Skip to content
Related Articles

Related Articles

Table of Contents

Improve Article
Save Article
Like Article

Smallest element from all square submatrices of size K from a given Matrix

  • Difficulty Level : Expert
  • Last Updated : 16 Sep, 2021

Given a matrix arr[][] and an integer K, the task is to find the smallest element from all possible square submatrices of size K from the given matrix.

Examples:

Input: K = 2, arr[][] ={ {1, 2, 3}, {4, 5, 6},  {7, 8, 9} }
Output: 
1 2 
4 5
Explanation:
Smallest elements from all possible square submatrices of size 2 are as follows:
{ {1, 2}, {4, 5} } -> 1
{ {2, 3}, {5, 6} } -> 2
{ {4, 5}, {7, 8} } -> 4
{ {5, 6}, {8, 9} } -> 5 

Input: K = 3,  
arr[][] = { {-1, 5, 4, 1, -3}, 
{4, 3, 1, 1, 6}, 
{2, -2, 5, 3, 1}, 
{8, 5, 1, 9, -4}, 
{12, 3, 5, 8, 1} }
Output: 
-2 -2 -3
-2 -2 -4
-2 -2 -4

Naive Approach: The simplest approach to solve the problem is to generate all possible square submatrices of size K from the given matrix and print the smallest element from each such submatrices.

Time Complexity: O(N * M * K2)
Auxiliary Space: O(1)

Efficient Approach: Follow the steps below to optimize the above approach:

  • Traverse over each row of the matrix and for every arr[i][j], update in-place the smallest element present between indices arr[i][j] to arr[i][j + K – 1] (0 <= j < M – K + 1).
  • Similarly, traverse over each column of the matrix and for every arr[i][j], update in-place the smallest element present between indices arr[i][j] to arr[i+K-1][j] (0 <= i < N – K + 1).
  • After performing the above operations, the top-left submatrix of size (N – K + 1)*(M – K + 1) of the matrix arr[][] consists of all the smallest elements of all the K x K sub-matrices of the given matrix.
  • Therefore, print the submatrix of size (N – K + 1)*(M – K + 1) as the required output.

Below is the implementation of the above approach:

C++




// C++ Program for
// the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to returns a smallest
// elements of all KxK submatrices
// of a given NxM matrix
vector<vector<int> > matrixMinimum(
       vector<vector<int> > nums, int K)
{
  // Stores the dimensions
  // of the given matrix
  int N = nums.size();
  int M = nums[0].size();
 
  // Stores the required
  // smallest elements
  vector<vector<int> > res(N - K + 1,
                           vector<int>(M - K + 1));
 
  // Update the smallest elements row-wise
  for (int i = 0; i < N; i++)
  {
    for (int j = 0; j < M - K + 1; j++)
    {
      int mini = INT_MAX;
      for (int k = j; k < j + K; k++)
      {
        mini = min(mini, nums[i][k]);
      }
      nums[i][j] = mini;
    }
  }
 
  // Update the minimum column-wise
  for (int j = 0; j < M; j++)
  {
    for (int i = 0; i < N - K + 1; i++)
    {
      int mini = INT_MAX;
      for (int k = i; k < i + K; k++)
      {
        mini = min(mini, nums[k][j]);
      }
      nums[i][j] = mini;
    }
  }
 
  // Store the final submatrix with
  // required minimum values
  for (int i = 0; i < N - K + 1; i++)
    for (int j = 0; j < M - K + 1; j++)
      res[i][j] = nums[i][j];
 
  // Return the resultant matrix
  return res;
}
 
void smallestinKsubmatrices(vector<vector<int> > arr,
                            int K)
{
  // Function Call
  vector<vector<int> > res = matrixMinimum(arr, K);
 
  // Print resultant matrix with the
  // minimum values of KxK sub-matrix
  for (int i = 0; i < res.size(); i++)
  {
    for (int j = 0; j < res[0].size(); j++)
    {
      cout << res[i][j] << " ";
    }
    cout << endl;
  }
}
 
