# Check whether a Matrix is a Latin Square or not

Given a square matrix of size N x N, the task is to check if it is Latin square or not.

A square matrix is a Latin Square if each cell of the matrix contains one of N different values (in the range [1, N]), and no value is repeated within a row or a column.

Examples:

```Input: 1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
Output: YES

Input: 2 2 2 2
2 3 2 3
2 2 2 3
2 2 2 2
Output: NO

```

Naive Approach:

1. For every element, we first check whether the given element is already present in the given row and given column by iterating over all the elements of the given row and given column.
2. If not, then check whether the value is less than or equal to N, if yes, move for the next element.
3. If any of the above points are false, then the matrix is not a Latin square.

Efficient Approach: Here is the more efficient approach using a Set data structure in C++:

1. Define sets for each row and each column and create two arrays of sets, one for all the rows and the other for columns.
2. Iterate over all the elements and insert the value of the given element in the corresponding row set and in the corresponding column set.
3. Also, check whether the given value is less than N or not. If not, Print “NO” and return.
4. Now, Iterate over all row sets and column sets and check if the size of the set is less than N or not.
5. If Yes, Print “YES”. Otherwise, Print “NO”.

Below is the implementation of the above approach.

 `// C++ program to check if given matrix` `// is natural latin square or not`   `#include ` `using` `namespace` `std;`   `void` `CheckLatinSquare(``int` `mat[4][4])` `{` `    ``// Size of square matrix is NxN` `    ``int` `N = ``sizeof``(mat[0]) / ``sizeof``(mat[0][0]);`   `    ``// Vector of N sets corresponding` `    ``// to each row.` `    ``vector > rows(N);`   `    ``// Vector of N sets corresponding` `    ``// to each column.` `    ``vector > cols(N);`   `    ``// Number of invalid elements` `    ``int` `invalid = 0;`   `    ``for` `(``int` `i = 0; i < N; i++) {` `        ``for` `(``int` `j = 0; j < N; j++) {` `            ``rows[i].insert(mat[i][j]);` `            ``cols[j].insert(mat[i][j]);`   `            ``if` `(mat[i][j] > N || mat[i][j] <= 0) {` `                ``invalid++;` `            ``}` `        ``}` `    ``}` `    ``// Number of rows with` `    ``// repeatative elements.` `    ``int` `numrows = 0;`   `    ``// Number of columns with` `    ``// repeatative elements.` `    ``int` `numcols = 0;`   `    ``// Checking size of every row` `    ``// and column` `    ``for` `(``int` `i = 0; i < N; i++) {` `        ``if` `(rows[i].size() != N) {` `            ``numrows++;` `        ``}` `        ``if` `(cols[i].size() != N) {` `            ``numcols++;` `        ``}` `    ``}`   `    ``if` `(numcols == 0 && numrows == 0` `        ``&& invalid == 0)` `        ``cout << ``"YES"` `<< endl;` `    ``else` `        ``cout << ``"NO"` `<< endl;`   `    ``return``;` `}`   `// Driver code` `int` `main()` `{`   `    ``int` `Matrix[4][4] = { { 1, 2, 3, 4 },` `                         ``{ 2, 1, 4, 3 },` `                         ``{ 3, 4, 1, 2 },` `                         ``{ 4, 3, 2, 1 } };`   `    ``// Funtion call` `    ``CheckLatinSquare(Matrix);`   `    ``return` `0;` `}`

