# 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) + length(arr) + … + 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) + O(n + length(word) + O(n + length(word) + … 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.

```
1. 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. Trie

1. This part fills entries in goto g[][] and output o[].
2. Next we extend Trie into an automaton to support linear time matching. 1. 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

 `// C++ program for implementation of Aho Corasick algorithm` `// for string matching` `using` `namespace` `std;` `#include `   `// 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[ch] == -1)` `            ``g[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[ch] != 0)` `        ``{` `            ``f[g[ch]] = 0;` `            ``q.push(g[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` `answer = currentState;` `    ``int` `ch = nextInput - ``'a'``;`   `    ``// If goto is not defined, use failure function` `    ``while` `(g[answer][ch] == -1)` `        ``answer = f[answer];`   `    ``return` `g[answer][ch];` `}`   `// 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 match not found, move to next state` `        ``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);`   `    ``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

```

Source:
http://www.cs.uku.fi/~kilpelai/BSA05/lectures/slides04.pdf

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Improved By : PawelWolowiec, 2001guljain

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