Find a specific pair in Matrix

3.8

Given an n x n matrix mat[n][n] of integers, find the maximum value of mat(c, d) – mat(a, b) over all choices of indexes such that both c > a and d > b.

Example:

Input:
mat[N][N] = {{ 1, 2, -1, -4, -20 },
             { -8, -3, 4, 2, 1 }, 
             { 3, 8, 6, 1, 3 },
             { -4, -1, 1, 7, -6 },
             { 0, -4, 10, -5, 1 }};
Output: 18
The maximum value is 18 as mat[4][2] 
- mat[1][0] = 18 has maximum difference. 

The program should do only ONE traversal of the matrix. i.e. expected time complexity is O(n2)

A simple solution would be to apply Brute-Force. For all values mat(a, b) in the matrix, we find mat(c, d) that has maximum value such that c > a and d > b and keeps on updating maximum value found so far. We finally return the maximum value.

Below is its implementation.

C++

// A Naive method to find maximum value of mat1[d]
// - ma[a][b] such that c > a and d > b
#include <bits/stdc++.h>
using namespace std;
#define N 5

// The function returns maximum value A(c,d) - A(a,b)
// over all choices of indexes such that both c > a
// and d > b.
int findMaxValue(int mat[][N])
{
    // stores maximum value
    int maxValue = INT_MIN;

    // Consider all possible pairs mat[a][b] and
    // mat1[d]
    for (int a = 0; a < N - 1; a++)
      for (int b = 0; b < N - 1; b++)
         for (int c = a + 1; c < N; c++)
           for (int d = b + 1; d < N; d++)
              if (maxValue < (mat1[d] - mat[a][b]))
                  maxValue = mat1[d] - mat[a][b];

    return maxValue;
}

// Driver program to test above function
int main()
{
   int mat[N][N] = {
                  { 1, 2, -1, -4, -20 },
                  { -8, -3, 4, 2, 1 },
                  { 3, 8, 6, 1, 3 },
                  { -4, -1, 1, 7, -6 },
                  { 0, -4, 10, -5, 1 }
               };
    cout << "Maximum Value is " 
         << findMaxValue(mat);

    return 0;
}

Java

// A Naive method to find maximum value of mat1[d]
// - ma[a][b] such that c > a and d > b
import java.io.*;
import java.util.*;
 
class GFG 
{
    // The function returns maximum value A(c,d) - A(a,b)
    // over all choices of indexes such that both c > a
    // and d > b.
    static int findMaxValue(int N,int mat[][])
    {
        // stores maximum value
        int maxValue = Integer.MIN_VALUE;
     
        // Consider all possible pairs mat[a][b] and
        // mat1[d]
        for (int a = 0; a < N - 1; a++)
          for (int b = 0; b < N - 1; b++)
             for (int c = a + 1; c < N; c++)
               for (int d = b + 1; d < N; d++)
                  if (maxValue < (mat1[d] - mat[a][b]))
                      maxValue = mat1[d] - mat[a][b];
     
        return maxValue;
    }
     
    // Driver code
    public static void main (String[] args) 
    {
        int N = 5;

        int mat[][] = {
                      { 1, 2, -1, -4, -20 },
                      { -8, -3, 4, 2, 1 },
                      { 3, 8, 6, 1, 3 },
                      { -4, -1, 1, 7, -6 },
                      { 0, -4, 10, -5, 1 }
                   };

        System.out.print("Maximum Value is " + 
                         findMaxValue(N,mat));
    }
}
 
// Contributed by Prakriti Gupta


Output:
Maximum Value is 18

The above program runs in O(n^4) time which is nowhere close to expected time complexity of O(n^2)

An efficient solution uses extra space. We pre-process the matrix such that index(i, j) stores max of elements in matrix from (i, j) to (N-1, N-1) and in the process keeps on updating maximum value found so far. We finally return the maximum value.

