## Introduction

The best way to explain this topic is with a problem:

You are given a text T (1 <= |T| <= 1,000,000) and n (1 <= n <= 1000) patterns, each pattern Pi has at least 1 and at most 500 characters. Find how many times each pattern occurs in T.

Although we can find the number of occurrences of a pattern in linear time (for example, using the Z Algorithm) , there are many patterns and so this approach will not be fast enough to finish in a reasonably amount of time.

In short and as you probably have already guessed, there exist a more efficient solution to this problem than our first attempt, the string matching algorithm invented by Alfred V. Aho and Margaret J. Corasick. I will not try to explain the algorithm because the authors explain it very well in their paper, I'm only going to provide you with an implementation and some practice problems.

This is the paper Efficient String Matching: An Aid to Bibliographic Search

Usually these papers are very complex and hard to understand without enough background, but to my surprise, this one is well explained and easy to understand.

## Aho-Corasick implementation in C++

        #include <iostream>
#include <vector>
#include <queue>
#include <cstdio>
#include <cstdlib>
#include <cstring>
using namespace std;

// Maximum number of states, at least as large as the sum of all key word's length.
const int MAX = 505 * 505;

// Alphabet size, 26 for lower case English letters.
const int ALPHA_SIZE = 26 + 1;
const int FAIL = -1;

// Maximum length of the text.
const int MAX_LENGTH = 1000010;

int g[MAX][ALPHA_SIZE];
int f[MAX];
vector<int> output[MAX];
char text[MAX_LENGTH];
int new_state;

void enter(char *word, int index)
{
int length = strlen(word);
int j, state = 0;
for (j = 0; j < length; j++) {
int c = word[j] - 'a';
if (g[state][c] == FAIL)
break;
state = g[state][c];
}

for ( ; j < length; j++) {
int c = word[j] - 'a';
new_state++;
g[state][c] = new_state;
state = new_state;
}

output[state].push_back(index);
}

int main(int argc, char *argv[])
{
int  n;
char word[505];
int count[505];

scanf("%d", &n);
scanf("%s", text);

// Build goto function
new_state = 0;
memset(g, -1, sizeof g);
memset(f, -1, sizeof f);
for (int i = 0; i < MAX; i++)
output[i].clear();

// Read patterns and build the Trie
for (int i = 0; i < n; i++) {
scanf("%s", word);
count[i] = 0;
enter(word, i);
}

for (char c = 'a'; c <= 'z'; c++) {
int a = c - 'a';
if (g[0][a] == FAIL)
g[0][a] = 0;
}

// Build failure function
queue<int> Q;
for (char c = 'a'; c <= 'z'; c++) {
int a = c - 'a';
int s = g[0][a];
if (s != 0) {
Q.push(s);
f[s] = 0;
}
}

while (!Q.empty()) {
int r = Q.front();
Q.pop();
for (char c = 'a'; c <= 'z'; c++) {
int a = c - 'a';
int s = g[r][a];
if (s != FAIL) {
Q.push(s);
int state = f[r];
while (g[state][a] == FAIL)
state = f[state];
f[s] = g[state][a];
for (unsigned k = 0; k < output[f[s]].size(); k++)
output[s].push_back(output[f[s]][k]);
}
}
}

// Count occurrences
int state = 0;
int length = strlen(text);
for (int i = 0; i < length; i++) {
int a = text[i] - 'a';
while (g[state][a] == FAIL)
state = f[state];
state = g[state][a];
for (unsigned k = 0; k < output[state].size(); k++)
count[output[state][k]]++;
}

for (int i = 0; i < n; i++)
printf("%d\n", count[i]);

return 0;
}



## References

 [1] ALFRED V. AHO AND MARGARET J. CORASICK, Efficient String Matching: An Aid to Bibliographic Search, http://dl.acm.org/citation.cfm?id=360855 [2] PEKKA KILPEL AINEN, Lecture 4: Set Matching and. Aho-Corasick Algorithm, //www.cs.uku.fi/~kilpelai/BSA05/lectures/slides04.pdf