# Saddle point in a matrix

• Difficulty Level : Medium
• Last Updated : 15 May, 2022

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 maximum in its column.
Examples :

```Input: Mat = { {1, 2, 3},
{4, 5, 6},
{7, 8, 9}}
Output: 7
7 is minimum in its row and maximum in its column.

Input: Mat = {{1, 2, 3},
{4, 5, 6},
{10, 18, 4}}

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 ``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], 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 ``#include ` `#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], 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

 ` ``\$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

 ``

Output :

`Value of Saddle Point 7`

Exercise :
Can there be more than one Saddle Points in a Matrix?
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