From b399b8a0984dfb1de8b6379b9f98257116fdd776 Mon Sep 17 00:00:00 2001 From: sshwy Date: Tue, 16 Jul 2019 15:59:12 +0800 Subject: [PATCH] =?utf8?q?=E6=8C=81=E7=BB=AD=E6=9B=B4=E6=96=B0=E5=86=85?= =?utf8?q?=E5=AE=B9?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- docs/string/ac-automaton.md | 391 ++++++++++++----------------------- docs/string/images/ac-automaton3.gif | Bin 0 -> 779683 bytes docs/string/images/ac-automaton3.png | Bin 0 -> 79961 bytes docs/string/images/ac-automaton4.png | Bin 0 -> 68106 bytes 4 files changed, 132 insertions(+), 259 deletions(-) create mode 100644 docs/string/images/ac-automaton3.gif create mode 100644 docs/string/images/ac-automaton3.png create mode 100644 docs/string/images/ac-automaton4.png diff --git a/docs/string/ac-automaton.md b/docs/string/ac-automaton.md index e18323a1..67c59730 100644 --- a/docs/string/ac-automaton.md +++ b/docs/string/ac-automaton.md @@ -52,9 +52,7 @@ AC 自动机在做匹配时,同一位上可匹配多个模式串。 ### 例子 -下面放一张 GIF 帮助大家理解: - -对字符串 `i` `he` `his` `she` `hers` 组成的字典树构建 fail 指针: +下面放一张 GIF 帮助大家理解。对字符串 `i` `he` `his` `she` `hers` 组成的字典树构建 fail 指针: 1. 黄色结点:当前的结点 $u$。 2. 绿色结点:表示已经 BFS 遍历完毕的结点, @@ -67,7 +65,9 @@ AC 自动机在做匹配时,同一位上可匹配多个模式串。 ![AC_automation_6_9.png](./images/ac-automaton1.png) -找到 6 的父结点 5,$fail[5]=10$。然而 10 结点没有字母`s`连出的边;继续跳到 10 的 fail 指针,$fail[10]=0$。发现 0 结点有字母`s`连出的边,指向 7 结点;所以 $fail[6]=7$。 +找到 6 的父结点 5,$fail[5]=10$。然而 10 结点没有字母`s`连出的边;继续跳到 10 的 fail 指针,$fail[10]=0$。发现 0 结点有字母`s`连出的边,指向 7 结点;所以 $fail[6]=7$。最后放一张建出来的图 + +![finish](./images/ac-automaton4.png) ## 字典树与字典图 @@ -125,7 +125,7 @@ void build(){ ## 多模式匹配 -接下来分析匹配函数 `query()`: +接下来分析匹配函数 `query()`: ```cpp int query(char *t){ @@ -139,168 +139,148 @@ int query(char *t){ return res; } ``` -声明 u 作为字典树上当前匹配到的结点,res 即返回的答案。循环遍历匹配串,u 在字典树上跟踪当前字符。利用 fail 指针找出所有匹配的模式串,累加到答案中。然后清 0。对 $e[j]$ 取反的操作用来判断 $e[j]$ 是否等于 -1。 - -Q- 读者可能纳闷了:你这里的 u 一直在往字典树后面走,没有跳 fail 指针啊!这和 KMP 的思想不一样啊,怎么匹配得出来啊 - -读者表示:我 TM 一点也不纳闷 emm +声明 $u$ 作为字典树上当前匹配到的结点,$res$ 即返回的答案。循环遍历匹配串,$u$ 在字典树上跟踪当前字符。利用 fail 指针找出所有匹配的模式串,累加到答案中。然后清 0。对 $e[j]$ 取反的操作用来判断 $e[j]$ 是否等于 -1。在上文中我们分析过,字典树的结构其实就是一个 $trans$ 函数,而构建好这个函数后,在匹配字符串的过程中,我们会舍弃部分前缀达到最低限度的匹配。fail指针则指向了更多的匹配状态。最后上一份图。对于刚才的自动机: - - -### Answer to Q - -还记得刚才的字典图吗?事实上你并不是一直在往后跳,而是在图上穿梭跳动。比如,刚才的字典图: - -![AC_automation_b_13.png][5] +![AC_automation_b_13.png](./images/ac-automaton3.png) 我们从根结点开始尝试匹配 `ushersheishis`,那么 p 的变化将是: -![AC_automation_gif_c.gif][6] - -红色结点表示 p 结点,粉色箭头表示 p 在字典图上的跳转,浅蓝色的边表示成功匹配的模式串,深蓝色的结点表示跳 fail 指针时的结点。 +![AC_automation_gif_c.gif](./images/ac-automaton3.gif) -其中的部分跳转,我们利用的就是新构建的字典图上的边,它也满足后缀相同(sher 和 her),所以自动跳转到下一个位置。 - -综上,$fail$ 指针的意义是,在匹配串**同一个位置**失配时的跳转指针,这样就利用 fail 指针在同一位置上进行多模式匹配,匹配完了,就在字典图上自动跳转到下一位置。 +1. 红色结点: p 结点 +2. 粉色箭头: p 在自动机上的跳转, +3. 蓝色的边:成功匹配的模式串 +4. 蓝色结点:示跳 fail 指针时的结点(状态)。 ## 总结 -到此,你已经理解了整个 AC 自动机的内容。我们一句话总结 AC 自动机的运行原理: - -**构建字典图实现自动跳转,构建失配指针实现多模式匹配。** +~~希望~~大家看懂了文章。其实总结一下,你只需要知道AC自动机的板子很好背就行啦。 -所以 AC 自动机到底是啥 +???+ note "模板1" -## 模板 + [LuoguP3808【模板】AC 自动机(简单版)](https://www.luogu.org/problemnew/show/P3808) -[LuoguP3808【模板】AC 自动机(简单版)](https://www.luogu.org/problemnew/show/P3808) + ```cpp + #include + using namespace std; + const int N=1e6+6; + int n; -```cpp -##include -using namespace std; -const int N=1e6+6; -int n; - -namespace AC{ - int tr[N][26],tot; - int e[N],fail[N]; - void insert(char *s){ - int u=0; - for(int i=1;s[i];i++){ - if(!tr[u][s[i]-'a'])tr[u][s[i]-'a']=++tot; - u=tr[u][s[i]-'a']; - } - e[u]++; - } - queue q; - void build(){ - for(int i=0;i<26;i++)if(tr[0][i])q.push(tr[0][i]); - while(q.size()){ - int u=q.front();q.pop(); - for(int i=0;i<26;i++){ - if(tr[u][i])fail[tr[u][i]]=tr[fail[u]][i],q.push(tr[u][i]); - else tr[u][i]=tr[fail[u]][i]; - } - } - } - int query(char *t){ - int u=0,res=0; - for(int i=1;t[i];i++){ - u=tr[u][t[i]-'a'];// 转移 - for(int j=u;j&&e[j]!=-1;j=fail[j]){ - res+=e[j],e[j]=-1; - } - } - return res; - } -} - -char s[N]; -int main(){ - scanf("%d",&n); - for(int i=1;i<=n;i++)scanf("%s",s+1),AC::insert(s); - scanf("%s",s+1); - AC::build(); - printf("%d",AC::query(s)); - return 0; -} -``` - -## 模板 2 - -[P3796 【模板】AC 自动机(加强版)](https://www.luogu.org/problemnew/show/P3796) - -```cpp -##include -using namespace std; -const int N=156,L=1e6+6; -namespace AC{ - const int SZ=N*80; - int tot,tr[SZ][26]; - int fail[SZ],idx[SZ],val[SZ]; - int cnt[N];// 记录第 i 个字符串的出现次数 - void init(){ - memset(fail,0,sizeof(fail)); - memset(tr,0,sizeof(tr)); - memset(val,0,sizeof(val)); - memset(cnt,0,sizeof(cnt)); - memset(idx,0,sizeof(idx)); - tot=0; - } - void insert(char *s,int id){//id 表示原始字符串的编号 - int u=0; - for(int i=1;s[i];i++){ - if(!tr[u][s[i]-'a'])tr[u][s[i]-'a']=++tot; - u=tr[u][s[i]-'a']; - } - idx[u]=id; - } - queue q; - void build(){ - for(int i=0;i<26;i++)if(tr[0][i])q.push(tr[0][i]); - while(q.size()){ - int u=q.front();q.pop(); - for(int i=0;i<26;i++){ - if(tr[u][i])fail[tr[u][i]]=tr[fail[u]][i],q.push(tr[u][i]); - else tr[u][i]=tr[fail[u]][i]; - } - } - } - int query(char *t){// 返回最大的出现次数 - int u=0,res=0; - for(int i=1;t[i];i++){ - u=tr[u][t[i]-'a']; - for(int j=u;j;j=fail[j])val[j]++; - } - for(int i=0;i<=tot;i++)if(idx[i])res=max(res,val[i]),cnt[idx[i]]=val[i]; - return res; - } -} -int n; -char s[N][100],t[L]; -int main(){ - while(~scanf("%d",&n)){if(n==0)break; - AC::init(); - for(int i=1;i<=n;i++)scanf("%s",s[i]+1),AC::insert(s[i],i); - AC::build(); - scanf("%s",t+1); - int x=AC::query(t); - printf("%d\n",x); - for(int i=1;i<=n;i++)if(AC::cnt[i]==x)printf("%s\n",s[i]+1); - } - return 0; -} -/* - * BUG##1 build 的时候忘了 push(tr[u][i]) - * BUG##2 误以为倒序遍历 AC 自动机就是 BFS 的倒序,实际上不是这样 - */ + namespace AC{ + int tr[N][26],tot; + int e[N],fail[N]; + void insert(char *s){ + int u=0; + for(int i=1;s[i];i++){ + if(!tr[u][s[i]-'a'])tr[u][s[i]-'a']=++tot; + u=tr[u][s[i]-'a']; + } + e[u]++; + } + queue q; + void build(){ + for(int i=0;i<26;i++)if(tr[0][i])q.push(tr[0][i]); + while(q.size()){ + int u=q.front();q.pop(); + for(int i=0;i<26;i++){ + if(tr[u][i])fail[tr[u][i]]=tr[fail[u]][i],q.push(tr[u][i]); + else tr[u][i]=tr[fail[u]][i]; + } + } + } + int query(char *t){ + int u=0,res=0; + for(int i=1;t[i];i++){ + u=tr[u][t[i]-'a'];// 转移 + for(int j=u;j&&e[j]!=-1;j=fail[j]){ + res+=e[j],e[j]=-1; + } + } + return res; + } + } -``` + char s[N]; + int main(){ + scanf("%d",&n); + for(int i=1;i<=n;i++)scanf("%s",s+1),AC::insert(s); + scanf("%s",s+1); + AC::build(); + printf("%d",AC::query(s)); + return 0; + } + ``` + +???+ note "模板2" + + [P3796 【模板】AC 自动机(加强版)](https://www.luogu.org/problemnew/show/P3796) + + ```cpp + #include + using namespace std; + const int N=156,L=1e6+6; + namespace AC{ + const int SZ=N*80; + int tot,tr[SZ][26]; + int fail[SZ],idx[SZ],val[SZ]; + int cnt[N];// 记录第 i 个字符串的出现次数 + void init(){ + memset(fail,0,sizeof(fail)); + memset(tr,0,sizeof(tr)); + memset(val,0,sizeof(val)); + memset(cnt,0,sizeof(cnt)); + memset(idx,0,sizeof(idx)); + tot=0; + } + void insert(char *s,int id){//id 表示原始字符串的编号 + int u=0; + for(int i=1;s[i];i++){ + if(!tr[u][s[i]-'a'])tr[u][s[i]-'a']=++tot; + u=tr[u][s[i]-'a']; + } + idx[u]=id; + } + queue q; + void build(){ + for(int i=0;i<26;i++)if(tr[0][i])q.push(tr[0][i]); + while(q.size()){ + int u=q.front();q.pop(); + for(int i=0;i<26;i++){ + if(tr[u][i])fail[tr[u][i]]=tr[fail[u]][i],q.push(tr[u][i]); + else tr[u][i]=tr[fail[u]][i]; + } + } + } + int query(char *t){// 返回最大的出现次数 + int u=0,res=0; + for(int i=1;t[i];i++){ + u=tr[u][t[i]-'a']; + for(int j=u;j;j=fail[j])val[j]++; + } + for(int i=0;i<=tot;i++)if(idx[i])res=max(res,val[i]),cnt[idx[i]]=val[i]; + return res; + } + } + int n; + char s[N][100],t[L]; + int main(){ + while(~scanf("%d",&n)){if(n==0)break; + AC::init(); + for(int i=1;i<=n;i++)scanf("%s",s[i]+1),AC::insert(s[i],i); + AC::build(); + scanf("%s",t+1); + int x=AC::query(t); + printf("%d\n",x); + for(int i=1;i<=n;i++)if(AC::cnt[i]==x)printf("%s\n",s[i]+1); + } + return 0; + } + ``` ## KMP 自动机 -为了介绍 AC 自动机这种神奇的算法,先介绍自动机和 KMP 自动机。 +最后介绍自动机和 KMP 自动机,供学有余力的朋友观赏。 有限状态自动机 (DFA):字符集,有限状态控制,初始状态,接受状态。 @@ -322,114 +302,7 @@ $$ 我们发现 $trans_{i}$ 只依赖于之前的值,所以可以跟[KMP](/string/prefix-function/##knuth-morris-pratt)一起求出来。 -时间和空间复杂度: $O(m|\Sigma|)$ - -一些细节:走到接受状态之后立即转移到该状态的 $next$ 。 - -## AC 自动机 - -AC 自动机其实就是 Trie 上的自动机。 - -注意在[BFS](/search/bfs)的同时求出 $trans$ 数组即可。 - -AC 自动机一般用来解决多串匹配问题。 - -注意细节:AC 自动机的时间复杂度在需要找到所有匹配位置时是 $O(|s|+m)$ ,其中 $|s|$ 表示文本串的长度, $m$ 表示模板串的总匹配次数;而只需要求是否匹配时时间复杂度为 $O(|s|)$ 。 - -## AC 自动机的实现 - -```cpp -// luogu P3808 -//注:这并不是标准的AC自动机,而是trie图。标准的AC自动机实际应用并不多 -##include - -using namespace std; - -class ACAM { - private: - struct Node { - int ptr[26]; - int fail; - - int cnt; - - Node() : fail(0), cnt(0) { memset(ptr, 0, sizeof(ptr)); } - } nd[1000010]; - int cnt; - - queue q; - - public: - ACAM() : cnt(0) {} - - void insert(const string &s) { - int len = s.size(), now = 0; - for (int i = 0; i < len; i++) { - int x = s[i] - 'a'; - if (!nd[now].ptr[x]) { - nd[now].ptr[x] = ++cnt; - } - now = nd[now].ptr[x]; - } - nd[now].cnt++; - } - - void build() { - for (int i = 0; i < 26; i++) { - if (nd[0].ptr[i]) { - nd[nd[0].ptr[i]].fail = 0; - q.push(nd[0].ptr[i]); - } - } - while (!q.empty()) { - int now = q.front(); - q.pop(); - for (int i = 0; i < 26; i++) { - if (nd[now].ptr[i]) { - nd[nd[now].ptr[i]].fail = nd[nd[now].fail].ptr[i]; - q.push(nd[now].ptr[i]); - } else { - nd[now].ptr[i] = nd[nd[now].fail].ptr[i]; - } - } - } - } - - int query(const string &s) { - int now = 0, ans = 0; - int len = s.size(); - for (int i = 0; i < len; i++) { - int x = s[i] - 'a'; - now = nd[now].ptr[x]; - for (int p = now; p && ~nd[p].cnt; p = nd[p].fail) { - ans += nd[p].cnt; - nd[p].cnt = -1; - } - } - return ans; - } -} A; - -int main() { - int n; - cin >> n; - for (int i = 1; i <= n; i++) { - string temp; - cin >> temp; - A.insert(temp); - } - A.build(); - string s; - cin >> s; - cout << A.query(s) << '\n'; -} -``` - -[1]: https://hexo-source-1257756441.cos.ap-chengdu.myqcloud.com/2018/08/2858847684.gif +时间和空间复杂度: $O(m|\Sigma|)$。 -[2]: https://hexo-source-1257756441.cos.ap-chengdu.myqcloud.com/2018/08/3946915055.png -[3]: https://hexo-source-1257756441.cos.ap-chengdu.myqcloud.com/2018/08/1745118561.gif -[4]: https://hexo-source-1257756441.cos.ap-chengdu.myqcloud.com/2018/08/1426947356.png -[5]: https://hexo-source-1257756441.cos.ap-chengdu.myqcloud.com/2018/08/1085042377.png -[6]: https://hexo-source-1257756441.cos.ap-chengdu.myqcloud.com/2018/08/24151497.gif +一些细节:走到接受状态之后立即转移到该状态的 $next$。 diff --git a/docs/string/images/ac-automaton3.gif b/docs/string/images/ac-automaton3.gif new file mode 100644 index 0000000000000000000000000000000000000000..a3c2c3cd576991ded0c65874b945e4b9dd539437 GIT binary patch literal 779683 zcmXt8b97zJ*N&YuY0%iVZQHiZ#_z@>4!F|9`NCfUxHOB)|2i`actb|3B~3|3orB#5Eyh zvNc4zHAJ#Cf4en*@@KRcjl4+_qS2hZ#euxZTf9k8`I7^R#!ZTz&7X#|XTPK8v~znh zKQNg;c^IvI0>N}|&9*p1FNE{w600NUghNC`!$vY>BA18^!Lrp+n`vuESNm&^ zpS)W8_~Zw0cy@nwZGU*}YIf~vbMX?m2m%5>K6}mXU#;z5?c5#C?mw>WKLYpf*RCGd zu0DXEzd+C%@Z;lZfB))J_OCwGCmyf%AFr-H9`-*z4j&({K0f)`hJ=`^6gx93G~{Q9 z@lzoFTP6?)V2F1_LfOWjgW(96^oD@O;^Am?3i*85rjpTkTn_s!KvU`X4^oLZLb>L$ zi8NY`8pDa^@~JFVi^+VsmWr7?UiXWwiI&Q_BGGUpBKg*;g;LoxdZWqK>ZMB6QuzY; 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