Pattern Searching | Set 6 (Efficient Construction of Finite Automata)

Difficulty Level : Hard
Last Updated : 29 Nov, 2021

In the previous post, we discussed the Finite Automata-based pattern searching algorithm. The FA (Finite Automata) construction method discussed in the previous post takes O((m^3)*NO_OF_CHARS) time. FA can be constructed in O(m*NO_OF_CHARS) time. In this post, we will discuss the O(m*NO_OF_CHARS) algorithm for FA construction. The idea is similar to LPs (longest prefix suffix) array construction discussed in the KMP algorithm. We use previously filled rows to fill a new row.

Efficient Construction of Finite Automata

Pattern Searching

The above diagrams represent graphical and tabular representations of pattern ACACAGA.

Algorithm:
1) Fill the first row. All entries in the first row are always 0 except the entry for the pat[0] character. For pat[0] character, we always need to go to state 1.
2) Initialize lps as 0. lps for the first index is always 0.
3) Do following for rows at index i = 1 to M. (M is the length of the pattern)
…..a) Copy the entries from the row at index equal to lps.
…..b) Update the entry for pat[i] character to i+1.
…..c) Update lps “lps = TF[lps][pat[i]]” where TF is the 2D array which is being constructed.

Following is the implementation for the above algorithm.

Implementation

C++

#include <bits/stdc++.h>
using namespace std;
#define NO_OF_CHARS 256
  
/* This function builds the TF table 
which represents Finite Automata for a 
given pattern */
void computeTransFun(char* pat, int M, int TF[][NO_OF_CHARS])
{
    int i, lps = 0, x;
  
    // Fill entries in first row
    for (x = 0; x < NO_OF_CHARS; x++)
        TF[0][x] = 0;
    TF[0][pat[0]] = 1;
  
    // Fill entries in other rows
    for (i = 1; i <= M; i++) {
        // Copy values from row at index lps
        for (x = 0; x < NO_OF_CHARS; x++)
            TF[i][x] = TF[lps][x];
  
        // Update the entry corresponding to this character
        TF[i][pat[i]] = i + 1;
  
        // Update lps for next row to be filled
        if (i < M)
            lps = TF[lps][pat[i]];
    }
}
  
/* Prints all occurrences of pat in txt */
void search(char pat[], char txt[])
{
    int M = strlen(pat);
    int N = strlen(txt);
  
    int TF[M + 1][NO_OF_CHARS];
  
    computeTransFun(pat, M, TF);
  
    // process text over FA.
    int i, j = 0;
    for (i = 0; i < N; i++) {
        j = TF[j][txt[i]];
        if (j == M) {
            cout << "pattern found at index " << i - M + 1 << endl;
        }
    }
}
  
/* Driver code */
int main()
{
    char txt[] = "ACACACACAGAAGA ACACAGAACACAGA GEEKS";
    char pat[] = "ACACAGA";
    search(pat, txt);
    return 0;
}
  
// This is code is contributed by rathbhupendra


#include <stdio.h>
#include <string.h>
#define NO_OF_CHARS 256

/* This function builds the TF table which represents Finite Automata for a
   given pattern  */
void computeTransFun(char* pat, int M, int TF[][NO_OF_CHARS])
{
    int i, lps = 0, x;

    // Fill entries in first row
    for (x = 0; x < NO_OF_CHARS; x++)
        TF[0][x] = 0;
    TF[0][pat[0]] = 1;

    // Fill entries in other rows
    for (i = 1; i <= M; i++) {
        // Copy values from row at index lps
        for (x = 0; x < NO_OF_CHARS; x++)
            TF[i][x] = TF[lps][x];

        // Update the entry corresponding to this character
        TF[i][pat[i]] = i + 1;

        // Update lps for next row to be filled
        if (i < M)
            lps = TF[lps][pat[i]];
    }
}

/* Prints all occurrences of pat in txt */
void search(char* pat, char* txt)
{
    int M = strlen(pat);
    int N = strlen(txt);

    int TF[M + 1][NO_OF_CHARS];

    computeTransFun(pat, M, TF);

