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

Last Updated : 09 Sep, 2023

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

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

/* 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).

Introduction to Pattern Searching - Data Structure and Algorithm Tutorial

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

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