# Search in a row wise and column wise sorted matrix

Given an n x n matrix and an integer x, find the position of x in the matrix if it is present. Otherwise, print “Element not found”. Every row and column of the matrix is sorted in increasing order. The designed algorithm should have linear time complexity

Examples:

Input: mat[4][4] = { {10, 20, 30, 40},  x = 29
{15, 25, 35, 45},
{27, 29, 37, 48},
{32, 33, 39, 50}}

Output: Found at (2, 1)
Explanation: Element at (2,1) is 29

Input : mat[4][4] = { {10, 20, 30, 40},   x = 100
{15, 25, 35, 45},
{27, 29, 37, 48},
{32, 33, 39, 50}};

Explanation: Element 100 does not exist in the matrix

Naive approach: To solve the problem follow the below idea:

The simple idea is to traverse the array and search elements one by one

Follow the given steps to solve the problem:

• Run a nested loop, outer loop for the row, and inner loop for the column
• Check every element with x and if the element is found then print “element found”

Below is the implementation of the above approach:

## C++

 `// C++ program to search an element in row-wise``// and column-wise sorted matrix``#include ` `using` `namespace` `std;` `/* Searches the element x in mat[][]. If the``element is found, then prints its position``and returns true, otherwise prints "not found"``and returns false */``int` `search(``int` `mat[4][4], ``int` `n, ``int` `x)``{``    ``if` `(n == 0)``        ``return` `-1;` `    ``// traverse through the matrix``    ``for` `(``int` `i = 0; i < n; i++) {``        ``for` `(``int` `j = 0; j < n; j++)``            ``// if the element is found``            ``if` `(mat[i][j] == x) {``                ``cout << ``"Element found at ("` `<< i << ``", "``                     ``<< j << ``")\n"``;``                ``return` `1;``            ``}``    ``}` `    ``cout << ``"n Element not found"``;``    ``return` `0;``}` `// Driver code``int` `main()``{``    ``int` `mat[4][4] = { { 10, 20, 30, 40 },``                      ``{ 15, 25, 35, 45 },``                      ``{ 27, 29, 37, 48 },``                      ``{ 32, 33, 39, 50 } };` `    ``// Function call``    ``search(mat, 4, 29);` `    ``return` `0;``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## C

 `// C program to search an element in row-wise``// and column-wise sorted matrix``#include ` `/* Searches the element x in mat[][]. If the``element is found, then prints its position``and returns true, otherwise prints "not found"``and returns false */``int` `search(``int` `mat[4][4], ``int` `n, ``int` `x)``{``    ``if` `(n == 0)``        ``return` `-1;` `    ``// traverse through the matrix``    ``for` `(``int` `i = 0; i < n; i++) {``        ``for` `(``int` `j = 0; j < n; j++)``            ``// if the element is found``            ``if` `(mat[i][j] == x) {``                ``printf``(``"Element found at (%d, %d)\n"``, i, j);``                ``return` `1;``            ``}``    ``}` `    ``printf``(``"Element not found"``);``    ``return` `0;``}` `// Driver code``int` `main()``{``    ``int` `mat[4][4] = { { 10, 20, 30, 40 },``                      ``{ 15, 25, 35, 45 },``                      ``{ 27, 29, 37, 48 },``                      ``{ 32, 33, 39, 50 } };` `    ``// Function call``    ``search(mat, 4, 29);` `    ``return` `0;``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## Java

 `// Java program to search an element in row-wise``// and column-wise sorted matrix` `class` `GFG {``    ``static` `int` `search(``int``[][] mat, ``int` `n, ``int` `x)``    ``{``        ``if` `(n == ``0``)``            ``return` `-``1``;` `        ``// traverse through the matrix``        ``for` `(``int` `i = ``0``; i < n; i++) {``            ``for` `(``int` `j = ``0``; j < n; j++)``                ``// if the element is found``                ``if` `(mat[i][j] == x) {``                    ``System.out.print(``"Element found at ("``                                     ``+ i + ``", "` `+ j``                                     ``+ ``")\n"``);``                    ``return` `1``;``                ``}``        ``}` `        ``System.out.print(``" Element not found"``);``        ``return` `0``;``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `mat[][] = { { ``10``, ``20``, ``30``, ``40` `},``                        ``{ ``15``, ``25``, ``35``, ``45` `},``                        ``{ ``27``, ``29``, ``37``, ``48` `},``                        ``{ ``32``, ``33``, ``39``, ``50` `} };` `        ``// Function call``        ``search(mat, ``4``, ``29``);``    ``}``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## Python3

