AC自动机模板及对多模式匹配的理解

在接触AC自动机之前, 只仅仅掌握单模式匹配的算法: 比如KMP, BMH等算法; 经过优化后, KMP和BMH都具有线性时间复杂度, 而实际情况下, 一般的匹配问题BMH具有亚线性的表现; 而昨天接触的AC自动机则是一种结合了字典树和KMP的一种算法, 使得在多模式匹配下, 时间复杂度达到O(Σmi + n), 其中n为原串长度, mi为第i个模式串的长度;

匹配过程中类似于KMP, 原串不走回头路, 利用之前已经匹配过的结果来构造特殊的字典树从而形成AC自动机;

创建自动机的过程中, 最为重要的是fail指针的构造; 我是从这篇文章中学会的: AC自动机算法详解; fail指针的作用类似于KMP中的next数组, 而AC自动机的实质是一个特殊的哈希树;

应用:

-- 在很多计算机算法竞赛中, 多模式匹配的AC自动机通常与树形动态规划相结合, 动态规划的过程就是在自动机上走过路径形成的字串与给定字串的比较关系(最少修改字串, 避免病毒串等);

-- 多模式精确匹配;

优化:

-- 我的这个模板中并没有考虑对自动机的优化, 比如ptr->fail->next[i]与ptr->next[i]若同时不存在, 则ptr->fail其实是可以直接指向ptr->fail->fail的(原因很简单, 因为ptr->next[i]发生失配时, ptr = ptr->fail, 此时肯定仍然失配, 需要继续ptr->fail), 当然优化的代价是增加对存储空间的占用, fail需要变为vector<trieTreeNode*>
fail, 每个字母都应对应一个fail指针)

模板:

输入参数: patterns是模式串集合, s为待匹配原串, answer是成功匹配的模式串集合, 返回值为成功匹配的模式串个数;

////////////////////////////////////////////////////////////////////////////////
/*
//readme//
interfaces:
multiPatternsMatchingByAcAutomation:
1 const vector<string> &patterns: several pattern strings;
2 const string &s: original strings;
3 vector<string> &answer: the patterns which are matched in the original strings;
4 return the number of patterns which are matched.
*/
//made by HaoyuHu
//Tsinghua University
////////////////////////////////////////////////////////////////////////////////
#include <iostream>
#include <vector>
#include <queue>
#include <string>
#include <unordered_set>
#define ALPH_NUM 26
using namespace std;

struct trieTreeNode {
vector<trieTreeNode*> next;
bool mark;
trieTreeNode *fail;
trieTreeNode(): next(26, nullptr), mark(false), fail(nullptr) {}
};

trieTreeNode *createAcAutomation(const vector<string> &patterns);
int findPatterns(vector<string> &answer, trieTreeNode *root, const string &s);
void makeFoundPatterns(vector<string> &answer, unordered_set<trieTreeNode*> &save, trieTreeNode *root, string pattern);
inline char turn_char(int index);
int multiPatternsMatchingByAcAutomation(const vector<string> &patterns, const string &s, vector<string> &answer);

trieTreeNode *createAcAutomation(const vector<string> &patterns) {
trieTreeNode *root = new trieTreeNode(), *ptr, *cur;
for (int i = 0; i != patterns.size(); ++i) {
cur = root;
for (int k = 0; k != patterns[i].size(); ++k) {
int index = patterns[i][k] - 'a';
if (!cur->next[index])
cur->next[index] = new trieTreeNode();
cur = cur->next[index];
}
cur->mark = true;
}
queue<trieTreeNode*> makeFail;
makeFail.push(root);
while (!makeFail.empty()) {
cur = makeFail.front(); makeFail.pop();
for (int i = 0; i != ALPH_NUM; ++i) {
if (cur->next[i]) {
for (ptr = cur->fail; ptr && !ptr->next[i]; ptr = ptr->fail);
cur->next[i]->fail = ptr ? ptr->next[i] : root;
makeFail.push(cur->next[i]);
}
}
}
return root;
}

int findPatterns(vector<string> &answer, trieTreeNode *root, const string &s) {
int count = 0;
string pattern;
unordered_set<trieTreeNode*> save;
trieTreeNode *cur = root;
for (int i = 0; i != s.size(); ) {
int index = s[i] - 'a';
if (!cur) {
cur = root; ++i;
}
else if (cur->next[index]) {
cur = cur->next[index];
if (cur->mark) {
++count;
save.insert(cur);
}
++i;
}
else {
cur = cur->fail;
if (cur && cur->mark) {
++count;
save.insert(cur);
}
}
}
makeFoundPatterns(answer, save, root, pattern);
return count;
}

void makeFoundPatterns(vector<string> &answer, unordered_set<trieTreeNode*> &save,
trieTreeNode *root, string pattern) {
unordered_set<trieTreeNode*>::iterator it = save.find(root);
if (it != save.end())
answer.push_back(pattern);
for (int i = 0; i != ALPH_NUM; ++i) {
if (root->next[i]) {
string t(pattern);
t.push_back(turn_char(i));
makeFoundPatterns(answer, save, root->next[i], t);
}
}
}

inline char turn_char(int index) {
return 'a' + index;
}

int multiPatternsMatchingByAcAutomation(const vector<string> &patterns, const string &s, vector<string> &answer) {
trieTreeNode *root = createAcAutomation(patterns);
return findPatterns(answer, root, s);
}

Enjoy it!

如果有错误请指出, 谢谢!