bbe\u524d $r$ \u4e2a Lyndon \u56e0\u5b50\u5b8c\u5168\u76f8\u7b49\uff0c\u5373 $w_1 = w_2 = \\dots = w<em>r &gt; w<\/em>{r+1}$\uff08\u82e5 $r = k$\uff0c\u5219 $w_{r+1}$ \u89c6\u4e3a\u7a7a\u4e32\uff09\u3002<\/p>\n<p>\u6b64\u65f6\u6211\u4eec\u53ea\u9700\u8bc1 $w<em>1^{r-1} w<\/em>{r+1} \\dots w_k&lt;w<em>1^r w<\/em>{r+1} \\dots w_k$\u3002<\/p>\n<p>\u5b83\u4eec\u62e5\u6709\u516c\u5171\u524d\u7f00 $w_1^{r-1}$\u3002\u5c06\u5176\u6d88\u53bb\u540e\uff0c\u95ee\u9898\u8f6c\u5316\u4e3a\u6bd4\u8f83\u4ee5\u4e0b\u4e24\u4e2a\u5b57\u7b26\u4e32\uff1a<\/p>\n<p>$$A = w<em>{r+1} w<\/em>{r+2} \\dots w_k,B = w<em>1 w<\/em>{r+1} \\dots w_k$$<\/p>\n<p>\u5982\u679c $A$ \u662f\u7a7a\u4e32\uff0c\u90a3\u4e48 $A &lt; B$ \u663e\u7136\u6210\u7acb\u3002<\/p>\n<p>\u5982\u679c $A$ \u4e0d\u662f\u7a7a\u4e32\uff0c\u6211\u4eec\u5df2\u77e5 $w<em>1 &gt; w<\/em>{r+1}$\uff0c\u6b64\u65f6\u9700\u8981\u5206\u4e24\u79cd\u60c5\u51b5\uff1a<\/p>\n<p>\u60c5\u51b5\u4e00\uff1a$w_{r+1}$ \u4e0d\u662f $w_1$ \u7684\u771f\u524d\u7f00<\/p>\n<p>\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7531\u4e8e $w<em>1 &gt; w<\/em>{r+1}$ \u4e14\u4e0d\u5305\u542b\u524d\u7f00\u5173\u7cfb\uff0c\u7531\u5f15\u7406 1.2.1\uff0c\u663e\u7136\u6709 $A &lt; B$\u3002<\/p>\n<p>\u60c5\u51b5\u4e8c\uff1a$w_{r+1}$ \u662f $w_1$ \u7684\u771f\u524d\u7f00<\/p>\n<p>\u8bbe $w<em>1 = w<\/em>{r+1} u$\uff08$u$ \u4e0d\u80fd\u4e3a\u7a7a\uff09\u3002<\/p>\n<p>\u6b64\u65f6<\/p>\n<p>$$A = w<em>{r+1} w<\/em>{r+2} \\dots w<em>k,B = w<\/em>{r+1} u w_{r+1} \\dots w_k$$<\/p>\n<p>\u56e0\u4e3a $w_1$ \u662f Lyndon \u4e32\uff0c\u6839\u636e\u5b9a\u4e49\u6709 $u &gt; w_1$\u3002<\/p>\n<p>\u4e8e\u662f\u6709 $u &gt; w<em>1 \\ge w<\/em>{r+1} \\ge w<em>{r+2}$\uff0c\u5373 $u &gt; w<\/em>{r+2}$\u3002<\/p>\n<p>\u73b0\u5728\uff0c\u6211\u4eec\u5728 $A$ \u548c $B$ \u4e2d\u540c\u65f6\u6d88\u53bb\u516c\u5171\u524d\u7f00 $w_{r+1}$\uff0c\u5269\u4e0b\uff1a<\/p>\n<p>$$A' = w_{r+2} \\dots w<em>k,B' = u w<\/em>{r+1} \\dots w_k$$<\/p>\n<p>\u6b64\u65f6 $A'$ \u4ee5 $w<em>{r+2}$ \u5f00\u5934\uff0c$B'$ \u4ee5 $u$ \u5f00\u5934\uff0c\u4e14\u5df2\u77e5 $u &gt; w<\/em>{r+2}$\u3002\u4f60\u4f1a\u53d1\u73b0\uff0c\u8fd9\u4e0e\u6211\u4eec\u521a\u624d\u9762\u4e34\u7684\u5c40\u9762\u5b8c\u5168\u4e00\u81f4\u3002<\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u4e0d\u65ad\u91cd\u590d\u4e0a\u8ff0\u8fc7\u7a0b\u3002\u7531\u4e8e\u6bcf\u6b21 $B$ \u5269\u4f59\u90e8\u5206\u5f00\u5934\u7684\u5b57\u7b26\u4e32\uff0c\u5fc5\u7136\u662f\u4e0a\u4e00\u8f6e $u$ \u7684\u771f\u540e\u7f00\uff0c\u56e0\u6b64\u5b83\u4e5f\u5c31\u662f $w_1$ \u7684\u67d0\u4e2a\u771f\u540e\u7f00\u3002\u6839\u636e Lyndon \u6027\u8d28\uff0c\u5b83\u6c38\u8fdc\u4e25\u683c\u5927\u4e8e $w_1$\uff0c\u81ea\u7136\u4e5f\u4e25\u683c\u5927\u4e8e\u4e0b\u4e00\u4e2a\u5c06\u8981\u6bd4\u8f83\u7684 Lyndon \u56e0\u5b50 $w_j$\u3002<\/p>\n<p>\u6700\u7ec8\u8981\u4e48\u9047\u5230\u4e86\u60c5\u51b5\u4e00\u7ed3\u675f\uff0c\u8981\u4e48\u91cd\u590d\u60c5\u51b5\u4e8c\u76f4\u5230 $A$ \u7684\u56e0\u5b50\u88ab\u6d88\u8017\u6b86\u5c3d\u53d8\u6210\u7a7a\u4e32\uff0c\u800c\u6b64\u65f6 $B$ \u8fd8\u6709\u5269\u4f59\uff0c\u4e8e\u662f\u4e00\u5b9a\u6709 $A &lt; B$\u3002<\/p>\n<p>\u8bc1\u6bd5\u3002<\/p>\n<ul>\n<li><strong>\u63a8\u8bba 2.3.5.1\uff1a\u5bf9\u4e8e\u4efb\u610f\u5b57\u7b26\u4e32\u7684 Lyndon \u5206\u89e3\uff0c\u603b\u6709 $w_1\\dots w_k&gt;w_2 \\dots w<em>k&gt;\\dots&gt;w<\/em>{k-1} w_k&gt; w_k$\u3002<\/strong><\/li>\n<\/ul>\n<h3>2.3 \u7b97\u6cd5<\/h3>\n<h4>2.3.1 Duval \u7b97\u6cd5<\/h4>\n<p>Duval \u53ef\u4ee5\u5728 $\\mathcal{O}(n)$ \u7684\u65f6\u95f4\u5185\u6c42\u51fa\u4e00\u4e2a\u4e32\u7684 Lyndon \u5206\u89e3\u3002<\/p>\n<p>\u8be5\u7b97\u6cd5\u9700\u8981\u7ef4\u62a4\u4e09\u4e2a\u53d8\u91cf $i,j,k$\uff0c\u8fd9\u4e09\u4e2a\u6307\u9488\u5c06\u6574\u4e2a\u5b57\u7b26\u4e32\u5206\u6210\u4e86\u56db\u4e2a\u90e8\u5206 $s[1,i-1],s[i,k-1],s[j,k-1],s[k,n]$ :<\/p>\n<ul>\n<li>$s[1,i-1]$\uff1a\u8fd9\u90e8\u5206\u7684 Lyndon \u5206\u89e3\u5df2\u7ecf\u5b8c\u6210\u3002<\/li>\n<li>$s[i,k-1]$\uff1a\u8fd9\u90e8\u5206\u53ef\u4ee5\u88ab\u8868\u793a\u6210 $t^h+v$\uff0c\u5176\u4e2d $t$ \u662f\u4e00\u4e2a Lyndon \u4e32\uff0c$t^h$ \u8868\u793a $t$ \u5faa\u73af $h$ \u6b21\uff0c$v$\u662f $t$ \u7684\u4e00\u4e2a\u53ef\u7a7a\u771f\u524d\u7f00\u3002<\/li>\n<li>$s[j,k-1]$\uff1a\u6ce8\u610f\u8fd9\u90e8\u5206\u662f\u5305\u542b\u5728\u4e0a\u4e00\u4e2a\u90e8\u5206\u4e2d\u7684\uff0c\u5176\u4e2d$j=k\u2013|t|$\u3002<\/li>\n<li>$s[k,n]$\uff1a\u8fd9\u90e8\u5206\u8fd8\u672a\u5904\u7406\uff0c\u6b64\u65f6\u6b63\u5728\u8003\u8651 $k$\u3002<\/li>\n<\/ul>\n<p>\u8003\u8651k\u6709\u4e09\u79cd\u60c5\u51b5:<\/p>\n<ul>\n<li>$s_j=s_k$\uff1a\u53ef\u4ee5\u5c06 $t$ \u7ee7\u7eed\u5faa\u73af\u4e0b\u53bb\uff0c\u56e0\u6b64 $j$ \u5f80\u540e\u79fb\u4e00\u4f4d\uff0c\u8003\u8651\u4e0b\u4e00\u4e2a $k$\u3002<\/li>\n<li>$s_j&lt;s_k$\uff1a\u53ef\u4ee5\u5c06 $t^c+v+s_k$ \u5408\u5e76\u4e3a\u4e00\u4e2a Lyndon \u4e32\uff0c\u56e0\u6b64 $j$ \u8bbe\u4e3a $i$\uff0c\u8003\u8651\u4e0b\u4e00\u4e2a $k$\u3002<\/li>\n<li>$s_j&gt;s_k$\uff1a\u53ef\u4ee5\u5c06 $t$ \u5355\u72ec\u4f5c\u4e3a Lyndon \u5206\u89e3\u4e2d\u7684\u4e00\u4e2a\u4e32\uff0c\u7136\u540e\u4ece $v$ \u7684\u5f00\u5934\u5f00\u59cb\u91cd\u65b0\u8003\u8651\uff0c\u56e0\u6b64 $i$ \u8bbe\u4e3a $v$ \u7684\u5f00\u5934\uff0c$j,k$ \u5206\u522b\u8bbe\u4e3a $i,i+1$\u3002<\/li>\n<\/ul>\n<pre><code class=\"language-cpp\">string s;\ncin&amp;gt;&amp;gt;s;\nint n=s.length();\ns=&amp;#039; &amp;#039;+s;\nfor(int i=1;i&amp;lt;=n;){\n    int j=i,k=i+1;\n    while(k&amp;lt;=n and s[j]&amp;lt;=s[k]){\n        if(s[j]&amp;lt;s[k])j=i;\n        else j++;\n        k++;\n    }\n    while(i&amp;lt;=j){\n        cout&amp;lt;&amp;lt;i+k-j-1&amp;lt;&amp;lt;&amp;#039; &amp;#039;;  \/\/\u5206\u89e3\u540e\u6bcf\u4e2a Lyndon \u4e32\u7684\u53f3\u7aef\u70b9\n        i+=k-j;\n    }\n}<\/code><\/pre>\n<h4>2.3.2 SA + \u5355\u8c03\u6808\u7b97\u6cd5<\/h4>\n<p><strong>\u6ce8\u610f\uff1a\u8fd9\u4e2a\u7b97\u6cd5\u8fdc\u4e0d\u53ca Duval \u7b97\u6cd5\u9ad8\u6548\uff0c\u5e73\u65f6\u6700\u5e38\u7528\u8fd8\u662f Duval \u7b97\u6cd5\uff0c\u5199\u5728\u8fd9\u91cc\u4ec5\u4e3a\u4e86\u62d3\u5bbd\u601d\u8def\u3002<\/strong>\uff08\u4e8b\u5b9e\u4e0a\uff0c\u67d0\u4e9b\u9898\u751a\u81f3\u53ef\u4ee5\u7528 Lyndon \u5206\u89e3\u540a\u6253 SA\uff0c\u540c\u65f6\u4e5f\u5b58\u5728\u7528 Lyndon \u7406\u8bba\u6c42 SA \u7684\u7b97\u6cd5\uff09<\/p>\n<p>\u6211\u4eec\u6c42\u51fa\u540e\u7f00\u6570\u7ec4\uff08SA\uff09\u548c $rank$ \u6570\u7ec4\u540e\u3002<\/p>\n<p>\u6839\u636e\u5b9a\u7406 2.3.5\uff0c\u8fd9\u4e2a\u95ee\u9898\u8f6c\u5316\u4e3a\uff1a\u5728 $rank$ \u6570\u7ec4\u4e2d\uff0c\u5bfb\u627e $i$ \u540e\u9762\u7b2c\u4e00\u4e2a $j$ \u4f7f\u5f97 $rank[j] &lt; rank[i]$\u3002<\/p>\n<p>\u6211\u4eec\u7528\u5355\u8c03\u6808\u5bf9 $rank$ \u6570\u7ec4\u9884\u5904\u7406\u4e00\u4e0b\uff0c\u5c31\u80fd $O(1)$ \u627e\u5230 $j$ \u4e86\u3002<\/p>\n<p>\u590d\u6742\u5ea6\uff1a$O(n)$ \u6216 $O(n \\log n)$ \u53d6\u51b3\u4e8e SA \u7684\u5b9e\u73b0\u3002<\/p>\n<h3>2.4 \u5e94\u7528<\/h3>\n<ul>\n<li>\n<p>\u6c42\u6bcf\u4e2a\u524d\u7f00\u7684\u6700\u5c0f\u524d\u7f00\/\u540e\u7f00\u3002<\/p>\n<\/li>\n<li>\n<p>\u6c42\u5b57\u7b26\u4e32\u7684\u6700\u5c0f\u8868\u793a\u6cd5\u3002<\/p>\n<\/li>\n<li>\n<p>\u7b49\u7b49\u3002<\/p>\n<\/li>\n<\/ul>\n<h3>2.5 \u4f8b\u9898<\/h3>\n<h4><a href=\"https:\/\/acm.hdu.edu.cn\/showproblem.php?pid=6761\">HDU6761 Minimum Index<\/a><\/h4>\n<p>\u9898\u610f\uff1a\u6c42\u6bcf\u4e2a\u524d\u7f00\u7684\u6700\u5c0f\u540e\u7f00\u7684\u8d77\u59cb\u4f4d\u7f6e\u4e0b\u6807\u3002<\/p>\n<p>\u9898\u89e3\uff1a<\/p>\n<p>\u6839\u636e\u5b9a\u7406 