 `# Python program to search an element in row-wise``# and column-wise sorted matrix` `# Searches the element x in mat[][]. If the``# element is found, then prints its position``# and returns true, otherwise prints "not found"``# and returns false`  `def` `search(mat, n, x):``    ``if``(n ``=``=` `0``):``        ``return` `-``1` `    ``# Traverse through the matrix``    ``for` `i ``in` `range``(n):``        ``for` `j ``in` `range``(n):` `            ``# If the element is found``            ``if``(mat[i][j] ``=``=` `x):``                ``print``(``"Element found at ("``, i, ``","``, j, ``")"``)``                ``return` `1` `    ``print``(``" Element not found"``)``    ``return` `0`  `# Driver code``if` `__name__ ``=``=` `"__main__"``:``    ``mat ``=` `[[``10``, ``20``, ``30``, ``40``], [``15``, ``25``, ``35``, ``45``],``           ``[``27``, ``29``, ``37``, ``48``], [``32``, ``33``, ``39``, ``50``]]` `    ``# Function call``    ``search(mat, ``4``, ``29``)` `# This code is contributed by rag2127`

## C#

 `// C# program to search an element in row-wise``// and column-wise sorted matrix``using` `System;` `class` `GFG {` `    ``/* Searches the element x in mat[][]. If the``    ``element is found, then prints its position``    ``and returns true, otherwise prints "not found"``    ``and returns false */``    ``static` `int` `search(``int``[, ] mat, ``int` `n, ``int` `x)``    ``{``        ``if` `(n == 0)``            ``return` `-1;` `        ``// Traverse through the matrix``        ``for` `(``int` `i = 0; i < n; i++) {``            ``for` `(``int` `j = 0; j < n; j++)` `                ``// If the element is found``                ``if` `(mat[i, j] == x) {``                    ``Console.Write(``"Element found at ("` `+ i``                                  ``+ ``", "` `+ j + ``")\n"``);``                    ``return` `1;``                ``}``        ``}``        ``Console.Write(``" Element not found"``);``        ``return` `0;``    ``}` `    ``// Driver code``    ``static` `public` `void` `Main()``    ``{``        ``int``[, ] mat = { { 10, 20, 30, 40 },``                        ``{ 15, 25, 35, 45 },``                        ``{ 27, 29, 37, 48 },``                        ``{ 32, 33, 39, 50 } };` `        ``// Function call``        ``search(mat, 4, 29);``    ``}``}` `// This code is contributed by avanitrachhadiya2155`

## Javascript

 ``

Output
```Element found at (2, 1)

```

Time Complexity: O(N2)
Auxiliary Space: O(1), since no extra space has been taken

Note: A better solution is to use Divide and Conquer to find the element which has a time complexity of O(n1.58). Please refer here for details

## Search in a row-wise and column-wise sorted matrix in linear time complexity:

The simple idea is to remove a row or column in each comparison until an element is found. Start searching from the top-right corner of the matrix. There are three possible cases:-

1. The given number is greater than the current number: This will ensure that all the elements in the current row are smaller than the given number as the pointer is already at the right-most elements and the row is sorted. Thus, the entire row gets eliminated and continues the search for the next row. Here, elimination means that a row needs not to be searched.
2. The given number is smaller than the current number: This will ensure that all the elements in the current column are greater than the given number. Thus, the entire column gets eliminated and continues the search for the previous column, i.e. the column on the immediate left.
3. The given number is equal to the current number: This will end the search.

Follow the given steps to solve the problem:

• Let the given element be x, create two variable i = 0, j = n-1 as index of row and column.
• Run a loop until i < n.
• Check if the current element is greater than x then decrease the count of j. Exclude the current column.
• Check if the current element is less than x then increase the count of i. Exclude the current row.
• If the element is equal, then print the position and end.

Thanks to devendraiiit for suggesting the approach below

Below is the implementation of the above approach:

## C++

 `// C++ program to search an element in row-wise``// and column-wise sorted matrix``#include ``using` `namespace` `std;` `/* Searches the element x in mat[][]. If the``element is found, then prints its position``and returns true, otherwise prints "not found"``and returns false */``int` `search(``int` `mat[4][4], ``int` `n, ``int` `x)``{``    ``if` `(n == 0)``        ``return` `-1;` `    ``int` `smallest = mat[0][0], largest = mat[n - 1][n - 1];``    ``if` `(x < smallest || x > largest)``        ``return` `-1;` `    ``// set indexes for top right element``    ``int` `i = 0, j = n - 1;``    ``while` `(i < n && j >= 0) {``        ``if` `(mat[i][j] == x) {``            ``cout << ``"Element found at "` `<< i << ``", "` `<< j;``            ``return` `1;``        ``}``        ``if` `(mat[i][j] > x)``            ``j--;` `        ``// Check if mat[i][j] < x``        ``else``            ``i++;``    ``}` `    ``cout << ``"n Element not found"``;``    ``return` `0;``}` `// Driver code``int` `main()``{``    ``int` `mat[4][4] = { { 10, 20, 30, 40 },``                      ``{ 15, 25, 35, 45 },``                      ``{ 27, 29, 37, 48 },``                      ``{ 32, 33, 39, 50 } };` `    ``// Function call``    ``search(mat, 4, 29);` `    ``return` `0;``}` `// This code is contributed``// by Akanksha Rai(Abby_akku)`

## C

 `// C program to search an element in row-wise``// and column-wise sorted matrix``#include ` `/* Searches the element x in mat[][]. If the``element is found, then prints its position``and returns true, otherwise prints "not found"``and returns false */``int` `search(``int` `mat[4][4], ``int` `n, ``int` `x)``{``    ``if` `(n == 0)``        ``return` `-1;``    ``int` `smallest = mat[0][0], largest = mat[n - 1][n - 1];``    ``if` `(x < smallest || x > largest)``        ``return` `-1;` `    ``// set indexes for top right element``    ``int` `i = 0, j = n - 1;``    ``while` `(i < n && j >= 0) {``        ``if` `(mat[i][j] == x) {``            ``printf``(``"Element found at %d, %d"``, i, j);``            ``return` `1;``        ``}``        ``if` `(mat[i][j] > x)``            ``j--;``        ``else` `// if mat[i][j] < x``            ``i++;``    ``}` `    ``printf``(``"n Element not found"``);``    ``return` `0; ``// if ( i==n || j== -1 )``}` `// Driver code``int` `main()``{``    ``int` `mat[4][4] = {``        ``{ 10, 20, 30, 40 },``        ``{ 15, 25, 35, 45 },``        ``{ 27, 29, 37, 48 },``        ``{ 32, 33, 39, 50 },``    ``};` `    ``// Function call``    ``search(mat, 4, 29);``    ``return` `0;``}`

## Java

 `// JAVA Code for Search in a row wise and``// column wise sorted matrix` `class` `GFG {` `    ``/* Searches the element x in mat[][]. If the``    ``element is found, then prints its position``    ``and returns true, otherwise prints "not found"``    ``and returns false */``    ``private` `static` `void` `search(``int``[][] mat, ``int` `n, ``int` `x)``    ``{` `        ``// set indexes for top right``        ``int` `i = ``0``, j = n - ``1``;``        ``// element` `        ``while` `(i < n && j >= ``0``) {``            ``if` `(mat[i][j] == x) {``                ``System.out.print(``"Element found at "` `+ i``                                 ``+ ``" "` `+ j);``                ``return``;``            ``}``            ``if` `(mat[i][j] > x)``                ``j--;``            ``else` `// if mat[i][j] < x``                ``i++;``        ``}` `        ``System.out.print(``"n Element not found"``);``        ``return``; ``// if ( i==n || j== -1 )``    ``}``    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `mat[][] = { { ``10``, ``20``, ``30``, ``40` `},``                        ``{ ``15``, ``25``, ``35``, ``45` `},``                        ``{ ``27``, ``29``, ``37``, ``48` `},``                        ``{ ``32``, ``33``, ``39``, ``50` `} };` `        ``// Function call``        ``search(mat, ``4``, ``29``);``    ``}``}``// This code is contributed by Arnav Kr. Mandal.`

## Python3

 `# Python3 program to search an element``# in row-wise and column-wise sorted matrix` `# Searches the element x in mat[][]. If the``# element is found, then prints its position``# and returns true, otherwise prints "not found"``# and returns false`  `def` `search(mat, n, x):` `    ``i ``=` `0` `    ``# set indexes for top right element``    ``j ``=` `n ``-` `1``    ``while` `(i < n ``and` `j >``=` `0``):` `        ``if` `(mat[i][j] ``=``=` `x):` `            ``print``(``"Element found at "``, i, ``", "``, j)``            ``return` `1` `        ``if` `(mat[i][j] > x):``            ``j ``-``=` `1` `        ``# if mat[i][j] < x``        ``else``:``            ``i ``+``=` `1` `    ``print``(``"Element not found"``)``    ``return` `0`  `# if (i == n || j == -1 )`  `# Driver Code``if` `__name__ ``=``=` `"__main__"``:``    ``mat ``=` `[[``10``, ``20``, ``30``, ``40``],``           ``[``15``, ``25``, ``35``, ``45``],``           ``[``27``, ``29``, ``37``, ``48``],``           ``[``32``, ``33``, ``39``, ``50``]]` `    ``# Function call``    ``search(mat, ``4``, ``29``)` `# This code is contributed by Anant Agarwal.`