// Driver Code
int main()
{
 
  // Given matrix
  vector<vector<int> > arr = {{-1, 5, 4, 1, -3},
                              {4, 3, 1, 1, 6},
                              {2, -2, 5, 3, 1},
                              {8, 5, 1, 9, -4},
                              {12, 3, 5, 8, 1}};
 
  // Given K
  int K = 3;
 
  smallestinKsubmatrices(arr, K);
}
 
// This code is contributed by Chitranayal
Java



// Java Program for the above approach
 
import java.util.*;
import java.lang.*;
 
class GFG {
 
    // Function to returns a smallest
    // elements of all KxK submatrices
    // of a given NxM matrix
    public static int[][] matrixMinimum(
        int[][] nums, int K)
    {
        // Stores the dimensions
        // of the given matrix
        int N = nums.length;
        int M = nums[0].length;
 
        // Stores the required
        // smallest elements
        int[][] res
            = new int[N - K + 1][M - K + 1];
 
        // Update the smallest elements row-wise
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < M - K + 1; j++) {
 
                int min = Integer.MAX_VALUE;
                for (int k = j; k < j + K; k++) {
                    min = Math.min(min, nums[i][k]);
                }
                nums[i][j] = min;
            }
        }
 
        // Update the minimum column-wise
        for (int j = 0; j < M; j++) {
            for (int i = 0; i < N - K + 1; i++) {
                int min = Integer.MAX_VALUE;
                for (int k = i; k < i + K; k++) {
                    min = Math.min(min, nums[k][j]);
                }
                nums[i][j] = min;
            }
        }
 
        // Store the final submatrix with
        // required minimum values
        for (int i = 0; i < N - K + 1; i++)
            for (int j = 0; j < M - K + 1; j++)
                res[i][j] = nums[i][j];
 
        // Return the resultant matrix
        return res;
    }
 
    public static void smallestinKsubmatrices(
        int arr[][], int K)
    {
        // Function Call
        int[][] res = matrixMinimum(arr, K);
 
        // Print resultant matrix with the
        // minimum values of KxK sub-matrix
        for (int i = 0; i < res.length; i++) {
            for (int j = 0; j < res[0].length; j++) {
                System.out.print(res[i][j] + " ");
            }
            System.out.println();
        }
    }
 
    // Driver Code
    public static void main(String[] args)
    {
 
        // Given matrix
        int[][] arr = { { -1, 5, 4, 1, -3 },
                        { 4, 3, 1, 1, 6 },
                        { 2, -2, 5, 3, 1 },
                        { 8, 5, 1, 9, -4 },
                        { 12, 3, 5, 8, 1 } };
 
        // Given K
        int K = 3;
 
        smallestinKsubmatrices(arr, K);
    }
}
Python3



# Python3 program for the above approach
import sys
 
# Function to returns a smallest
# elements of all KxK submatrices
# of a given NxM matrix
def matrixMinimum(nums, K):
 
    # Stores the dimensions
    # of the given matrix
    N = len(nums)
    M = len(nums[0])
 
    # Stores the required
    # smallest elements
    res = [[0 for x in range(M - K + 1)]
              for y in range(N - K + 1)]
 
    # Update the smallest elements row-wise
    for i in range(N):
        for j in range(M - K + 1):
            mn = sys.maxsize
             
            for k in range(j, j + K):
                mn = min(mn, nums[i][k])
 
            nums[i][j] = mn
 
    # Update the minimum column-wise
    for j in range(M):
        for i in range(N - K + 1):
            mn = sys.maxsize
             
            for k in range(i, i + K):
                mn = min(mn, nums[k][j])
 
            nums[i][j] = mn
 
    # Store the final submatrix with
    # required minimum values
    for i in range(N - K + 1):
        for j in range(M - K + 1):
            res[i][j] = nums[i][j]
 
    # Return the resultant matrix
    return res
 
def smallestinKsubmatrices(arr, K):
 