    // process text over FA.
    int i, j = 0;
    for (i = 0; i < N; i++) {
        j = TF[j][txt[i]];
        if (j == M) {
            printf("\n pattern found at index %d", i - M + 1);
        }
    }
}

/* Driver program to test above function */
int main()
{
    char* txt = "GEEKS FOR GEEKS";
    char* pat = "GEEKS";
    search(pat, txt);
    getchar();
    return 0;
}

Java

/* A Java program to answer queries to check whether 
the substrings are palindrome or not efficiently */
  
class GFG
{
  
    static int NO_OF_CHARS = 256;
  
    /* This function builds the TF table 
    which represents Finite Automata for a 
    given pattern */
    static void computeTransFun(char[] pat, 
                                int M, int TF[][]) 
    {
        int i, lps = 0, x;
  
        // Fill entries in first row 
        for (x = 0; x < NO_OF_CHARS; x++) 
        {
            TF[0][x] = 0;
        }
        TF[0][pat[0]] = 1;
  
        // Fill entries in other rows 
        for (i = 1; i < M; i++) 
        {
            // Copy values from row at index lps 
            for (x = 0; x < NO_OF_CHARS; x++) 
            {
                TF[i][x] = TF[lps][x];
            }
  
            // Update the entry corresponding to this character 
            TF[i][pat[i]] = i + 1;
  
            // Update lps for next row to be filled 
            if (i < M) 
            {
                lps = TF[lps][pat[i]];
            }
        }
    }
  
    /* Prints all occurrences of pat in txt */
    static void search(char pat[], char txt[])
    {
        int M = pat.length;
        int N = txt.length;
  
        int[][] TF = new int[M + 1][NO_OF_CHARS];
  
        computeTransFun(pat, M, TF);
  
        // process text over FA. 
        int i, j = 0;
        for (i = 0; i < N; i++) 
        {
            j = TF[j][txt[i]];
            if (j == M) 
            {
                System.out.println("pattern found at index " + 
                                                (i - M + 1));
            }
        }
    }
  
    /* Driver code */
    public static void main(String[] args) 
    {
        char txt[] = "GEEKS FOR GEEKS".toCharArray();
        char pat[] = "GEEKS".toCharArray();
        search(pat, txt);
    }
}
  
// This code is contributed by Princi Singh

Python3

""" A Python3 program to answer queries to check whether  
the substrings are palindrome or not efficiently """
NO_OF_CHARS = 256
  
""" This function builds the TF table 
which represents Finite Automata for a 
given pattern """
  
  
def computeTransFun(pat, M, TF):
  
    lps = 0
  
    # Fill entries in first row
    for x in range(NO_OF_CHARS):
        TF[0][x] = 0
    TF[0][ord(pat[0])] = 1
  
    # Fill entries in other rows
    for i in range(1, M+1):
  
        # Copy values from row at index lps
        for x in range(NO_OF_CHARS):
            TF[i][x] = TF[lps][x]
  
        if (i < M):
            # Update the entry corresponding to this character
            TF[i][ord(pat[i])] = i + 1
  
            # Update lps for next row to be filled
  
            lps = TF[lps][ord(pat[i])]
  
# Prints all occurrences of pat in txt
  
  
def search(pat, txt):
    M = len(pat)
    N = len(txt)
    TF = [[0 for i in range(NO_OF_CHARS)] for j in range(M + 1)]
    computeTransFun(pat, M, TF)
  
    # process text over FA.
    j = 0
    for i in range(N):
        j = TF[j][ord(txt[i])]
        if (j == M):
            print("pattern found at index", i - M + 1)
  
  
# Driver code
txt = "ACACACACAGAAGA ACACAGAACACAGA GEEKS"
pat = "ACACAGA"
search(pat, txt)
  
# This code is contributed by divyeshrabadiya07

C#

/* A C# program to answer queries to check whether 
the substrings are palindrome or not efficiently */
using System;
      
class GFG
{
  
    static int NO_OF_CHARS = 256;
  
