# Aho-Corasick Algorithm for Pattern Searching

Given an input text and an array of k words, arr[], find all occurrences of all words in the input text. Let n be the length of text and m be the total number characters in all words, i.e. m = length(arr[0]) + length(arr[1]) + .. + O(n + length(arr[k-1]). Here k is total numbers of input words.

Example:

```Input: text = "ahishers"
arr[] = {"he", "she", "hers", "his"}

Output:
Word his appears from 1 to 3
Word he appears from 4 to 5
Word she appears from 3 to 5
Word hers appears from 4 to 7
```

If we use a linear time searching algorithm like KMP, then we need to one by one search all words in text[]. This gives us total time complexity as O(n + length(word[0]) + O(n + length(word[1]) + O(n + length(word[2]) + … O(n + length(word[k-1]). This time complexity can be written as O(n*k + m).
Aho-Corasick Algorithm finds all words in O(n + m + z) time where z is total number of occurrences of words in text. The Aho–Corasick string matching algorithm formed the basis of the original Unix command fgrep.

1. Prepocessing : Build an automaton of all words in arr[] The automaton has mainly three functions:
```Go To :   This function simply follows edges
of Trie of all words in arr[]. It is
represented as 2D array g[][] where
we store next state for current state
and character.

Failure : This function stores all edges that are
followed when current character doesn't
have edge in Trie.  It is represented as
1D array f[] where we store next state for
current state.

Output :  Stores indexes of all words that end at
current state. It is represented as 1D
array o[] where we store indexes
of all matching words as a bitmap for
current state.
```
2. Matching : Traverse the given text over built automaton to find all matching words.

Preprocessing:

1. We first Build a Trie (or Keyword Tree) of all words.

This part fills entries in goto g[][] and output o[].
2. Next we extend Trie into an automaton to support linear time matching.

This part fills entries in failure f[] and output o[].

Go to :
We build Trie. And for all characters which don’t have an edge at root, we add an edge back to root.

Failure :
For a state s, we find the longest proper suffix which is a proper prefix of some pattern. This is done using Breadth First Traversal of Trie.

Output :
For a state s, indexes of all words ending at s are stored. These indexes are stored as bitwise map (by doing bitwise OR of values). This is also computing using Breadth First Traversal with Failure.

Below is C++ implementation of Aho-Corasick Algorithm

```// C++ program for implementation of Aho Corasick algorithm
// for string matching
using namespace std;
#include <bits/stdc++.h>

// Max number of states in the matching machine.
// Should be equal to the sum of the length of all keywords.
const int MAXS = 500;

// Maximum number of characters in input alphabet
const int MAXC = 26;

// OUTPUT FUNCTION IS IMPLEMENTED USING out[]
// Bit i in this mask is one if the word with index i
// appears when the machine enters this state.
int out[MAXS];

// FAILURE FUNCTION IS IMPLEMENTED USING f[]
int f[MAXS];

// GOTO FUNCTION (OR TRIE) IS IMPLEMENTED USING g[][]
int g[MAXS][MAXC];

// Builds the string matching machine.
// arr -   array of words. The index of each keyword is important:
//         "out[state] & (1 << i)" is > 0 if we just found word[i]
//         in the text.
// Returns the number of states that the built machine has.
// States are numbered 0 up to the return value - 1, inclusive.
int buildMatchingMachine(string arr[], int k)
{
// Initialize all values in output function as 0.
memset(out, 0, sizeof out);

// Initialize all values in goto function as -1.
memset(g, -1, sizeof g);

// Initially, we just have the 0 state
int states = 1;

// Construct values for goto function, i.e., fill g[][]
// This is same as building a Trie for arr[]
for (int i = 0; i < k; ++i)
{
const string &word = arr[i];
int currentState = 0;

// Insert all characters of current word in arr[]
for (int j = 0; j < word.size(); ++j)
{
int ch = word[j] - 'a';

// Allocate a new node (create a new state) if a
// node for ch doesn't exist.
if (g[currentState][ch] == -1)
g[currentState][ch] = states++;

currentState = g[currentState][ch];
}

// Add current word in output function
out[currentState] |= (1 << i);
}

// For all characters which don't have an edge from
// root (or state 0) in Trie, add a goto edge to state
// 0 itself
for (int ch = 0; ch < MAXC; ++ch)
if (g[0][ch] == -1)
g[0][ch] = 0;

// Now, let's build the failure function

// Initialize values in fail function
memset(f, -1, sizeof f);

// Failure function is computed in breadth first order
// using a queue
queue<int> q;

// Iterate over every possible input
for (int ch = 0; ch < MAXC; ++ch)
{
// All nodes of depth 1 have failure function value
// as 0. For example, in above diagram we move to 0
// from states 1 and 3.
if (g[0][ch] != 0)
{
f[g[0][ch]] = 0;
q.push(g[0][ch]);
}
}

// Now queue has states 1 and 3
while (q.size())
{
// Remove the front state from queue
int state = q.front();
q.pop();

// For the removed state, find failure function for
// all those characters for which goto function is
// not defined.
for (int ch = 0; ch <= MAXC; ++ch)
{
// If goto function is defined for character 'ch'
// and 'state'
if (g[state][ch] != -1)
{
// Find failure state of removed state
int failure = f[state];

// Find the deepest node labeled by proper
// suffix of string from root to current
// state.
while (g[failure][ch] == -1)
failure = f[failure];

failure = g[failure][ch];
f[g[state][ch]] = failure;

// Merge output values
out[g[state][ch]] |= out[failure];

// Insert the next level node (of Trie) in Queue
q.push(g[state][ch]);
}
}
}

return states;
}

// Returns the next state the machine will transition to using goto
// and failure functions.
// currentState - The current state of the machine. Must be between
//                0 and the number of states - 1, inclusive.
// nextInput - The next character that enters into the machine.
int findNextState(int currentState, char nextInput)
{
int ch = nextInput - 'a';

// If goto is not defined, use failure function

}

// This function finds all occurrences of all array words
// in text.
void searchWords(string arr[], int k, string text)
{
// Preprocess patterns.
// Build machine with goto, failure and output functions
buildMatchingMachine(arr, k);

// Initialize current state
int currentState = 0;

// Traverse the text through the nuilt machine to find
// all occurrences of words in arr[]
for (int i = 0; i < text.size(); ++i)
{
currentState = findNextState(currentState, text[i]);

if (out[currentState] == 0)
continue;

// Match found, print all matching words of arr[]
// using output function.
for (int j = 0; j < k; ++j)
{
if (out[currentState] & (1 << j))
{
cout << "Word " << arr[j] << " appears from "
<< i - arr[j].size() + 1 << " to " << i << endl;
}
}
}
}

// Driver program to test above
int main()
{
string arr[] = {"he", "she", "hers", "his"};
string text = "ahishers";
int k = sizeof(arr)/sizeof(arr[0]);

searchWords(arr, k, text);

return 0;
}
```

Output:

```Word his appears from 1 to 3
Word he appears from 4 to 5
Word she appears from 3 to 5
Word hers appears from 4 to 7```

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