Divide Matrix into K groups of adjacent cells having minimum difference between maximum and minimum sized groups

Given N rows and M columns of a matrix, the task is to fill the matrix cells using first K integers such that:

• Each group of cells is denoted by a single integer.
• The difference between the group containing the maximum and the minimum number of cells should be minimum.
• All the cells of the same group should be contiguous i.e., for any group, two adjacent cells should follow the rule |x_i+1 – x_i| + |y_i+1 – y_i| = 1.

Examples:

Input: N = 5, M = 5, K = 6
Output:
1 1 1 1 1
3 2 2 2 2
3 3 3 4 4
5 5 5 4 4
5 6 6 6 6
Explanation:
The above matrix follows all the conditions above and dividing the matrix into K different groups.

Input: N = 2, M = 3, K = 3
Output:
1 1 2
3 3 2
Explanation:
For making three group of the matrix each should have the group of size two.
So, to reduce the difference between the group containing maximum and minimum no of cells and all the matrix cells are used to make the K different groups having all the adjacent elements of the same group follow the |x_{i + 1} – x_i| + |y_{i + 1} – y_i| = 1 as well.

Approach: Below are the steps:

• Create the matrix of size N * M.
• To reduce the difference between group containing max and min no of cells, fill all the part with at least (N*M)/ K cells.
• The remaining part will contain (N * M)/ K + 1 no of cells.
• To follow the given rules, traverse the matrix and fill the matrix with the different parts accordingly.
• Print the matrix after the above steps.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

// Function to fill the matrix with
// the given conditions
void fillMatrix(int** mat, int& row,
                int& col, int sizeOfpart,
                int noOfPart, int& start,
                int m, int& flag)
{

    // Count of parts with size sizeOfPart
    for (int i = 0; i < noOfPart; ++i) {

        int count = 0;

        while (count < sizeOfpart) {

            // Assigning the cell
            // with no of groups
            mat[row][col] = start;

            // Update row
            if (col == m - 1
                && flag == 1) {
                row++;
                col = m;
                flag = 0;
            }
            else if (col == 0
                     && flag == 0) {
                row++;
                col = -1;
                flag = 1;
            }

            // Update col
            if (flag == 1) {
                col++;
            }
            else {
                col--;
            }

            // Increment count
            count++;
        }

        // For new group increment start
        start++;
    }
}

// Function to return the reference of
// the matrix to be filled
int** findMatrix(int N, int M, int k)
{

    // Create matrix of size N*M
    int** mat = (int**)malloc(
        N * sizeof(int*));

    for (int i = 0; i < N; ++i) {
        mat[i] = (int*)malloc(
            M * sizeof(int));
    }

    // Starting index of the matrix
    int row = 0, col = 0;

    // Size of one group
    int size = (N * M) / k;
    int rem = (N * M) % k;

    // Element to assigned to matrix
    int start = 1, flag = 1;

    // Fill the matrix that have rem
    // no of parts with size size + 1
    fillMatrix(mat, row, col, size + 1,
               rem, start, M, flag);

    // Fill the remaining number of parts
    // with each part size is 'size'
    fillMatrix(mat, row, col, size,
               k - rem, start, M, flag);

    // Return the matrix
    return mat;
}

// Function to print the matrix
void printMatrix(int** mat, int N,
                 int M)
{
    // Traverse the rows
    for (int i = 0; i < N; ++i) {

        // Traverse the columns
        for (int j = 0; j < M; ++j) {
            cout << mat[i][j] << " ";
        }
        cout << endl;
    }
}

// Driver Code
int main()
{
    // Given N, M, K
    int N = 5, M = 5, K = 6;

    // Function Call
    int** mat = findMatrix(N, M, K);

    // Function Call to print matrix
    printMatrix(mat, N, M);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;

class GFG{

static int N, M;
static int [][]mat;
static int row, col, start, flag;

// Function to fill the matrix with
// the given conditions
static void fillMatrix(int sizeOfpart,
                       int noOfPart,
                       int m)
{

    // Count of parts with size sizeOfPart
    for(int i = 0; i < noOfPart; ++i)
    {
        int count = 0;

        while (count < sizeOfpart)
        {

            // Assigning the cell
            // with no of groups
            mat[row][col] = start;

