Rabin-Karp algorithm for Pattern Searching in Matrix

Given matrices txt[][] of dimensions m1 x m2 and pattern pat[][] of dimensions n1 x n2, the task is to check whether a pattern exists in the matrix or not, and if yes then print the top most indices of the pat[][] in txt[][]. It is assumed that m1, m2 ≥ n1, n2

Examples:

Input:
txt[][] = {{G, H, I, P}
           {J, K, L, Q}
           {R, G, H, I}  
           {S, J, K, L}
          }
pat[][] = {{G, H, I},
           {J, K, L}
          }
Output:
Pattern found at ( 0, 0 )
Pattern found at ( 2, 1 )
Explanation:


Input:
txt[][] = { {A, B, C},
            {D, E, F},
            {G, H, I}
          }
pat[][] = { {E, F},
            {H, I}
          }
Output:
Pattern found at (1, 1)

Approach: In order to find a pattern in a 2-D array using Rabin-Karp algorithm, consider an input matrix txt[m1][m2] and a pattern pat[n1][n2]. The idea is to find the hash of each columns of mat[][] and pat[][] and compare the hash values. For any column if hash values are equals than check for the corresponding rows values. Below are the steps:

1. Find the hash values of each column for the first N1 rows in both txt[][] and pat[][] matrix.
2. Apply Rabin-Karp Algorithm by finding hash values for the column hashes found in step 1.
3. If a match is found compare txt[][] and pat[][] matrices for the specific rows and columns.
4. Else slide down the column hashes by 1 row in the txt matrix using a rolling hash.
5. Repeat steps 2 to 4 for all the hash values and if we found any pat[][] match in txt[][] then print the indices of top most cell in the txt[][].

To find the hash value: In order to find the hash value of a substring of size N in a text using rolling hash follow below steps:  

1. Remove the first character from the string: hash(txt[s:s+n-1])-(radix**(n-1)*txt[s])%prime_number
2. Add the next character to the string: hash(txt[s:s+n-1])*radix + txt[n]

Below is the implementation of the above approach: 

C++

#include <bits/stdc++.h>
using namespace std;
long long mod = 257;  //The modular value
long long r = 256;    //radix
long long dr = 1;     //Highest power for row hashing
long long dc = 1;     //Highest power for col hashing

//func that return a power n under mod m in LogN
long long power(int a,int n,long long m){
  if(n == 0){
    return 1;
  }
  if(n == 1){
    return a%m;
  }
  long long pow = power(a,n/2,m);
  if(n&1){
    return ((a%m)*(pow)%m * (pow)%m)%m;
  }
  else{
    return ((pow)%m * (pow)%m)%m;
  }
}
//Checks if all values of pattern matches with the text
bool check(vector<vector<char>> &txt,vector<vector<char>> &pat, long long r,long long c){
  for(long long i=0;i<pat.size();i++){
    for(long long j=0;j<pat[0].size();j++){
      if(pat[i][j] != txt[i+r][j+c]) return false;
    }
  }
  return true;
}
//Finds the first hash of first n rows where n is no. of rows in pattern
vector<long long> findHash(vector<vector<char>> &mat,long long row){
  vector<long long> hash;
  long long col = mat[0].size();
  for(long long i=0;i<col;i++){
    long long h = 0;
    for(long long j=0;j<row;j++){
      h = ((h*r)%mod + mat[j][i]%mod)%mod;
    }
    hash.push_back(h);
  }
  return hash;
}
//rolling hash function for columns
void colRollingHash(vector<vector<char>> &txt, vector<long long> &t_hash, long long row, long long p_row){
  for(long long i=0;i<t_hash.size();i++){
    t_hash[i] = (t_hash[i]%mod - ((txt[row][i])%mod*(dr)%mod)%mod)%mod;
    t_hash[i] = ((t_hash[i]%mod) * (r%mod))%mod;
    t_hash[i] = (t_hash[i]%mod + txt[row+p_row][i]%mod)%mod;
  }
}
void rabinKarp(vector<vector<char>> &txt, vector<vector<char>> &pat){
  long long t_row = txt.size();
  long long t_col = txt[0].size();
  long long p_row = pat.size();
  long long p_col = pat[0].size();
  dr = power(r,p_row-1,mod);
  dc = power(r,p_col-1,mod);
  vector<long long> t_hash = findHash(txt,p_row); //column hash of p_row rows
  vector<long long> p_hash = findHash(pat,p_row); //column hash of p_row rows
  long long p_val = 0;  //hash of entire pattern matrix
  for(long long i=0;i<p_col;i++){
    p_val = (p_val*r + p_hash[i])%mod;
  }
  for(long long i=0;i<=(t_row-p_row);i++){
    long long t_val = 0;
    for(long long i=0;i<p_col;i++){
      t_val = ((t_val*r) + t_hash[i])%mod;
    }
    for(long long j=0;j<=(t_col-p_col);j++){
      if(p_val == t_val){
        if(check(txt,pat,i,j)){
          cout<<i<<" "<<j<<endl;
        }
      }
      //calculating t_val for next set of columns
      t_val = (t_val%mod - ((t_hash[j]%mod)*(dc%mod))%mod + mod)%mod;
      t_val = (t_val%mod * r%mod)%mod;
      t_val = (t_val%mod + t_hash[j+p_col]%mod)%mod;
    }
    if(i < t_row-p_row){
      //call this function for hashing form next row
      colRollingHash(txt,t_hash,i,p_row);
    }
  }
}
int main(){
  vector<vector<char>> txt{{'A','B','C','D','E'},{'A','B','C','D','E'},{'A','B','C','D','E'},{'A','B','C','D','E'},{'A','B','C','D','E'}};
  vector<vector<char>> pat{{'A','B','C','D','E'},{'A','B','C','D','E'},{'A','B','C','D','E'},{'A','B','C','D','E'}};
  //function prints the indices of row and col where its a match in txt
  rabinKarp(txt,pat);
  return 0;
}

