Minimum steps to convert all paths in matrix from top left to bottom right as palindromic paths | Set 2

Given a matrix mat[][] with N rows and M columns. The task is to find the minimum number of changes required in the matrix such that every path from top left to bottom right is a palindromic path. In a path only right and bottom movements are allowed from one cell to another cell.

Examples:

Input: M = 2, N = 2, mat[M][N] = {{0, 0}, {0, 1}} 
Output: 1 
Explanation:
Change matrix[0][0] from 0 to 1. The two paths from (0, 0) to (1, 1) become palindromic.

Input: M = 3, N = 7, mat[M][N] = {{1, 2, 3, 4, 5, 6, 7}, {2, 2, 3, 3, 4, 3, 2}, {1, 2, 3, 2, 5, 6, 4}} 
Output: 10

Naive Approach: For the Naive Approach please refer to this article. 

Time Complexity: O(N^3) 
Auxiliary Space: O(N)

Efficient Approach:
The following observations have to be made: 

• A unique diagonal exists for every value (i+j) where i is the row index and j is the column index.
• The task reduces to selecting two diagonals, which are at the same distance from cell (0, 0) and cell (M – 1, N – 1) respectively and making all their elements equal to a single number, which is repeated most number of times in both the chosen diagonals.
• If the number of elements between cell (0, 0) and (M – 1, N – 1) are odd, then a common diagonal equidistant from both the cells exists. Elements of that diagonal need not be modified as they do not affect the palindromic arrangement of a path.
• If the numbers of cells between (0, 0) and (M – 1, N – 1) are even, no such common diagonal exists.

Follow the below steps to solve the problem:  

1. Create a 2D array frequency_diagonal that stores the frequency of all numbers in each chosen diagonal.
2. Each diagonal can be uniquely represented as the sum of (i, j).
3. Initialize a count variable that stores the count of the total number of cells where the values have to be replaced.
4. Iterate over the mat[][] and increment the frequency of current element in the diagonal ((i + j) value) where it belongs to.
5. Initialize a variable number_of_elements to 1, which stores the number of elements in each of the currently chosen pair of diagonals.
6. Initialize start = 0 and end = M + N – 2 and repeat the steps below until start < end: 
   • Find the frequency of the number which appears maximum times in the two selected diagonals that are equidistant from (0, 0) and (M-1, N-1).
   • Let the frequency in the above step be X. Add the value (total number of elements in the two diagonals – X) to count the minimum number of changes.
   • Increment start by 1 and decrement end by 1 and if the number of elements in the current diagonal is less than the maximum possible elements in any diagonal of the matrix, then increment number_of_elements by 1.
7. Print the value of total count after the above steps.

Below is the implementation of the above approach:

C++

// C++ program of the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to calculate the minimum
// number of replacements
int MinReplacements(
    int M, int N, int mat[][30])
{

    // 2D array to store frequency
    // of all the numbers in
    // each diagonal

    int frequency_diagonal[100][10005];

    // Initialise all the elements
    // of 2D array with 0
    memset(frequency_diagonal, 0,
           sizeof(frequency_diagonal));

    // Initialise answer as 0
    int answer = 0;

    // Iterate over the matrix
    for (int i = 0; i < M; i++) {
        for (int j = 0; j < N; j++) {

            // Update the frequency of
            // number mat[i][j]
            // for the diagonal
            // identified by (i+j)
            frequency_diagonal[i + j]
                              [(mat[i][j])]++;
        }
    }

    // Initialize start as 0
    // which indicates the
    // first diagonal
    int start = 0;

    // Initialize end as
    // M + N - 2 which indicates
    // the last diagonal
    int end = M + N - 2;

    // Number of elements in
    // the current diagonal
    int no_of_elemnts = 1;

    // Maximum possible number
    // of elements in a diagonal
    // can be minimum of (number of
    // rows and number of columns)
    int max_elements = min(M, N);

    while (start < end) {

        // The frequency of number
        // which occurs for the
        // maximum number of times
        // in the two selected
        // diagonals
        int X = INT_MIN;

        for (int i = 0; i <= 10000; i++) {
            X = max(
                X,
                frequency_diagonal[start][i]
                    + frequency_diagonal[end][i]);
        }

        answer = answer + (2 * (no_of_elemnts)) - X;

        // Increment start
        start++;
        // Decrement end
        end--;

        // Increment current number
        // of elements until it reaches
        // the maximum possible value
        if (no_of_elemnts < max_elements)
            no_of_elemnts++;
    }

    // return the final answer
    return answer;
}

// Driver Code
int main()
{
    // Number of rows
    int M = 3;

