# 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 n^{2}-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, the 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 the 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++

`// 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;` `}` |

## Java

`// 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` |

## Python3

`# 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.` |

## C#

`// 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` |

## PHP

`<?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` `?>` |

## Javascript

`<script>` `// Javascript implementation to find` `// the total energy required` `// to rearrange the numbers` `let SIZE = 100;` ` ` ` ` `// function to find the total energy` ` ` `// required to rearrange the numbers` ` ` `function` `calculateEnergy(mat, n)` ` ` `{` ` ` `let i_des, j_des, q;` ` ` `let tot_energy = 0;` ` ` ` ` `// nested loops to access the elements` ` ` `// of the given matrix` ` ` `for` `(let i = 0; i < n; i++) {` ` ` ` ` `for` `(let j = 0; j < n; j++) {` ` ` ` ` `// store quotient` ` ` `q = Math.floor(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 program` ` ` `let mat = [[ 4, 7, 0, 3 ],` ` ` `[ 8, 5, 6, 1 ],` ` ` `[ 9, 11, 10, 2 ],` ` ` `[ 15, 13, 14, 12 ]];` ` ` ` ` `let n = 4;` ` ` ` ` `document.write(` `"Total energy required = "` ` ` `+ calculateEnergy(mat, n) + ` `" units"` `);` `</script>` |

**Output :**

Total energy required = 22 units

**Time Complexity :** O(n^{2})

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