    # Function call
    res = matrixMinimum(arr, K)
 
    # Print resultant matrix with the
    # minimum values of KxK sub-matrix
    for i in range(len(res)):
        for j in range(len(res[0])):
            print(res[i][j], end = " ")
             
        print()
 
# Driver Code
 
# Given matrix
arr = [ [ -1, 5, 4, 1, -3 ],
        [ 4, 3, 1, 1, 6 ],
        [ 2, -2, 5, 3, 1 ],
        [ 8, 5, 1, 9, -4 ],
        [ 12, 3, 5, 8, 1 ] ]
 
# Given K
K = 3
 
# Function call
smallestinKsubmatrices(arr, K)
 
# This code is contributed by Shivam Singh
C#



// C# program for the above approach
using System;
 
class GFG{
 
// Function to returns a smallest
// elements of all KxK submatrices
// of a given NxM matrix
public static int[,] matrixMinimum(int[,] nums,
                                   int K)
{
     
    // Stores the dimensions
    // of the given matrix
    int N = nums.GetLength(0);
    int M = nums.GetLength(1);
 
    // Stores the required
    // smallest elements
    int[,] res = new int[N - K + 1,
                         M - K + 1];
 
    // Update the smallest elements row-wise
    for(int i = 0; i < N; i++)
    {
        for(int j = 0; j < M - K + 1; j++)
        {
            int min = int.MaxValue;
            for(int k = j; k < j + K; k++)
            {
                min = Math.Min(min, nums[i, k]);
            }
            nums[i, j] = min;
        }
    }
 
    // Update the minimum column-wise
    for(int j = 0; j < M; j++)
    {
        for(int i = 0; i < N - K + 1; i++)
        {
            int min = int.MaxValue;
            for(int k = i; k < i + K; k++)
            {
                min = Math.Min(min, nums[k, j]);
            }
            nums[i, j] = min;
        }
    }
 
    // Store the readonly submatrix with
    // required minimum values
    for(int i = 0; i < N - K + 1; i++)
        for(int j = 0; j < M - K + 1; j++)
            res[i, j] = nums[i, j];
 
    // Return the resultant matrix
    return res;
}
 
public static void smallestinKsubmatrices(int [,]arr,
                                          int K)
{
     
    // Function call
    int[,] res = matrixMinimum(arr, K);
 
    // Print resultant matrix with the
    // minimum values of KxK sub-matrix
    for(int i = 0; i < res.GetLength(0); i++)
    {
        for(int j = 0; j < res.GetLength(1); j++)
        {
            Console.Write(res[i, j] + " ");
        }
        Console.WriteLine();
    }
}
 
// Driver Code
public static void Main(String[] args)
{
 
    // Given matrix
    int[,] arr = { { -1, 5, 4, 1, -3 },
                   { 4, 3, 1, 1, 6 },
                   { 2, -2, 5, 3, 1 },
                   { 8, 5, 1, 9, -4 },
                   { 12, 3, 5, 8, 1 } };
 
    // Given K
    int K = 3;
 
    smallestinKsubmatrices(arr, K);
}
}
 
// This code is contributed by Amit Katiyar
Javascript



<script>
// javascript program for the
// above approach
 
// Function to returns a smallest
    // elements of all KxK submatrices
    // of a given NxM matrix
    function matrixMinimum(
        nums, K)
    {
        // Stores the dimensions
        // of the given matrix
        let N = nums.length;
        let M = nums[0].length;
  
        // Stores the required
        // smallest elements
        let res
            = new Array(N - K + 1);
        for (var i = 0; i < res.length; i++) {
            res[i] = new Array(2);
        }
  
        // Update the smallest elements row-wise
        for (let i = 0; i < N; i++) {
            for (let j = 0; j < M - K + 1; j++) {
  
                let min = Number.MAX_VALUE;
                for (let k = j; k < j + K; k++) {
                    min = Math.min(min, nums[i][k]);
                }
                nums[i][j] = min;
            }
        }
  