C++

// An efficient method to find maximum value of mat1[d]
// - ma[a][b] such that c > a and d > b
#include <bits/stdc++.h>
using namespace std;
#define N 5

// The function returns maximum value A(c,d) - A(a,b)
// over all choices of indexes such that both c > a
// and d > b.
int findMaxValue(int mat[][N])
{
    //stores maximum value
    int maxValue = INT_MIN;

    // maxArr[i][j] stores max of elements in matrix
    // from (i, j) to (N-1, N-1)
    int maxArr[N][N];

    // last element of maxArr will be same's as of
    // the input matrix
    maxArr[N-1][N-1] = mat[N-1][N-1];

    // preprocess last row
    int maxv = mat[N-1][N-1];  // Initialize max
    for (int j = N - 2; j >= 0; j--)
    {
        if (mat[N-1][j] > maxv)
            maxv = mat[N - 1][j];
        maxArr[N-1][j] = maxv;
    }

    // preprocess last column
    maxv = mat[N - 1][N - 1];  // Initialize max
    for (int i = N - 2; i >= 0; i--)
    {
        if (mat[i][N - 1] > maxv)
            maxv = mat[i][N - 1];
        maxArr[i][N - 1] = maxv;
    }

    // preprocess rest of the matrix from bottom
    for (int i = N-2; i >= 0; i--)
    {
        for (int j = N-2; j >= 0; j--)
        {
            // Update maxValue
            if (maxArr[i+1][j+1] - mat[i][j] >
                                            maxValue)
                maxValue = maxArr[i + 1][j + 1] - mat[i][j];

            // set maxArr (i, j)
            maxArr[i][j] = max(mat[i][j],
                               max(maxArr[i][j + 1],
                                   maxArr[i + 1][j]) );
        }
    }

    return maxValue;
}

// Driver program to test above function
int main()
{
    int mat[N][N] = {
                      { 1, 2, -1, -4, -20 },
                      { -8, -3, 4, 2, 1 },
                      { 3, 8, 6, 1, 3 },
                      { -4, -1, 1, 7, -6 },
                      { 0, -4, 10, -5, 1 }
                    };
    cout << "Maximum Value is " 
         << findMaxValue(mat);

    return 0;
}

Java

// An efficient method to find maximum value of mat1[d]
// - ma[a][b] such that c > a and d > b
import java.io.*;
import java.util.*;
 
class GFG 
{
    // The function returns maximum value A(c,d) - A(a,b)
    // over all choices of indexes such that both c > a
    // and d > b.
    static int findMaxValue(int N,int mat[][])
    {
        //stores maximum value
        int maxValue = Integer.MIN_VALUE;
     
        // maxArr[i][j] stores max of elements in matrix
        // from (i, j) to (N-1, N-1)
        int maxArr[][] = new int[N][N];
     
        // last element of maxArr will be same's as of
        // the input matrix
        maxArr[N-1][N-1] = mat[N-1][N-1];
     
        // preprocess last row
        int maxv = mat[N-1][N-1];  // Initialize max
        for (int j = N - 2; j >= 0; j--)
        {
            if (mat[N-1][j] > maxv)
                maxv = mat[N - 1][j];
            maxArr[N-1][j] = maxv;
        }
     
        // preprocess last column
        maxv = mat[N - 1][N - 1];  // Initialize max
        for (int i = N - 2; i >= 0; i--)
        {
            if (mat[i][N - 1] > maxv)
                maxv = mat[i][N - 1];
            maxArr[i][N - 1] = maxv;
        }
     
        // preprocess rest of the matrix from bottom
        for (int i = N-2; i >= 0; i--)
        {
            for (int j = N-2; j >= 0; j--)
            {
                // Update maxValue
                if (maxArr[i+1][j+1] - mat[i][j] > maxValue)
                    maxValue = maxArr[i + 1][j + 1] - mat[i][j];
     
                // set maxArr (i, j)
                maxArr[i][j] = Math.max(mat[i][j],
                                   Math.max(maxArr[i][j + 1],
                                       maxArr[i + 1][j]) );
            }
        }
     
        return maxValue;
    }
    
    // Driver code
    public static void main (String[] args) 
    {
        int N = 5;

        int mat[][] = {
                      { 1, 2, -1, -4, -20 },
                      { -8, -3, 4, 2, 1 },
                      { 3, 8, 6, 1, 3 },
                      { -4, -1, 1, 7, -6 },
                      { 0, -4, 10, -5, 1 }
                   };

        System.out.print("Maximum Value is " + 
                           findMaxValue(N,mat));
    }
}
 
// Contributed by Prakriti Gupta


Output:
Maximum Value is 18

If we are allowed to modify of the matrix, we can avoid using extra space and use input matrix instead.

Exercise: Print index (a, b) and (c, d) as well.

This article is contributed by Aditya Goel. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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