# Minimum changes required to make each path in a matrix palindrome

Given a matrix with N rows and M columns, the task is make all possible paths from the cell (N, M) to (1, 1) palindrome by minimum changes in the cell values.

Possible moves from any cell (x, y) is either move Left(x – 1, y) or move Down (x, y – 1).

Examples:

Input: mat[ ][ ] = { { 1, 2, 2 }, { 1, 0, 0 } }
Output:
Explanation:
For each path in matrix to be Palindrome, possible matrices (after changes) are
{ { 0, 2, 2 }, { 2, 2, 0 } } or { { 1, 2, 2 }, { 2, 2, 1 } }.
Input: mat[ ][ ] = { { 5, 3 }, { 0, 5 } }
Output:
Explanation:
No change required in above matrix. Each path from (N, M) to (1, 1) is already Palindrome

Approach:

• A path is called palindromic if the value of the last cell is equal to the value of the first cell, the value of the second last cell is equal to the value of the second cell, and so on.

• So we can conclude that, to make a path palindromic, the cells at distance (Manhatten distance) x from (N, M) must be equal to the cells at distance x from (1, 1)

• To minimize the number of changes, convert each cell at distance x from (1, 1) and (N, M) to the most frequent among all values present in those cells.

Below is the implementation of the above approach.

## C++

 `// C++ Program to implement the` `// above approach` `#include ` `using` `namespace` `std;`   `#define N 7`   `// Function for counting changes` `int` `countChanges(``int` `matrix[][N],` `                 ``int` `n, ``int` `m)` `{` `    ``// Maximum distance possible` `    ``// is (n - 1 + m - 1)` `    ``int` `dist = n + m - 1;`   `    ``// Stores the maximum element` `    ``int` `Max_element = 0;` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``for` `(``int` `j = 0; j < m; j++) {` `            ``// Update the maximum` `            ``Max_element = max(Max_element,` `                              ``matrix[i][j]);` `        ``}` `    ``}`   `    ``// Stores frequencies of` `    ``// values for respective` `    ``// distances` `    ``int` `freq[dist][Max_element + 1];`   `    ``// Initialize frequencies of` `    ``// cells as 0` `    ``for` `(``int` `i = 0; i < dist; i++) {` `        ``for` `(``int` `j = 0; j < Max_element + 1;` `             ``j++)` `            ``freq[i][j] = 0;` `    ``}`   `    ``// Count frequencies of cell` `    ``// values in the matrix` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``for` `(``int` `j = 0; j < m; j++) {`   `            ``// Increment frequency of` `            ``// value at distance i+j` `            ``freq[i + j][matrix[i][j]]++;` `        ``}` `    ``}`   `    ``int` `min_changes_sum = 0;` `    ``for` `(``int` `i = 0; i < dist / 2; i++) {`   `        ``// Store the most frequent` `        ``// value at i-th distance` `        ``// from (0, 0) and (N - 1, M - 1)` `        ``int` `maximum = 0;` `        ``int` `total_values = 0;`   `        ``// Calculate max frequency` `        ``// and total cells at distance i` `        ``for` `(``int` `j = 0; j < Max_element + 1;` `             ``j++) {`   `            ``maximum` `                ``= max(maximum,` `                      ``freq[i][j]` `                          ``+ freq[n + m - 2` `                                 ``- i][j]);`   `            ``total_values` `                ``+= freq[i][j]` `                   ``+ freq[n + m - 2 - i][j];` `        ``}`   `        ``// Count changes required` `        ``// to convert all cells` `        ``// at i-th distance to` `        ``// most frequent value` `        ``min_changes_sum` `            ``+= total_values - maximum;` `    ``}`   `    ``return` `min_changes_sum;` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `mat[][N] = { { 7, 0, 3, 1, 8, 1, 3 },` `                     ``{ 0, 4, 0, 1, 0, 4, 0 },` `                     ``{ 3, 1, 8, 3, 1, 0, 7 } };`   `    ``int` `minChanges = countChanges(mat, 3, 7);`   `    ``cout << minChanges;`   `    ``return` `0;` `}`

