Saddle point in a matrix
Given a matrix of n x n size, the task is to find the saddle point of the matrix. A saddle point is an element of the matrix such that it is the minimum element in its row and the maximum in its column.
Examples :
Input: Mat[3][3] = { {1, 2, 3}, {4, 5, 6}, {7, 8, 9}} Output: 7 7 is minimum in its row and maximum in its column. Input: Mat[3][3] = {{1, 2, 3}, {4, 5, 6}, {10, 18, 4}} Output: No saddle point
A simple solution is to traverse all matrix elements one by one and check if the element is Saddle Point or not.
An efficient solution is based on the below steps.
Traverse all rows one by one and do the following for every row i.
- Find the minimum element of the current row and store the column index of the minimum element.
- Check if the row minimum element is also maximum in its column. We use the stored column index here.
- If yes, then saddle point else continues till the end of the matrix.
Below is the implementation of the above steps.
C++
// C++ program to illustrate Saddle point #include <bits/stdc++.h> using namespace std; const int MAX = 100; // Function to find saddle point bool findSaddlePoint( int mat[MAX][MAX], int n) { // Process all rows one by one for ( int i = 0; i < n; i++) { // Find the minimum element of row i. // Also find column index of the minimum element int min_row = mat[i][0], col_ind = 0; for ( int j = 1; j < n; j++) { if (min_row > mat[i][j]) { min_row = mat[i][j]; col_ind = j; } } // Check if the minimum element of row is also // the maximum element of column or not int k; for (k = 0; k < n; k++) // Note that col_ind is fixed if (min_row < mat[k][col_ind]) break ; // If saddle point is present in this row then // print it if (k == n) { cout << "Value of Saddle Point " << min_row; return true ; } } // If Saddle Point not found return false ; } // Driver code int main() { int mat[MAX][MAX] = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}; int n = 3; if (findSaddlePoint(mat, n) == false ) cout << "No Saddle Point " ; return 0; } |
C
// C program to illustrate Saddle point #include <stdio.h> #include <stdbool.h> #define MAX 100 // Function to find saddle point bool findSaddlePoint( int mat[MAX][MAX], int n) { // Process all rows one by one for ( int i = 0; i < n; i++) { // Find the minimum element of row i. // Also find column index of the minimum element int min_row = mat[i][0], col_ind = 0; for ( int j = 1; j < n; j++) { if (min_row > mat[i][j]) { min_row = mat[i][j]; col_ind = j; } } // Check if the minimum element of row is also // the maximum element of column or not int k; for (k = 0; k < n; k++) // Note that col_ind is fixed if (min_row < mat[k][col_ind]) break ; // If saddle point is present in this row then // print it if (k == n) { printf ( "Value of Saddle Point %d" ,min_row); return true ; } } // If Saddle Point not found return false ; } // Driver code int main() { int mat[MAX][MAX] = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}; int n = 3; if (findSaddlePoint(mat, n) == false ) printf ( "No Saddle Point " ); return 0; } // This code is contributed by kothavvsaakash. |
Java
// Java program to illustrate Saddle point class Test { // Method to find saddle point static boolean findSaddlePoint( int mat[][ ], int n) { // Process all rows one by one for ( int i = 0 ; i < n; i++) { // Find the minimum element of row i. // Also find column index of the minimum element int min_row = mat[i][ 0 ], col_ind = 0 ; for ( int j = 1 ; j < n; j++) { if (min_row > mat[i][j]) { min_row = mat[i][j]; col_ind = j; } } // Check if the minimum element of row is also // the maximum element of column or not int k; for (k = 0 ; k < n; k++) // Note that col_ind is fixed if (min_row < mat[k][col_ind]) break ; // If saddle point is present in this row then // print it if (k == n) { System.out.println( "Value of Saddle Point " + min_row); return true ; } } // If Saddle Point not found return false ; } // Driver method public static void main(String[] args) { int mat[][] = {{ 1 , 2 , 3 }, { 4 , 5 , 6 }, { 7 , 8 , 9 }}; int n = 3 ; if (findSaddlePoint(mat, n) == false ) System.out.println( "No Saddle Point " ); } } |
Python3