## C#

 `// C# Code for Search in a row wise and``// column wise sorted matrix``using` `System;` `class` `GFG {``    ``/* Searches the element x in mat[][]. If the``    ``element is found, then prints its position``    ``and returns true, otherwise prints "not found"``    ``and returns false */``    ``private` `static` `void` `search(``int``[, ] mat, ``int` `n, ``int` `x)``    ``{``        ``// set indexes for top right``        ``// element``        ``int` `i = 0, j = n - 1;` `        ``while` `(i < n && j >= 0) {``            ``if` `(mat[i, j] == x) {``                ``Console.Write(``"Element found at "` `+ i + ``", "``                              ``+ j);``                ``return``;``            ``}` `            ``if` `(mat[i, j] > x)``                ``j--;``            ``else` `// if mat[i][j] < x``                ``i++;``        ``}` `        ``Console.Write(``"n Element not found"``);``        ``return``; ``// if ( i==n || j== -1 )``    ``}``    ``// Driver code``    ``public` `static` `void` `Main()``    ``{` `        ``int``[, ] mat = { { 10, 20, 30, 40 },``                        ``{ 15, 25, 35, 45 },``                        ``{ 27, 29, 37, 48 },``                        ``{ 32, 33, 39, 50 } };` `        ``// Function call``        ``search(mat, 4, 29);``    ``}``}` `// This code is contributed by Sam007`

## Javascript

 ``

## PHP

 `= 0)``    ``{``        ``if` `(``\$mat``[``\$i``][``\$j``] == ``\$x``)``        ``{``            ``echo` `"Element found at "` `. ``\$i``. ``                        ``", "` `. ``\$j``;``            ``return` `1;``        ``}``        ``if` `(``\$mat``[``\$i``][``\$j``] > ``\$x``)``            ``\$j``--;``        ``else` `// if \$mat[\$i][\$j] < \$x``            ``\$i``++;``    ``}``    ` `    ``echo` `"n Element not found"``;``    ``return` `0; ``// if ( \$i==\$n || \$j== -1 )``}` `// Driver Code``\$mat` `= ``array``(``array``(10, 20, 30, 40),``            ``array``(15, 25, 35, 45),``            ``array``(27, 29, 37, 48),``            ``array``(32, 33, 39, 50));` `// Function call``search(``\$mat``, 4, 29);` `// This code is contributed``// by ChitraNayal``?>`

Output
```Element found at 2, 1

```

Time Complexity: O(N), Only one traversal is needed, i.e, i from 0 to n and j from n-1 to 0 with at most 2*N steps. The above approach will also work for the M x N matrix (not only for N x N). Complexity would be O(M + N)
Auxiliary Space: O(1), No extra space is required

Related Article: Search element in a sorted matrix

Please write comments if you find the above codes/algorithms incorrect, or find other ways to solve the same problem.

### Using Linear Search:

Approach:

In this approach, we traverse the matrix row by row and check each element until we find the target element.

Define a function search_element that takes a matrix mat and a target element x as input.
Traverse the matrix mat row by row using a nested loop.
For each element in the matrix, check if it is equal to the target element x.
If the target element is found, return its position as a string in the format “Found at (i, j)”, where i and j are the row and column indices of the element, respectively.

## C++

 `#include ``#include ``using` `namespace` `std;` `string search_element(vector> mat, ``int` `x) {``    ``for` `(``int` `i = 0; i < mat.size(); i++) {``        ``for` `(``int` `j = 0; j < mat[0].size(); j++) {``            ``if` `(mat[i][j] == x) {``                ``return` `"Found at ("` `+ to_string(i) + ``", "` `+ to_string(j) + ``")"``;``            ``}``        ``}``    ``}``    ``return` `"Element not found"``;``}` `int` `main() {``    ``vector> mat = {{10, 20, 30, 40},``                               ``{15, 25, 35, 45},``                               ``{27, 29, 37, 48},``                               ``{32, 33, 39, 50}};` `    ``int` `x = 29;``    ``cout << search_element(mat, x) << endl; ``// Output: Found at (2, 1)` `    ``x = 100;``    ``cout << search_element(mat, x) << endl; ``// Output: Element not found` `    ``return` `0;``}`

