Minimum cost to sort a matrix of numbers from 0 to n^2 – 1

Given an n x n matrix containing all the numbers in the range 0 to n2-1. The problem is to calculate the total energy required for rearranging all the numbers in the matrix in strictly increasing order, i.e., after the rearrangement the 1st row contains ‘n’ numbers from 0 to n-1, then 2nd row from n to 2n-1 and so on up to the last or nth row. A number can be moved in any of the four directions left, right, top or bottom from its current position to reach its destination in the final modified matrix. The number of steps moved in transferring a number from its current location to its required destination is the energy required by the number for its movement. For example, in a 4 x 4 matrix, number ‘6’ is present at location (2, 3). Its destination location in the modified matrix is (1, 1). So ‘6’ is moved 2 steps towards left and 1 step up to reach location (1, 1). Total 3 steps moved and thus energy required by ‘6’ is 3 units. In this way we have to sum up all the energies required in the movement / rearrangement of all the numbers.

Examples :

Input : mat[][] = { {3, 0},
                    {2, 1} }
Output : 4 units
For number '3':
Move it one step right and one step down.
Total 2 steps thus energy required = 2 units.

For number '0':
Move it one step left.
Total 1 step thus energy required = 1 unit.

For number '1':
Move it one step up.
Total 1 step thus energy required = 1 unit.

Total energy required = 4 units.

Input : mat[][] = { {4, 7, 10, 3},
                {8, 5, 6, 1},
                  {9, 11, 10, 2},
            {15, 13, 14, 12} }                      
Output : 22 units



Algorithm:

calculateEnergy(mat, n)
    Declare i_des, j_des, q
    Initialize tot_energy = 0
    for i = 0 to n-1
        for j = 0 to n-1
        q = mat[i][j] / n
        i_des = q
        j_des = mat[i][j] - (n * q)
        tot_energy += abs(i_des - i) + abs(j_des - j)    
    return tot_energy

C++

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// C++ implementation to find the total energy
// required to rearrange the numbers
#include <bits/stdc++.h>
  
using namespace std;
  
#define SIZE 100
  
// function to find the total energy
// required to rearrange the numbers
int calculateEnergy(int mat[SIZE][SIZE], int n)
{
    int i_des, j_des, q;
    int tot_energy = 0;
  
    // nested loops to access the elements
    // of the given matrix
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            // store quotient
            q = mat[i][j] / n;
  
            // final destination location (i_des, j_des) of
            // the element mat[i][j] is being calculated
            i_des = q;
            j_des = mat[i][j] - (n * q);
  
            // energy required for the movement of the
            // element mat[i][j] is calculated and then
            // accumulated in the 'tot_energy'
            tot_energy += abs(i_des - i) + abs(j_des - j);
        }
    }
  
    // required total energy
    return tot_energy;
}
  
// Driver program to test above
int main()
{
    int mat[SIZE][SIZE] = { { 4, 7, 0, 3 },
                            { 8, 5, 6, 1 },
                            { 9, 11, 10, 2 },
                            { 15, 13, 14, 12 } };
    int n = 4;
  
    cout << "Total energy required = "
         << calculateEnergy(mat, n) << " units";
  
    return 0;
}

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Java

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// Java implementation to find
// the total energy required 
// to rearrange the numbers
  
import java.util.*;
import java.lang.*;
  
public class GfG{
  
    private final static int SIZE  = 100;
   
    // function to find the total energy
    // required to rearrange the numbers
    public static int calculateEnergy(int mat[][],
    int n)
    {
        int i_des, j_des, q;
        int tot_energy = 0;
   
        // nested loops to access the elements
        // of the given matrix
        for (int i = 0; i < n; i++) {
  
            for (int j = 0; j < n; j++) {
  
                // store quotient
                q = mat[i][j] / n;
   
                // final destination location
                // (i_des, j_des) of
                // the element mat[i][j] is
                // being calculated
                i_des = q;
                j_des = mat[i][j] - (n * q);
   
                // energy required for the
                // movement of the
                // element mat[i][j] is
                // calculated and then
                // accumulated in the 'tot_energy'
                tot_energy += Math.abs(i_des - i) +
                Math.abs(j_des - j);
            }
        }
   