        // Update the minimum column-wise
        for (let j = 0; j < M; j++) {
            for (let i = 0; i < N - K + 1; i++) {
                let min = Number.MAX_VALUE;
                for (let k = i; k < i + K; k++) {
                    min = Math.min(min, nums[k][j]);
                }
                nums[i][j] = min;
            }
        }
  
        // Store the final submatrix with
        // required minimum values
        for (let i = 0; i < N - K + 1; i++)
            for (let j = 0; j < M - K + 1; j++)
                res[i][j] = nums[i][j];
  
        // Return the resultant matrix
        return res;
    }
  
    function smallestinKsubmatrices(arr, K)
    {
        // Function Call
        let res = matrixMinimum(arr, K);
  
        // Print resultant matrix with the
        // minimum values of KxK sub-matrix
        for (let i = 0; i < res.length; i++) {
            for (let j = 0; j < res[0].length; j++) {
                document.write(res[i][j] + " ");
            }
            document.write("<br/>");
        }
    }
  
// Driver Code
 
     // Given matrix
        let arr = [[ -1, 5, 4, 1, -3 ],
                        [ 4, 3, 1, 1, 6 ],
                        [ 2, -2, 5, 3, 1 ],
                        [ 8, 5, 1, 9, -4 ],
                        [ 12, 3, 5, 8, 1 ]];
  
        // Given K
        let K = 3;
  
        smallestinKsubmatrices(arr, K);
              
</script>
Output: 
-2 -2 -3 
-2 -2 -4 
-2 -2 -4
Time Complexity: O( max(N, M)3 )
Auxiliary Space: O( (N-K+1)*(M-K+1) ) 

My Personal Notes
arrow_drop_up
Previous
Maximize length of subarray of equal elements by performing at most K increment operations
Next

Minimum cost to generate any permutation of the given string
Recommended Articles
Page :
Maximum size of square such that all submatrices of that size have sum less than K
08, Apr 20
Reduce given Matrix by size 1 using sum of all 2x2 submatrices
26, Jan 22
Generate a matrix having each element equal to the sum of specified submatrices of a given matrix
03, Feb 21
Sum of all Submatrices of a Given Matrix
14, Feb 19
Count of square submatrices with average at least K
07, Jul 21
Generate a matrix having even sum of all diagonals in each 2 x 2 submatrices
19, Jan 21
Final Matrix after incrementing submatrices by K in range given by Q queries
27, Aug 20
Binary Matrix after flipping submatrices in given range for Q queries
27, Aug 20
Number of submatrices with all 1s
18, Feb 19
Sum of bitwise AND of all submatrices
20, Feb 19
Sum of Bitwise-OR of all Submatrices
20, Feb 19
Median of Bitwise XOR of all submatrices starting from the top left corner
03, Jan 21
Given an n x n square matrix, find sum of all sub-squares of size k x k
22, Sep 14
Construct a square Matrix whose parity of diagonal sum is same as size of matrix
09, Aug 20
Find smallest and largest element from square matrix diagonals
01, Dec 18
Size of smallest square that contains N non-overlapping rectangles of given dimensions
13, Apr 21
Smallest N digit number whose sum of square of digits is a Perfect Square
22, Apr 20
Find smallest perfect square number A such that N + A is also a perfect square number
02, May 20
Maximum size square sub-matrix with all 1s
04, Apr 10
Number of submatrices with OR value 1
18, Feb 19
Print maximum sum square sub-matrix of given size
13, May 16
Count squares of size K inscribed in a square of size N
14, Sep 20
Generate a Matrix such that given Matrix elements are equal to Bitwise OR of all corresponding row and column elements of generated Matrix
15, Jan 21
Maximum size square Sub-Matrix with sum less than or equals to K
21, Apr 20
Article Contributed By :
https://media.geeksforgeeks.org/auth/avatar.png
offbeat
@offbeat
Vote for difficulty
Current difficulty :
Expert





Improved By :
Article Tags :
Practice Tags :
Report Issue

We use cookies to ensure you have the best browsing experience on our website. By using our site, you
acknowledge that you have read and understood our
Cookie Policy &
        Privacy Policy

Lightbox

Start Your Coding Journey Now!