## Java

 `// Java Code for the above approach``import` `java.util.ArrayList;``import` `java.util.List;` `class` `GFG {``    ``public` `static` `String search_element(List> mat, ``int` `x) {``        ``for` `(``int` `i = ``0``; i < mat.size(); i++) {``            ``for` `(``int` `j = ``0``; j < mat.get(``0``).size(); j++) {``                ``if` `(mat.get(i).get(j) == x) {``                    ``return` `"Found at ("` `+ i + ``", "` `+ j + ``")"``;``                ``}``            ``}``        ``}``        ``return` `"Element not found"``;``    ``}` `    ``public` `static` `void` `main(String[] args) {``        ``List> mat = ``new` `ArrayList<>();``        ``mat.add(List.of(``10``, ``20``, ``30``, ``40``));``        ``mat.add(List.of(``15``, ``25``, ``35``, ``45``));``        ``mat.add(List.of(``27``, ``29``, ``37``, ``48``));``        ``mat.add(List.of(``32``, ``33``, ``39``, ``50``));` `        ``int` `x = ``29``;``        ``System.out.println(search_element(mat, x)); ``// Output: Found at (2, 1)` `        ``x = ``100``;``        ``System.out.println(search_element(mat, x)); ``// Output: Element not found``    ``}``}``// This code is contributed by Kanchan Agarwal`

## Python3

 `def` `search_element(mat, x):``    ``for` `i ``in` `range``(``len``(mat)):``        ``for` `j ``in` `range``(``len``(mat[``0``])):``            ``if` `mat[i][j] ``=``=` `x:``                ``return` `f``"Found at ({i}, {j})"``    ``return` `"Element not found"``        ` `mat ``=` `[``    ``[``10``, ``20``, ``30``, ``40``],``    ``[``15``, ``25``, ``35``, ``45``],``    ``[``27``, ``29``, ``37``, ``48``],``    ``[``32``, ``33``, ``39``, ``50``]``]` `x ``=` `29``print``(search_element(mat, x))  ``# Output: Found at (2, 1)` `x ``=` `100``print``(search_element(mat, x))  ``# Output: Element not found`

## C#

 `using` `System;``using` `System.Collections.Generic;` `class` `GFG``{``    ``static` `string` `SearchElement(List> mat, ``int` `x)``    ``{``        ``for` `(``int` `i = 0; i < mat.Count; i++)``        ``{``            ``for` `(``int` `j = 0; j < mat[0].Count; j++)``            ``{``                ``if` `(mat[i][j] == x)``                ``{``                    ``return` `"Found at ("` `+ i + ``", "` `+ j + ``")"``;``                ``}``            ``}``        ``}``        ``return` `"Element not found"``;``    ``}` `    ``static` `void` `Main(``string``[] args)``    ``{``        ``List> mat = ``new` `List>``        ``{``            ``new` `List<``int``> { 10, 20, 30, 40 },``            ``new` `List<``int``> { 15, 25, 35, 45 },``            ``new` `List<``int``> { 27, 29, 37, 48 },``            ``new` `List<``int``> { 32, 33, 39, 50 }``        ``};` `        ``int` `x = 29;``        ``Console.WriteLine(SearchElement(mat, x)); ``// Output: Found at (2, 1)` `        ``x = 100;``        ``Console.WriteLine(SearchElement(mat, x)); ``// Output: Element not found``    ``}``}`

## Javascript

 `function` `search_element(mat, x) {``    ``for` `(let i = 0; i < mat.length; i++) {``        ``for` `(let j = 0; j < mat[0].length; j++) {``            ``if` `(mat[i][j] === x) {``                ``return` `"Found at ("` `+ i + ``", "` `+ j + ``")"``;``            ``}``        ``}``    ``}``    ``return` `"Element not found"``;``}` `// Driver Program to test above function``const mat = [``    ``[10, 20, 30, 40],``    ``[15, 25, 35, 45],``    ``[27, 29, 37, 48],``    ``[32, 33, 39, 50]``];` `let x = 29;``console.log(search_element(mat, x)); ``// Output: Found at (2, 1)` `x = 100;``console.log(search_element(mat, x)); ``// Output: Element not found` `// THIS CODE IS CONTRIBUTED BY KANCHAN AGARWAL`

Output
```Found at (2, 1)

```

Time Complexity: O(n^2)
Space Complexity: O(1)