            // Update row
            if (col == m - 1 && flag == 1)
            {
                row++;
                col = m;
                flag = 0;
            }
            else if (col == 0 && flag == 0)
            {
                row++;
                col = -1;
                flag = 1;
            }

            // Update col
            if (flag == 1)
            {
                col++;
            }
            else
            {
                col--;
            }

            // Increment count
            count++;
        }

        // For new group increment start
        start++;
    }
}

// Function to return the reference of
// the matrix to be filled
static void findMatrix(int k)
{

    // Create matrix of size N*M
    mat = new int[M][N];

    // Starting index of the matrix
    row = 0;
    col = 0;

    // Size of one group
    int size = (N * M) / k;
    int rem = (N * M) % k;

    // Element to assigned to matrix
    start = 1;
    flag = 1;

    // Fill the matrix that have rem
    // no of parts with size size + 1
    fillMatrix(size + 1, rem, M);

    // Fill the remaining number of parts
    // with each part size is 'size'
    fillMatrix(size, k - rem, M);
}

// Function to print the matrix
static void printMatrix()
{

    // Traverse the rows
    for(int i = 0; i < N; ++i)
    {

        // Traverse the columns
        for(int j = 0; j < M; ++j)
        {
            System.out.print(mat[i][j] + " ");
        }
        System.out.println();
    }
}

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

    // Given N, M, K
    N = 5;
    M = 5;
    int K = 6;

    // Function Call
    findMatrix(K);

    // Function Call to print matrix
    printMatrix();
}
}

// This code is contributed by Amit Katiyar

C#

// C# program for the
// above approach
using System;
class GFG{

static int N, M;
static int [,]mat;
static int row, col,
           start, flag;

// Function to fill the
// matrix with the given
// conditions
static void fillMatrix(int sizeOfpart,
                       int noOfPart,
                       int m)
{
  // Count of parts with size
  // sizeOfPart
  for(int i = 0;
          i < noOfPart; ++i)
  {
    int count = 0;

    while (count < sizeOfpart)
    {
      // Assigning the cell
      // with no of groups
      mat[row, col] = start;

      // Update row
      if (col == m - 1 &&
          flag == 1)
      {
        row++;
        col = m;
        flag = 0;
      }
      else if (col == 0 &&
               flag == 0)
      {
        row++;
        col = -1;
        flag = 1;
      }

      // Update col
      if (flag == 1)
      {
        col++;
      }
      else
      {
        col--;
      }

      // Increment count
      count++;
    }

    // For new group increment
    // start
    start++;
  }
}

// Function to return the
// reference of the matrix
// to be filled
static void findMatrix(int k)
{
  // Create matrix of
  // size N*M
  mat = new int[M, N];

  // Starting index of the
  // matrix
  row = 0;
  col = 0;

  // Size of one group
  int size = (N * M) / k;
  int rem = (N * M) % k;

  // Element to assigned to
  // matrix
  start = 1;
  flag = 1;

  // Fill the matrix that have
  // rem no of parts with size
  // size + 1
  fillMatrix(size + 1,
             rem, M);

  // Fill the remaining number
  // of parts with each part
  // size is 'size'
  fillMatrix(size, k - rem, M);
}

// Function to print the
// matrix
static void printMatrix()
{
  // Traverse the rows
  for(int i = 0; i < N; ++i)
  {
    // Traverse the columns
    for(int j = 0; j < M; ++j)
    {
      Console.Write(mat[i, j] +
                    " ");
    }
    Console.WriteLine();
  }
}

// Driver Code
public static void Main(String[] args)
{
  // Given N, M, K
  N = 5;
  M = 5;
  int K = 6;

  // Function Call
  findMatrix(K);

  // Function Call to
  // print matrix
  printMatrix();
}
}

// This code is contributed by 29AjayKumar

Output:

1 1 1 1 1 
3 2 2 2 2 
3 3 3 4 4 
5 5 5 4 4 
5 6 6 6 6

Time Complexity: O(N * M)
Auxiliary Space: O(N * M)