# Python3 program to illustrate # Saddle point # Method to find saddle point def findSaddlePoint(mat, n): # Process all rows one # by one for i in range (n): # Find the minimum element # of row i. # Also find column index of # the minimum element min_row = mat[i][ 0 ]; col_ind = 0 ; for j in range ( 1 , n): if (min_row > mat[i][j]): min_row = mat[i][j]; col_ind = j; # Check if the minimum element # of row is also the maximum # element of column or not k = 0 ; for k in range (n): # Note that col_ind is fixed if (min_row < mat[k][col_ind]): break ; k + = 1 ; # If saddle point present in this # row then print if (k = = n): print ( "Value of Saddle Point " , min_row); return True ; # If Saddle Point found return False ; # Driver method if __name__ = = '__main__' : mat = [[ 1 , 2 , 3 ], [ 4 , 5 , 6 ], [ 7 , 8 , 9 ]]; n = 3 ; if (findSaddlePoint(mat, n) = = False ): print ( "No Saddle Po" ); # This code is contributed by 29AjayKumar |
C#
// C# program to illustrate Saddle point using System; class GFG { // Method to find saddle point static bool findSaddlePoint( int [,] mat, int n) { // Process all rows one by one for ( int i = 0; i < n; i++) { // Find the minimum element of // row i. Also find column index // of the minimum element int min_row = mat[i, 0], col_ind = 0; for ( int j = 1; j < n; j++) { if (min_row > mat[i, j]) { min_row = mat[i, j]; col_ind = j; } } // Check if the minimum element // of row is also the maximum // element of column or not int k; for (k = 0; k < n; k++) // Note that col_ind is fixed if (min_row < mat[k, col_ind]) break ; // If saddle point is present in this row then // print it if (k == n) { Console.WriteLine( "Value of Saddle Point " + min_row); return true ; } } // If Saddle Point not found return false ; } // Driver code public static void Main() { int [,] mat = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}; int n = 3; if (findSaddlePoint(mat, n) == false ) Console.WriteLine( "No Saddle Point " ); } } // This code is contributed by KRV. |
PHP
<?php // PHP program to illustrate // Saddle point $MAX = 100; // Function to find saddle point function findSaddlePoint( $mat , $n ) { // Process all rows one by one for ( $i = 0; $i < $n ; $i ++) { // Find the minimum element // of row i. Also find column // index of the minimum element $min_row = $mat [ $i ][0]; $col_ind = 0; for ( $j = 1; $j < $n ; $j ++) { if ( $min_row > $mat [ $i ][ $j ]) { $min_row = $mat [ $i ][ $j ]; $col_ind = $j ; } } // Check if the minimum element of // row is also the maximum element // of column or not $k ; for ( $k = 0; $k < $n ; $k ++) // Note that col_ind is fixed if ( $min_row < $mat [ $k ][ $col_ind ]) break ; // If saddle point is present in // this row then print it if ( $k == $n ) { echo "Value of Saddle Point " , $min_row ; return true; } } // If Saddle Point not found return false; } // Driver code $mat = array ( array (1, 2, 3), array (4, 5, 6), array (7, 8, 9)); $n = 3; if (findSaddlePoint( $mat , $n ) == false) echo "No Saddle Point " ; // This code is contributed by anuj_67. ?> |
Javascript
<script> // Javascript program to illustrate Saddle point // Method to find saddle point function findSaddlePoint(mat, n) { // Process all rows one by one for (let i = 0; i < n; i++) { // Find the minimum element of row i. // Also find column index of the minimum element let min_row = mat[i][0], col_ind = 0; for (let j = 1; j < n; j++) { if (min_row > mat[i][j]) { min_row = mat[i][j]; col_ind = j; } } // Check if the minimum element of row is also // the maximum element of column or not let k; for (k = 0; k < n; k++) // Note that col_ind is fixed if (min_row < mat[k][col_ind]) break ; // If saddle point is present in this row then // print it if (k == n) { document.write( "Value of Saddle Point " + min_row+ "<br>" ); return true ; } } // If Saddle Point not found return false ; } // Driver method let mat = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]; let n = 3; if (findSaddlePoint(mat, n) == false ) document.write( "No Saddle Point " ); // This code is contributed by rag2127 </script> |
Output
Value of Saddle Point 7
Time Complexity: O(n*n)
Auxiliary Space: O(1)
Exercise :
Can there be more than one Saddle Points in a Matrix?