2.3.1\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u524d\u7f00\u6211\u4eec\u53d6\u6700\u540e\u4e00\u4e2a Lyndon \u56e0\u5b50\u5c31\u662f\u7b54\u6848\u3002Duval \u8fc7\u7a0b\u4e2d\u8fb9\u627e\u8fb9\u8bb0\u5f55\u7b54\u6848\u5373\u53ef\u3002<\/p>\n<pre><code class=\"language-cpp\">#include &amp;lt;bits\/stdc++.h&amp;gt;\nusing namespace std;\ntypedef unsigned long long ll;\n#define int ll\nconst int maxn = 20000005, mod = 1000000007, INF = 1e9;\nint ans[maxn];\n\nvoid solve() {\n    string s;\n    cin&amp;gt;&amp;gt;s;\n    int n=s.length();\n    s=&amp;#039; &amp;#039;+s;\n    ans[1]=1;\n    for(int i=1;i&amp;lt;=n;){\n        int j=i,k=i+1;\n        while(k&amp;lt;=n and s[j]&amp;lt;=s[k]){\n            if(s[j]&amp;lt;s[k])ans[k]=j=i;\n            else ans[k]=ans[j]+k-j,j++;\n            k++;\n        }\n        while(i&amp;lt;=j){\n            i+=k-j;\n        }\n        ans[k]=i;\n    }\n    int sum=0;\n    for(int i=1,x=1;i&amp;lt;=n;i++,x=x*1112LL%mod){\n        sum+=ans[i]*x%mod;\n        sum%=mod;\n    }\n    cout&amp;lt;&amp;lt;sum&amp;lt;&amp;lt;&amp;#039;\\n&amp;#039;;\n}\n\nsigned main() {\n    ios::sync_with_stdio(0);\n    int t = 1;\n    cin &amp;gt;&amp;gt; t;\n    while (t--) {\n        solve();\n    }\n\n    return 0;\n}<\/code><\/pre>\n<p><a href=\"https:\/\/www.luogu.com.cn\/problem\/P13270\">P13270 \u3010\u6a21\u677f\u3011\u6700\u5c0f\u8868\u793a\u6cd5<\/a><\/p>\n<p>\u9898\u610f\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a $n$ \u7684\u5b57\u7b26\u4e32 $s$\uff0c\u6c42\u4e0e $s$ \u5faa\u73af\u540c\u6784\u7684\u5b57\u5178\u5e8f\u6700\u5c0f\u7684\u4e32\uff08\u5373\u5b57\u7b26\u4e32 $s$ \u7684\u6700\u5c0f\u8868\u793a\uff09\u3002<\/p>\n<p>\u9898\u89e3\uff1a<\/p>\n<p>\u8bbe $t = ss$\uff0c\u5219\u9700\u8981\u627e\u51fa $t$ \u7684\u957f\u5ea6\u4e3a $n$ \u7684\u6700\u5c0f\u5b50\u4e32\u3002<\/p>\n<p>\u5bf9 $t$ \u505a Lyndon \u5206\u89e3\uff0c\u7136\u540e\u5bfb\u627e\u8fd9\u4e2a\u5206\u89e3\u4e2d\u7684\u4e00\u4e2a Lyndon \u4e32 $w_i$\uff0c\u4f7f\u5f97\u5b83\u7684\u8d77\u70b9\u5c0f\u4e8e $n$ \u4e14\u7ec8\u70b9\u5927\u4e8e\u7b49\u4e8e $n$\u3002<\/p>\n<p>\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4f7f\u7528 Lyndon \u5206\u89e3\u7684\u6027\u8d28\u8bc1\u660e\uff0c\u5b50\u4e32 $w_i$ \u7684\u9996\u5b57\u7b26\u5c31\u662f $s$ \u7684\u6700\u5c0f\u8868\u793a\u6cd5\u7684\u9996\u5b57\u7b26\uff0c\u5373\u6211\u4eec\u6cbf\u7740 $w_i$ \u7684\u5f00\u5934\u5f80\u540e $n$ \u4e2a\u5b57\u7b26\u7ec4\u6210\u7684\u4e32\u5c31\u662f $s$ \u7684\u6700\u5c0f\u8868\u793a\u6cd5\u3002<\/p>\n<pre><code class=\"language-cpp\">#include &amp;lt;bits\/stdc++.h&amp;gt;\nusing namespace std;\ntypedef long long ll;\nconst int maxn = 5000050, mod = 1e9 + 7, INF = 1e9;\n\nint main() {\n    ios::sync_with_stdio(0);\n    int n;\n    cin&amp;gt;&amp;gt;n; \n    string s;\n    cin&amp;gt;&amp;gt;s;\n    s=&amp;#039; &amp;#039;+s+s;\n    int ans=0;\n    for(int i=1;i&amp;lt;=2*n;){\n        int j=i,k=i+1;\n        while(k&amp;lt;=2*n and s[j]&amp;lt;=s[k]){\n            if(s[j]&amp;lt;s[k])j=i;\n            else j++;\n            k++;\n        }\n        while(i&amp;lt;=j){\n            if(i+k-j-1&amp;lt;n) ans=max(ans,i+k-j-1)+1;\n            i+=k-j;\n        }\n    }\n    cout&amp;lt;&amp;lt;s.substr(ans,n)&amp;lt;&amp;lt;&amp;#039;\\n&amp;#039;;\n\n    return 0;\n}<\/code><\/pre>\n<p><a href=\"https:\/\/www.luogu.com.cn\/problem\/P14190\">P14190 [ICPC 2024 Hangzhou R] Dividing Sequence<\/a><\/p>\n<p>\u9898\u610f\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a $n$ \u7684\u5b57\u7b26\u4e32 $s$\uff0c\u5c06\u5176\u62c6\u6210\u4e24\u4e2a<strong>\u5b50\u5e8f\u5217<\/strong> $a$ \u548c $b$\uff0c\u4f7f\u5f97\u5b57\u5178\u5e8f $a&lt;b$ \u4e14\u6700\u5c0f\u5316 $b$ \u7684\u5b57\u5178\u5e8f\u3002<\/p>\n<p>\u9898\u89e3\uff1a<\/p>\n<p>\u8003\u8651\u5bf9\u539f\u4e32 $s$ \u505a Lyndon \u5206\u89e3 $s = w_1 w_2 \\dots w_k$\uff0c\u4e14 $w_1\\geq w_2\\geq\\dots\\geq w_k$\u3002<\/p>\n<p>$i$ \u4ece 1 \u5f00\u59cb\uff0c<\/p>\n<p>\u82e5 $w<em>{2i-1}=w<\/em>{2i}$\uff0c\u5219\u5c06\u8fd9\u4e24\u4e2a\u4e32\u5206\u522b\u63a5\u5728 $a,b$ \u7684\u672b\u5c3e\uff0c$i++$\u3002<\/p>\n<p>\u82e5 $w<em>{2i-1}\\neq w<\/em>{2i}$\uff0c\u5c06 $w_{2i-1}$ \u63a5\u5230 $b$ \u4e4b\u540e\uff0c\u5269\u4e0b\u7684\u6240\u6709\u4e32\u6309\u987a\u5e8f\u63a5\u5230 $a$ \u7684\u672b\u5c3e\uff0c\u6b64\u65f6\u7684$b$\u5c31\u662f\u7b54\u6848\u3002<\/p>\n<p>\u8be6\u7ec6\u8bc1\u660e\u53ef\u4ee5\u89c1\u9898\u89e3\u533a\uff0c\u8fd9\u91cc\u5c31\u4e0d\u8bc1\u4e86\u3002<\/p>\n<pre><code class=\"language-cpp\">#include &amp;lt;bits\/stdc++.h&amp;gt;\nusing namespace std;\ntypedef long long ll;\nconst int maxn = 5000050, mod = 1e9 + 7, INF = 1e9;\n\nvoid solve() {\n    int n;\n    cin&amp;gt;&amp;gt;n;\n    vector&amp;lt;int&amp;gt; s(n+1);\n    for(int i=1;i&amp;lt;=n;i++){\n        cin&amp;gt;&amp;gt;s[i];\n    }\n    vector&amp;lt;int&amp;gt; ans;\n    for(int i=1;i&amp;lt;=n;){\n        int j=i,k=i+1;\n        while(k&amp;lt;=n and