Python3

# Python implementation for the
# pattern matching in 2-D matrix

# Function to find the hash-value
# of the given columns of text
def findHash(arr, col, row):
    hashCol = []
    add = 0
    radix = 256

    # For each column
    for i in range(0, col):

        for j in reversed(range(0, row)):
            add = add + (radix**(row-j-1) *
                         ord(arr[j][i]))% 101
        hashCol.append(add % 101);
        add = 0
    return hashCol

# Function to check equality of the
# two strings
def checkEquality(txt, row, col, flag):
    txt = [txt[i][col:patCol + col]
           for i in range(row, patRow + row)]

# If pattern found
    if txt == pat:
        flag = 1
        print("Pattern found at", \
              "(", row, ", ", col, ")")
    return flag

# Function to find the hash value of
# of the next column using rolling-hash
# of the Rabin-karp
def colRollingHash(txtHash, nxtRow):

    radix = 256

    # Find the hash of the matrix
    for j in range(len(txtHash)):
        txtHash[j] = (txtHash[j]*radix \
                      + ord(txt[nxtRow][j]))% 101
        txtHash[j] = txtHash[j] - (radix**(patRow) *
                     ord(txt[nxtRow-patRow][j]))% 101
        txtHash[j] = txtHash[j]% 101
    return txtHash


# Function to match a pattern in
# the given 2D Matrix
def search(txt, pat):

# List of the hashed value for
    # the text and pattern columns
    patHash = []
    txtHash = []

    # Hash value of the
    # pat_hash and txt_hash
    patVal = 0
    txtVal = 0

    # Radix value for the input characters
    radix = 256

    # Variable to determine if
    # pattern was found or not
    flag = 0

    # Function call to find the
    # hash value of columns
    txtHash = findHash(txt, txtCol, patRow)
    patHash = findHash(pat, patCol, patRow)

    # Calculate hash value for patHash
    for i in range(len(patHash)):
        patVal = patVal \
                 + (radix**(len(patHash)-i-1) *
                 patHash[i]% 101)
    patVal = patVal % 101


    # Applying Rabin-Karp to compare
    # txtHash and patHash
    for i in range(patRow-1, txtRow):
        col = 0
        txtVal = 0

        # Find the hash value txtHash
        for j in range(len(patHash)):
            txtVal = txtVal\
                     + (radix**(len(patHash)-j-1) *
                     txtHash[j])% 101
        txtVal = txtVal % 101

        if txtVal == patVal:
            flag = checkEquality(\
                     txt, i + 1-patRow, col, flag)

        else:

            # Roll the txt window by one character
            for k in range(len(patHash), len(txtHash)):

                txtVal = txtVal \
                         * radix + (txtHash[k])% 101
                txtVal = txtVal \
                         - (radix**(len(patHash)) *
                         (txtHash[k-len(patHash)]))% 101
                txtVal = txtVal % 101
                col = col + 1

                # Check if txtVal and patVal are equal
                if patVal == txtVal:
                    flag = checkEquality(\
                           txt, i + 1-patRow, col, flag)
                else:
                    continue

        # To make sure i does not exceed txtRow
        if i + 1<txtRow:
            txtHash = colRollingHash(txtHash, i + 1)

    if flag == 0:
        print("Pattern not found")


# Driver Code
if __name__ == "__main__":

  # Given Text
  txt = [['A', 'B', 'C'], \
         ['D', 'E', 'F'], \
         ['G', 'H', 'I']]

  # Given Pattern
  pat = [['E', 'F'], ['H', 'I']]

  # Dimensions of the text
  txtRow = 3
  txtCol = 3

  # Dimensions for the pattern
  patRow = 2
  patCol = 2

  # Function Call
  search(txt, pat)

Output

Pattern found at ( 1 ,  1 )