        // required total energy
        return tot_energy;
    }
      
    // Driver function
    public static void main(String argc[]){
  
        int[][] mat = new int[][] {{ 4, 7, 0, 3 },
                                   { 8, 5, 6, 1 },
                                   { 9, 11, 10, 2 },
                                   { 15, 13, 14, 12 }};
  
        int n = 4;
   
    System.out.println("Total energy required = "
         + calculateEnergy(mat, n) + " units");
    }
      
}
// This code is contributed by Sagar Shukla 

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Python3

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# implementation to find the total
# energy required to rearrange the
# numbers
n = 4
  
# function to find the total energy
# required to rearrange the numbers
def calculateEnergy(mat,n):
  
    tot_energy = 0
  
    # nested loops to access the 
    # elements of the given matrix
    for i in range(n):
        for j in range(n):
  
            #store quotient
            q = mat[i][j]//n
  
            # final destination location
            # (i_des, j_des) of the 
            # element mat[i][j] is being
            # calculated
            i_des = q
            j_des = mat[i][j]- (n*q)
  
            # energy required for the
            # movement of the element
            # mat[i][j] is calculated 
            # and then accumulated in
            # the 'tot_energy'
            tot_energy += (abs(i_des-i) 
                         + abs(j_des-j))
  
    # required total energy
    return tot_energy
  
  
# Driver Program
mat = [[4, 7, 0, 3],
       [8, 5, 6, 1],
         [9, 11, 10, 2],
       [15, 13, 14, 12]]
print("Total energy required = ",
        calculateEnergy(mat,n), "units")
  
# This code is contributed by Shrikant13.

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C#

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// C# implementation to find
// the total energy required
// to rearrange the numbers
using System;
  
class GFG {
  
    // function to find the total energy
    // required to rearrange the numbers
    public static int calculateEnergy(int[, ] mat,
                                            int n)
    {
        int i_des, j_des, q;
        int tot_energy = 0;
  
        // nested loops to access the elements
        // of the given matrix
        for (int i = 0; i < n; i++) {
  
            for (int j = 0; j < n; j++) {
  
                // store quotient
                q = mat[i, j] / n;
  
                // final destination location
                // (i_des, j_des) of
                // the element mat[i][j] is
                // being calculated
                i_des = q;
                j_des = mat[i, j] - (n * q);
  
                // energy required for the
                // movement of the
                // element mat[i][j] is
                // calculated and then
                // accumulated in the 'tot_energy'
                tot_energy += Math.Abs(i_des - i) + 
                              Math.Abs(j_des - j);
            }
        }
  
        // required total energy
        return tot_energy;
    }
  
    // Driver function
    public static void Main()
    {
  
        int[, ] mat = new int[, ]{ { 4, 7, 0, 3 },
                                   { 8, 5, 6, 1 },
                                   { 9, 11, 10, 2 },
                                   { 15, 13, 14, 12 } };
  
        int n = 4;
  
        Console.Write("Total energy required = "
              calculateEnergy(mat, n) + " units");
    }
}
  
// This code is contributed by Sam007

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PHP

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<?php
// PHP implementation to find 
// the total energy required 
// to rearrange the numbers
  
// function to find the total energy
// required to rearrange the numbers
function calculateEnergy($mat, $n)
{
    $i_des; $j_des; $q;
    $tot_energy = 0;
  
    // nested loops to access the 
    // elements of the given matrix
    for ($i = 0; $i < $n; $i++) 
    {
        for ($j = 0; $j < $n; $j++) 
        {
            // store quotient
            $q = (int)($mat[$i][$j] / $n);
  
            // final destination location 
            // (i_des, j_des) of the element
            // mat[i][j] is being calculated
            $i_des = $q;
            $j_des = $mat[$i][$j] - ($n * $q);
  
            // energy required for the movement 
            // of the element mat[i][j] is 
            // calculated and then accumulated 
            // in the 'tot_energy'
            $tot_energy += abs($i_des - $i) + 
                           abs($j_des - $j);
        }
    }
  
    // required total energy
    return $tot_energy;
}
  
// Driver Code
$mat = array(array (4, 7, 0, 3),
             array (8, 5, 6, 1),
             array (9, 11, 10, 2),
             array (15, 13, 14, 12));
$n = 4;
  
echo "Total energy required = ",
calculateEnergy($mat, $n) , " units";
  
// This code is contributed by ajit
?>

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Output :

Total energy required = 22 units

Time Complexity : O(n2)

This article is contributed by Ayush Jauhari. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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