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

In the previous post, we discussed Finite Automata based pattern searching algorithm. The FA (Finite Automata) construction method discussed in 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.

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

Algorithm:
1) Fill the first row. All entries in first row are always 0 except the entry for 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 C/C++ 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[] = "GEEKS FOR GEEKS"; 
    char pat[] = "GEEKS"; 
    search(pat, txt); 
    return 0; 
} 
  
// This is code is contributed by rathbhupendra 

C

#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 

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 

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 same as the previous post and time complexity for it is O(n).

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