    /* This function builds the TF table 
    which represents Finite Automata for a 
    given pattern */
    static void computeTransFun(char[] pat, 
                                int M, int [,]TF) 
    {
        int i, lps = 0, x;
  
        // Fill entries in first row 
        for (x = 0; x < NO_OF_CHARS; x++) 
        {
            TF[0,x] = 0;
        }
        TF[0,pat[0]] = 1;
  
        // Fill entries in other rows 
        for (i = 1; i < M; i++) 
        {
            // Copy values from row at index lps 
            for (x = 0; x < NO_OF_CHARS; x++) 
            {
                TF[i,x] = TF[lps,x];
            }
  
            // Update the entry corresponding to this character 
            TF[i,pat[i]] = i + 1;
  
            // Update lps for next row to be filled 
            if (i < M) 
            {
                lps = TF[lps,pat[i]];
            }
        }
    }
  
    /* Prints all occurrences of pat in txt */
    static void search(char []pat, char []txt)
    {
        int M = pat.Length;
        int N = txt.Length;
  
        int[,] TF = new int[M + 1,NO_OF_CHARS];
  
        computeTransFun(pat, M, TF);
  
        // process text over FA. 
        int i, j = 0;
        for (i = 0; i < N; i++) 
        {
            j = TF[j,txt[i]];
            if (j == M) 
            {
                Console.WriteLine("pattern found at index " + 
                                                (i - M + 1));
            }
        }
    }
  
    /* Driver code */
    public static void Main(String[] args) 
    {
        char []txt = "GEEKS FOR GEEKS".ToCharArray();
        char []pat = "GEEKS".ToCharArray();
        search(pat, txt);
    }
}
  
// This code is contributed by Rajput-Ji

Javascript

<script>
/* A Javascript program to answer queries to check whether
the substrings are palindrome or not efficiently */
  
let NO_OF_CHARS = 256;
  
/* This function builds the TF table
    which represents Finite Automata for a
    given pattern */
function computeTransFun(pat,M,TF)
{
    let i, lps = 0, x;
   
        // Fill entries in first row
        for (x = 0; x < NO_OF_CHARS; x++)
        {
            TF[0][x] = 0;
        }
        TF[0][pat[0].charCodeAt(0)] = 1;
   
        // Fill entries in other rows
        for (i = 1; i < M; i++)
        {
            // Copy values from row at index lps
            for (x = 0; x < NO_OF_CHARS; x++)
            {
                TF[i][x] = TF[lps][x];
            }
   
            // Update the entry corresponding to this character
            TF[i][pat[i].charCodeAt(0)] = i + 1;
   
            // Update lps for next row to be filled
            if (i < M)
            {
                lps = TF[lps][pat[i].charCodeAt(0)];
            }
        }
}
  
/* Prints all occurrences of pat in txt */
function search(pat,txt)
{
    let M = pat.length;
        let N = txt.length;
   
        let TF = new Array(M + 1);
        for(let i=0;i<M+1;i++)
        {
            TF[i]=new Array(NO_OF_CHARS);
            for(let j=0;j<NO_OF_CHARS;j++)
            {
                TF[i][j]=0;
            }
        }
   
        computeTransFun(pat, M, TF);
   
        // process text over FA.
        let i, j = 0;
        for (i = 0; i < N; i++)
        {
            j = TF[j][txt[i].charCodeAt(0)];
            if (j == M)
            {
                document.write("pattern found at index " +
                                                (i - M + 1)+"<br>");
            }
        }
}
  
 /* Driver code */
let txt = "GEEKS FOR GEEKS".split("");
let pat = "GEEKS".split("");
search(pat, txt);
  
// This code is contributed by avanitrachhadiya2155
</script>

Output:

 pattern found at index 0
 pattern found at index 10

Time Complexity for FA construction is O(M*NO_OF_CHARS). The code for search is the same as the previous post and the time complexity for it is O(n).
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.