## Java

 `// Java program to implement the` `// above approach` `import` `java.util.*;`   `class` `GFG{` `static` `int` `N = ``7``;`   `// Function for counting changes` `static` `int` `countChanges(``int` `matrix[][],` `                        ``int` `n, ``int` `m)` `{` `    `  `    ``// Maximum distance possible` `    ``// is (n - 1 + m - 1)` `    ``int` `i, j, dist = n + m - ``1``;`   `    ``// Stores the maximum element` `    ``int` `Max_element = ``0``;` `    ``for``(i = ``0``; i < n; i++) ` `    ``{` `        ``for``(j = ``0``; j < m; j++)` `        ``{` `            `  `            ``// Update the maximum` `            ``Max_element = Math.max(Max_element,` `                                   ``matrix[i][j]);` `        ``}` `    ``}`   `    ``// Stores frequencies of` `    ``// values for respective` `    ``// distances` `    ``int` `freq[][] = ``new` `int``[dist][Max_element + ``1``];`   `    ``// Initialize frequencies of` `    ``// cells as 0` `    ``for``(i = ``0``; i < dist; i++) ` `    ``{` `        ``for``(j = ``0``; j < Max_element + ``1``;` `            ``j++)` `            ``freq[i][j] = ``0``;` `    ``}`   `    ``// Count frequencies of cell` `    ``// values in the matrix` `    ``for``(i = ``0``; i < n; i++)` `    ``{` `        ``for``(j = ``0``; j < m; j++)` `        ``{` `            `  `            ``// Increment frequency of` `            ``// value at distance i+j` `            ``freq[i + j][matrix[i][j]]++;` `        ``}` `    ``}` `    `  `    ``int` `min_changes_sum = ``0``;` `    ``for``(i = ``0``; i < dist / ``2``; i++)` `    ``{` `        `  `        ``// Store the most frequent` `        ``// value at i-th distance` `        ``// from (0, 0) and (N - 1, M - 1)` `        ``int` `maximum = ``0``;` `        ``int` `total_values = ``0``;`   `        ``// Calculate max frequency` `        ``// and total cells at distance i` `        ``for``(j = ``0``; j < Max_element + ``1``; j++)` `        ``{` `            ``maximum = Math.max(maximum,` `                               ``freq[i][j] +` `                               ``freq[n + m -` `                                    ``2` `- i][j]);` `            ``total_values += freq[i][j] + ` `                            ``freq[n + m -` `                                 ``2` `- i][j];` `        ``}`   `        ``// Count changes required` `        ``// to convert all cells` `        ``// at i-th distance to` `        ``// most frequent value` `        ``min_changes_sum += total_values -` `                           ``maximum;` `    ``}` `    ``return` `min_changes_sum;` `}`   `// Driver Code` `public` `static` `void` `main (String []args)` `{` `    ``int` `mat[][] = { { ``7``, ``0``, ``3``, ``1``, ``8``, ``1``, ``3` `},` `                    ``{ ``0``, ``4``, ``0``, ``1``, ``0``, ``4``, ``0` `},` `                    ``{ ``3``, ``1``, ``8``, ``3``, ``1``, ``0``, ``7` `} };`   `    ``int` `minChanges = countChanges(mat, ``3``, ``7``);`   `    ``System.out.print(minChanges);` `}` `}`   `// This code is contributed by chitranayal`

## Python3

 `# Python3 program to implement the ` `# above approach `   `# Function for counting changes` `def` `countChanges(matrix, n, m):`   `    ``# Maximum distance possible` `    ``# is (n - 1 + m - 1)` `    ``dist ``=` `n ``+` `m ``-` `1`   `    ``# Stores the maximum element` `    ``Max_element ``=` `0` `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `range``(m):` `            `  `            ``# Update the maximum` `            ``Max_element ``=` `max``(Max_element,` `                              ``matrix[i][j])`   `    ``# Stores frequencies of` `    ``# values for respective` `    ``# distances` `    ``freq ``=` `[[``0` `for` `i ``in` `range``(Max_element ``+` `1``)]` `               ``for` `j ``in` `range``(dist)]`   `    ``# Initialize frequencies of` `    ``# cells as 0` `    ``for` `i ``in` `range``(dist):` `        ``for` `j ``in` `range``(Max_element ``+` `1``):` `            ``freq[i][j] ``=` `0`   `    ``# Count frequencies of cell` `    ``# values in the matrix` `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `range``(m):`   `            ``# Increment frequency of` `            ``# value at distance i+j` `            ``freq[i ``+` `j][matrix[i][j]] ``+``=` `1`   `    ``min_changes_sum ``=` `0` `    ``for` `i ``in` `range``(dist ``/``/` `2``):`   `        ``# Store the most frequent` `        ``# value at i-th distance` `        ``# from (0, 0) and (N - 1, M - 1)` `        ``maximum ``=` `0` `        ``total_values ``=` `0`   `        ``# Calculate max frequency` `        ``# and total cells at distance i` `        ``for` `j ``in` `range``(Max_element ``+` `1``):` `            ``maximum ``=` `max``(maximum,` `                          ``freq[i][j] ``+` `                          ``freq[n ``+` `m ``-` `2` `-` `i][j])`   `            ``total_values ``+``=` `(freq[i][j] ``+` `                             ``freq[n ``+` `m ``-` `2` `-` `i][j])`   `        ``# Count changes required` `        ``# to convert all cells` `        ``# at i-th distance to` `        ``# most frequent value ` `        ``min_changes_sum ``+``=` `total_values ``-` `maximum`   `    ``return` `min_changes_sum`   `# Driver code` `if` `__name__ ``=``=` `'__main__'``:`   `    ``mat ``=` `[ [ ``7``, ``0``, ``3``, ``1``, ``8``, ``1``, ``3` `],` `            ``[ ``0``, ``4``, ``0``, ``1``, ``0``, ``4``, ``0` `],` `            ``[ ``3``, ``1``, ``8``, ``3``, ``1``, ``0``, ``7` `] ]`   `    ``minChanges ``=` `countChanges(mat, ``3``, ``7``)`   `    ``print``(minChanges)`   `# This code is contributed by Shivam Singh`

