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.  

1. Find the minimum element of the current row and store the column index of the minimum element.
2. Check if the row minimum element is also maximum in its column. We use the stored column index here.
3. 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?