s[j]&amp;lt;=s[k]){\n            if(s[j]&amp;lt;s[k])j=i;\n            else j++;\n            k++;\n        }\n        int f=0;\n        int len=k-j;\n        while(i&amp;lt;=j){\n            f^=1;\n            if(f){\n                for(int t=i;t&amp;lt;=i+len-1;t++){\n                    ans.push_back(s[t]);\n                }\n            }\n            i+=len;\n        }\n        if(f) break;\n    }\n    cout&amp;lt;&amp;lt;ans.size()&amp;lt;&amp;lt;&amp;#039;\\n&amp;#039;;\n\n    for(int x:ans){\n        cout&amp;lt;&amp;lt;x&amp;lt;&amp;lt;&amp;#039; &amp;#039;;\n    }\n    cout&amp;lt;&amp;lt;&amp;#039;\\n&amp;#039;;\n}\n\nint main() {\n    ios::sync_with_stdio(0);\n    int t = 1;\n    cin &amp;gt;&amp;gt; t;\n    while (t--) {\n        solve();\n    }\n\n    return 0;\n}<\/code><\/pre>\n<h2>\u53c2\u8003\u8d44\u6599 \uff1a<\/h2>\n<ul>\n<li><a href=\"https:\/\/www.luogu.com\/article\/d4y3zqqv\">Lyndon &amp; Runs - command_block<\/a><\/li>\n<li><a href=\"https:\/\/oi-wiki.org\/string\/lyndon\/\">Lyndon \u5206\u89e3 - OI Wiki<\/a><\/li>\n<li><a href=\"https:\/\/www.cnblogs.com\/5k-sync-closer\/p\/18516159\">Lyndon \u7406\u8bba\u5b66\u4e60\u7b14\u8bb0 - Sk_sync_closer<\/a><\/li>\n<li><a href=\"https:\/\/www.cnblogs.com\/Neal-lee\/p\/16353936.html\">The Runs Theorem \u548c Lyndon Tree \u5b66\u4e60\u7b14\u8bb0 - Neal_lee<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u5b57\u7b26\u4e32\uff0c\u592a\u6df1\u523b\u4e86\u3002 Lyndon \u7406\u8bba\uff0c\u592a\u795e\u5947\u4e86\u3002 \u672c\u6587\u8be6\u7ec6\u4ecb\u7ecd\u4e86 Lyndon \u7406\u8bba\u7684\u76f8\u5173\u5185\u5bb9\uff0c\u5e76\u4e14\u51e0\u4e4e\u6240\u6709 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,3,6],"tags":[],"class_list":["post-563","post","type-post","status-publish","format-standard","hentry","category-c","category-oi","category-6"],"_links":{"self":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts\/563","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/comments?post=563"}],"version-history":[{"count":8,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts\/563\/revisions"}],"predecessor-version":[{"id":572,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts\/563\/revisions\/572"}],"wp:attachment":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/media?parent=563"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/categories?post=563"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/tags?post=563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}