 `// Java program to check if given matrix` `// is natural latin square or not` `import` `java.util.*;`   `class` `GFG{` `    `  `@SuppressWarnings``(``"unchecked"``)` `static` `void` `CheckLatinSquare(``int` `mat[][])` `{` `    `  `    ``// Size of square matrix is NxN` `    ``int` `N = mat.length;` `    `  `    ``// Vector of N sets corresponding` `    ``// to each row.` `    ``HashSet[] rows = ``new` `HashSet[N];` `    `  `    ``// Vector of N sets corresponding` `    ``// to each column.` `    ``HashSet[] cols = ``new` `HashSet[N];` `    `  `    ``for``(``int` `i = ``0``; i < N; i++)` `    ``{` `        ``rows[i] = ``new` `HashSet();` `        ``cols[i] = ``new` `HashSet();` `    ``}` `    `  `    ``// Number of invalid elements` `    ``int` `invalid = ``0``;` `    `  `    ``for``(``int` `i = ``0``; i < N; i++)` `    ``{` `        ``for``(``int` `j = ``0``; j < N; j++)` `        ``{` `            ``rows[i].add(mat[i][j]);` `            ``cols[j].add(mat[i][j]);` `    `  `            ``if` `(mat[i][j] > N || mat[i][j] <= ``0``)` `            ``{` `                ``invalid++;` `            ``}` `        ``}` `    ``}` `    `  `    ``// Number of rows with` `    ``// repeatative elements.` `    ``int` `numrows = ``0``;` `    `  `    ``// Number of columns with` `    ``// repeatative elements.` `    ``int` `numcols = ``0``;` `    `  `    ``// Checking size of every row` `    ``// and column` `    ``for``(``int` `i = ``0``; i < N; i++)` `    ``{` `        ``if` `(rows[i].size() != N)` `        ``{` `            ``numrows++;` `        ``}` `        ``if` `(cols[i].size() != N)` `        ``{` `            ``numcols++;` `        ``}` `    ``}` `    `  `    ``if` `(numcols == ``0` `&&` `        ``numrows == ``0` `&& invalid == ``0``)` `        ``System.out.print(``"YES"` `+ ``"\n"``);` `    ``else` `        ``System.out.print(``"NO"` `+ ``"\n"``);` `    `  `    ``return``;` `}` `    `  `// Driver code` `public` `static` `void` `main(String[] args)` `{` `    `  `    ``int` `Matrix[][] = { { ``1``, ``2``, ``3``, ``4` `},` `                       ``{ ``2``, ``1``, ``4``, ``3` `},` `                       ``{ ``3``, ``4``, ``1``, ``2` `},` `                       ``{ ``4``, ``3``, ``2``, ``1` `} };` `    `  `    ``// Funtion call` `    ``CheckLatinSquare(Matrix);` `}` `}`   `// This code is contributed by 29AjayKumar`

 `// C# program to check if given matrix` `// is natural latin square or not` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG{` `    ``static` `void` `CheckLatinSquare(``int``[, ] mat)` `    ``{`   `        ``// Size of square matrix is NxN` `        ``int` `N = mat.GetLength(0);`   `        ``// List of N sets corresponding` `        ``// to each row.` `        ``HashSet<``int``>[] rows = ``new` `HashSet<``int``>[ N ];`   `        ``// List of N sets corresponding` `        ``// to each column.` `        ``HashSet<``int``>[] cols = ``new` `HashSet<``int``>[ N ];`   `        ``for` `(``int` `i = 0; i < N; i++) ` `        ``{` `            ``rows[i] = ``new` `HashSet<``int``>();` `            ``cols[i] = ``new` `HashSet<``int``>();` `        ``}`   `        ``// Number of invalid elements` `        ``int` `invalid = 0;`   `        ``for` `(``int` `i = 0; i < N; i++) ` `        ``{` `            ``for` `(``int` `j = 0; j < N; j++) ` `            ``{` `                ``rows[i].Add(mat[i, j]);` `                ``cols[j].Add(mat[i, j]);`   `                ``if` `(mat[i, j] > N || mat[i, j] <= 0) ` `                ``{` `                    ``invalid++;` `                ``}` `            ``}` `        ``}`   `        ``// Number of rows with` `        ``// repeatative elements.` `        ``int` `numrows = 0;`   `        ``// Number of columns with` `        ``// repeatative elements.` `        ``int` `numcols = 0;`   `        ``// Checking size of every row` `        ``// and column` `        ``for` `(``int` `i = 0; i < N; i++) ` `        ``{` `            ``if` `(rows[i].Count != N) ` `            ``{` `                ``numrows++;` `            ``}` `            ``if` `(cols[i].Count != N) ` `            ``{` `                ``numcols++;` `            ``}` `        ``}` `      `  `        ``if` `(numcols == 0 && numrows == 0 && invalid == 0)` `            ``Console.Write(``"YES"` `+ ``"\n"``);` `        ``else` `            ``Console.Write(``"NO"` `+ ``"\n"``);` `        ``return``;` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `Main(String[] args)` `    ``{` `        ``int``[, ] Matrix = {{1, 2, 3, 4},` `                          ``{2, 1, 4, 3},` `                          ``{3, 4, 1, 2},` `                          ``{4, 3, 2, 1}};`   `        ``// Funtion call` `        ``CheckLatinSquare(Matrix);` `    ``}` `}`   `// This code is contributed by Amit Katiyar`

Output:
```YES

```

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