    // Number of columns
    int N = 7;

    int mat[30][30]
        = { { 1, 2, 3, 4, 5, 6, 7 },
            { 2, 2, 3, 3, 4, 3, 2 },
            { 1, 2, 3, 2, 5, 6, 4 } };

    cout << MinReplacements(M, N, mat)
         << endl;
}

Java

// Java program of the above approach
class GFG{

// Function to calculate the minimum
// number of replacements
static int MinReplacements(int M, int N,
                           int mat[][])
{

    // 2D array to store frequency
    // of all the numbers in
    // each diagonal
    int [][]frequency_diagonal = new int[100][10005];

    // Initialise answer as 0
    int answer = 0;

    // Iterate over the matrix
    for(int i = 0; i < M; i++)
    {
        for(int j = 0; j < N; j++)
        {

            // Update the frequency of
            // number mat[i][j]
            // for the diagonal
            // identified by (i+j)
            frequency_diagonal[i + j][(mat[i][j])]++;
        }
    }

    // Initialize start as 0
    // which indicates the
    // first diagonal
    int start = 0;

    // Initialize end as
    // M + N - 2 which indicates
    // the last diagonal
    int end = M + N - 2;

    // Number of elements in
    // the current diagonal
    int no_of_elemnts = 1;

    // Maximum possible number
    // of elements in a diagonal
    // can be minimum of (number of
    // rows and number of columns)
    int max_elements = Math.min(M, N);

    while (start < end)
    {

        // The frequency of number
        // which occurs for the
        // maximum number of times
        // in the two selected
        // diagonals
        int X = Integer.MIN_VALUE;

        for(int i = 0; i <= 10000; i++)
        {
            X = Math.max(X,
                frequency_diagonal[start][i] +
                frequency_diagonal[end][i]);
        }

        answer = answer + (2 * (no_of_elemnts)) - X;

        // Increment start
        start++;

        // Decrement end
        end--;

        // Increment current number
        // of elements until it reaches
        // the maximum possible value
        if (no_of_elemnts < max_elements)
            no_of_elemnts++;
    }

    // return the final answer
    return answer;
}

// Driver Code
public static void main(String[] args)
{

    // Number of rows
    int M = 3;

    // Number of columns
    int N = 7;

    int mat[][] = { { 1, 2, 3, 4, 5, 6, 7 },
                    { 2, 2, 3, 3, 4, 3, 2 },
                    { 1, 2, 3, 2, 5, 6, 4 } };

    System.out.print(MinReplacements(M, N, mat) + "\n");
}
}

// This code is contributed by 29AjayKumar

Python3

# Python3 program of the above approach
import sys

# Function to calculate the minimum
# number of replacements
def MinReplacements(M, N, mat):

    # 2D array to store frequency
    # of all the numbers in
    # each diagonal
    frequency_diagonal = [[0 for x in range(10005)]
                             for y in range (100)];

    # Initialise answer as 0
    answer = 0

    # Iterate over the matrix
    for i in range(M):
        for j in range(N):

            # Update the frequency of
            # number mat[i][j]
            # for the diagonal
            # identified by (i+j)
            frequency_diagonal[i + j][(mat[i][j])] += 1

    # Initialize start as 0
    # which indicates the
    # first diagonal
    start = 0

    # Initialize end as
    # M + N - 2 which indicates
    # the last diagonal
    end = M + N - 2

    # Number of elements in
    # the current diagonal
    no_of_elemnts = 1

    # Maximum possible number
    # of elements in a diagonal
    # can be minimum of (number of
    # rows and number of columns)
    max_elements = min(M, N)

    while (start < end):

        # The frequency of number
        # which occurs for the
        # maximum number of times
        # in the two selected
        # diagonals
        X = -sys.maxsize - 1

        for i in range(10001):
            X = max(X,
                    frequency_diagonal[start][i] +
                    frequency_diagonal[end][i])

        answer = answer + (2 * (no_of_elemnts)) - X

        # Increment start
        start += 1

        # Decrement end
        end -= 1

        # Increment current number
        # of elements until it reaches
        # the maximum possible value
        if (no_of_elemnts < max_elements):
            no_of_elemnts += 1

    # Return the final answer
    return answer

# Driver Code

# Number of rows
M = 3

# Number of columns
N = 7

mat = [ [ 1, 2, 3, 4, 5, 6, 7 ],
        [ 2, 2, 3, 3, 4, 3, 2 ],
        [ 1, 2, 3, 2, 5, 6, 4 ] ]

print(MinReplacements(M, N, mat))

# This code is contributed by chitranayal

C#

// C# program of the above approach
using System;

class GFG{

// Function to calculate the minimum
// number of replacements
static int MinReplacements(int M, int N,
                           int [,]mat)
{

    // 2D array to store frequency
    // of all the numbers in
    // each diagonal
    int [,]frequency_diagonal = new int[100, 10005];

    // Initialise answer as 0
    int answer = 0;

    // Iterate over the matrix
    for(int i = 0; i < M; i++)
    {
        for(int j = 0; j < N; j++)
        {