## C#

 `// C# program to implement the` `// above approach` `using` `System;` `class` `GFG{` `  `  `//static int N = 7;`   `// Function for counting changes` `static` `int` `countChanges(``int` `[,]matrix,` `                        ``int` `n, ``int` `m)` `{` `    `  `    ``// Maximum distance possible` `    ``// is (n - 1 + m - 1)` `    ``int` `i, j, dist = n + m - 1;`   `    ``// Stores the maximum element` `    ``int` `Max_element = 0;` `    ``for``(i = 0; i < n; i++) ` `    ``{` `        ``for``(j = 0; j < m; j++)` `        ``{` `            `  `            ``// Update the maximum` `            ``Max_element = Math.Max(Max_element,` `                                   ``matrix[i, j]);` `        ``}` `    ``}`   `    ``// Stores frequencies of` `    ``// values for respective` `    ``// distances` `    ``int` `[,]freq = ``new` `int``[dist, Max_element + 1];`   `    ``// Initialize frequencies of` `    ``// cells as 0` `    ``for``(i = 0; i < dist; i++) ` `    ``{` `        ``for``(j = 0; j < Max_element + 1;` `            ``j++)` `            ``freq[i, j] = 0;` `    ``}`   `    ``// Count frequencies of cell` `    ``// values in the matrix` `    ``for``(i = 0; i < n; i++)` `    ``{` `        ``for``(j = 0; j < m; j++)` `        ``{` `            `  `            ``// Increment frequency of` `            ``// value at distance i+j` `            ``freq[i + j, matrix[i, j]]++;` `        ``}` `    ``}` `    `  `    ``int` `min_changes_sum = 0;` `    ``for``(i = 0; i < dist / 2; i++)` `    ``{` `        `  `        ``// Store the most frequent` `        ``// value at i-th distance` `        ``// from (0, 0) and (N - 1, M - 1)` `        ``int` `maximum = 0;` `        ``int` `total_values = 0;`   `        ``// Calculate max frequency` `        ``// and total cells at distance i` `        ``for``(j = 0; j < Max_element + 1; j++)` `        ``{` `            ``maximum = Math.Max(maximum,` `                               ``freq[i, j] +` `                               ``freq[n + m -` `                                    ``2 - i, j]);` `            ``total_values += freq[i, j] + ` `                            ``freq[n + m -` `                                 ``2 - i, j];` `        ``}`   `        ``// Count changes required` `        ``// to convert all cells` `        ``// at i-th distance to` `        ``// most frequent value` `        ``min_changes_sum += total_values -` `                           ``maximum;` `    ``}` `    ``return` `min_changes_sum;` `}`   `// Driver Code` `public` `static` `void` `Main(String []args)` `{` `    ``int` `[,]mat = { { 7, 0, 3, 1, 8, 1, 3 },` `                    ``{ 0, 4, 0, 1, 0, 4, 0 },` `                    ``{ 3, 1, 8, 3, 1, 0, 7 } };`   `    ``int` `minChanges = countChanges(mat, 3, 7);`   `    ``Console.Write(minChanges);` `}` `}`   `// This code is contributed by sapnasingh4991`

Output:

```6

```

Time Complexity: O(N * M)
Auxiliary Space: O((N + M)*maxm), where maxm is the maximum element present in the matrix.

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