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:
    1. 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.
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  1. Preprocessing:
    1. We first Build a Trie (or Keyword Tree) of all words.C++ Programming for Aho-Corasick Algorithm for Pattern Searching
    2. Next we extend Trie into an automaton to support linear time matching.Next we extend Trie into an automaton to support linear time matching.

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.

C++
// 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 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[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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