            // Update the frequency of
            // number mat[i,j]
            // for the diagonal
            // identified by (i+j)
            frequency_diagonal[i + j, mat[i, j]]++;
        }
    }

    // Initialize start as 0
    // which indicates the
    // first diagonal
    int start = 0;

    // Initialize end as
    // M + N - 2 which indicates
    // the last diagonal
    int end = M + N - 2;

    // Number of elements in
    // the current diagonal
    int no_of_elemnts = 1;

    // Maximum possible number
    // of elements in a diagonal
    // can be minimum of (number of
    // rows and number of columns)
    int max_elements = Math.Min(M, N);

    while (start < end)
    {

        // The frequency of number
        // which occurs for the
        // maximum number of times
        // in the two selected
        // diagonals
        int X = int.MinValue;

        for(int i = 0; i <= 10000; i++)
        {
            X = Math.Max(X,
                frequency_diagonal[start, i] +
                frequency_diagonal[end, i]);
        }

        answer = answer + (2 * (no_of_elemnts)) - X;

        // Increment start
        start++;

        // Decrement end
        end--;

        // Increment current number
        // of elements until it reaches
        // the maximum possible value
        if (no_of_elemnts < max_elements)
            no_of_elemnts++;
    }

    // Return the readonly answer
    return answer;
}

// Driver Code
public static void Main(String[] args)
{

    // Number of rows
    int M = 3;

    // Number of columns
    int N = 7;

    int [,]mat = { { 1, 2, 3, 4, 5, 6, 7 },
                   { 2, 2, 3, 3, 4, 3, 2 },
                   { 1, 2, 3, 2, 5, 6, 4 } };

    Console.Write(MinReplacements(M, N, mat) + "\n");
}
}

// This code is contributed by amal kumar choubey

Javascript

<script>

// Javascript program of the above approach

// Function to calculate the minimum
// number of replacements
function MinReplacements(M,N,mat)
{
    // 2D array to store frequency
    // of all the numbers in
    // each diagonal
    let frequency_diagonal = new Array(100);
      for(let i=0;i<100;i++)
    {
        frequency_diagonal[i]=new Array(10005);
        for(let j=0;j<10005;j++)
        {
            frequency_diagonal[i][j]=0;
        }
    }

    // Initialise answer as 0
    let answer = 0;

    // Iterate over the matrix
    for(let i = 0; i < M; i++)
    {
        for(let j = 0; j < N; j++)
        {

            // Update the frequency of
            // number mat[i][j]
            // for the diagonal
            // identified by (i+j)
            frequency_diagonal[i + j][(mat[i][j])]++;
        }
    }

    // Initialize start as 0
    // which indicates the
    // first diagonal
    let start = 0;

    // Initialize end as
    // M + N - 2 which indicates
    // the last diagonal
    let end = M + N - 2;

    // Number of elements in
    // the current diagonal
    let no_of_elemnts = 1;

    // Maximum possible number
    // of elements in a diagonal
    // can be minimum of (number of
    // rows and number of columns)
    let max_elements = Math.min(M, N);

    while (start < end)
    {

        // The frequency of number
        // which occurs for the
        // maximum number of times
        // in the two selected
        // diagonals
        let X = Number.MIN_VALUE;

        for(let i = 0; i <= 10000; i++)
        {
            X = Math.max(X,
                frequency_diagonal[start][i] +
                frequency_diagonal[end][i]);
        }

        answer = answer + (2 * (no_of_elemnts)) - X;

        // Increment start
        start++;

        // Decrement end
        end--;

        // Increment current number
        // of elements until it reaches
        // the maximum possible value
        if (no_of_elemnts < max_elements)
            no_of_elemnts++;
    }

    // return the final answer
    return answer;
}

// Driver Code
// Number of rows
    let M = 3;

    // Number of columns
    let N = 7;

    let mat = [[ 1, 2, 3, 4, 5, 6, 7 ],
                    [ 2, 2, 3, 3, 4, 3, 2 ],
                    [ 1, 2, 3, 2, 5, 6, 4 ]];

    document.write(MinReplacements(M, N, mat) + "<br>");

// This code is contributed by avanitrachhadiya2155
</script>

Output:

10

Time Complexity: O(M * N), The program iterates over the matrix using two nested loops, so the time complexity of the program is O(M*N), where M is the number of rows and N is the number of columns of the matrix. In addition, it has a loop that iterates over the frequency array, which has a fixed size of 10000, so it can be considered as a constant time operation. Therefore, the overall time complexity of the program is O(M*N).
Auxiliary Space: O(M * N), The space complexity of the program is O(M*N), as the program uses a 2D array of size 100×10005 to store the frequency of each element in each diagonal. Therefore, the space complexity of the program is also linear in terms of the size of the input.