Minimum cost to sort an Array such that swapping X and Y costs XY

Given an array of N distinct positive integers. The task is to find the minimum cost to sort the given array. The cost of swapping two elements X and Y is X*Y.

Examples:

Input: arr[] = {8, 4, 5, 3, 2, 7} 
Output: 57 
Explanation:
Swap element at index 4 with index 5 – cost(arr[4]*arr[5]) = (2*7) = 14, 
Array becomes {8, 4, 5, 3, 7, 2} 
then, swap element at index 0 with 5 – cost(arr[0]*arr[5]) = (8*2) = 16, 
Array becomes {2, 4, 5, 3, 7, 8} 
then, swap element at index 2 with 3 – cost(arr[2]*arr[3]) = (5*3) = 15, 
Array becomes {2, 4, 3, 5, 7, 8} 
then, swap element at index 1 with 2 – cost(arr[1]*arr[2]) = (4*3) = 12, 
Array becomes {2, 3, 4, 5, 7, 8} 
Array is now sorted and total cost = 14+16+15+12 = 57.

Input: arr[] = {1, 8, 9, 7, 6} 
Output: 36

Approach: The idea is that for sorting a cycle we have two choices either to use only the local minimum of the cycle or to use both local and overall minimum of the array. Choose the one swap element that gives a lower cost. Below are the steps:

1. Calculate the local minimum (say local_minimum) which is the minimum element in the present cycle and the overall minimum (say overall_minimum) which is the minimum element in the whole array.
2. Calculate and store the cost to sort the cycle (say cost1) by using only local minimum value.
3. Also, calculate and store the cost to sort the cycle (say cost2) by using both local minimum value and the overall minimum value.
4. Now the minimum cost to sort this cycle will be minimum of the costs cost1 and cost2. Add this cost to the total cost.

Below is the illustration for the array arr[] = {1, 8, 9, 7, 6}: 

• In the above figure, cycle {8, 9, 7, 6} can be sorted using the local minimum element 6 or with overall minimum element 1. By using only local minimum element i.e., swap 6 and 9, swap 6 and 7, swap 6 and 8. Therefore, the total cost is 6*9 + 6*7 + 6*8 = 144.
• By using both overall minimum and local minimum element i.e., swap 1 and 6, swap 1 and 9, swap 1 and 7, swap 1 and 8, swap 1 and 6. Therefore, the total cost is 1*6 +1*9 +1*7 +1*8 +1*6 = 36.
• The minimum of the above cost is 36.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function returns the minimum cost
// to sort the given array
int minCost(int arr[], int n)
{
    // Create array of pairs in which
    // 1st element is the array element
    // and 2nd element is index of first
    pair<int, int> sorted[n];

    // Initialize the total cost
    int total_cost = 0;

    for (int i = 0; i < n; i++) {
        sorted[i].first = arr[i];
        sorted[i].second = i;
    }
    // Sort the array with respect to
    // array value
    sort(sorted, sorted + n);

    // Initialize the overall minimum
    // which is the 1st element
    int overall_minimum = sorted[0].first;

    // To keep track of visited elements
    // create a visited array & initialize
    // all elements as not visited
    bool vis[n] = { false };

    // Iterate over every element
    // of the array
    for (int i = 0; i < n; i++) {

        // If the element is visited or
        // in the sorted position, and
        // check for next element
        if (vis[i] && sorted[i].second == i)
            continue;

        // Create a vector which stores
        // all elements of a cycle
        vector<int> v;
        int j = i;

        // It covers all the elements
        // of a cycle
        while (!vis[j]) {

            vis[j] = true;
            v.push_back(sorted[j].first);
            j = sorted[j].second;
        }

        // If cycle is found then the
        // swapping is required
        if (v.size() > 0) {

            // Initialize local minimum with
            // 1st element of the vector as
            // it contains the smallest
            // element in the beginning
            int local_minimum = v[0], result1 = 0,
                result2 = 0;

            // Stores the cost with using only
            // local minimum value.
            for (int k = 1; k < v.size(); k++)
                result1 += (local_minimum * v[k]);

            // Stores the cost of using both
            // local minimum and overall minimum
            for (int k = 0; k < v.size(); k++)
                result2 += (overall_minimum * v[k]);

            // Update the result2
            result2 += (overall_minimum
                        * local_minimum);

            // Store the minimum of the
            // two result to total cost
            total_cost += min(result1, result2);
        }
    }

    // Return the minimum cost
    return total_cost;
}

// Driver Code
int main()
{
    // Given array arr[]
    int arr[] = { 1, 8, 9, 7, 6 };
    int n = (sizeof(arr) / sizeof(int));

    // Function Call
    cout << minCost(arr, n);
    return 0;
}

Java

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

class GFG{

// Function returns the minimum cost
// to sort the given array
static int minCost(int arr[], int n)
{

    // Create array of pairs in which
    // 1st element is the array element
    // and 2nd element is index of first
    int[][] sorted = new int[n][2];

    // Initialize the total cost
    int total_cost = 0;

    for(int i = 0; i < n; i++)
    {
        sorted[i][0] = arr[i];
        sorted[i][1] = i;
    }

    // Sort the array with respect to
    // array value
    Arrays.sort(sorted, (a, b) -> a[0] - b[0]);

    // Initialize the overall minimum
    // which is the 1st element
    int overall_minimum = sorted[0][0];

    // To keep track of visited elements
    // create a visited array & initialize
    // all elements as not visited
    boolean[] vis = new boolean[n];

    // Iterate over every element
    // of the array
    for(int i = 0; i < n; i++)
    {

        // If the element is visited or
        // in the sorted position, and
        // check for next element
        if (vis[i] && sorted[i][1] == i)
            continue;

        // Create a vector which stores
        // all elements of a cycle
        ArrayList<Integer> v = new ArrayList<>();
        int j = i;

        // It covers all the elements
        // of a cycle
        while (!vis[j])
        {
            vis[j] = true;
            v.add(sorted[j][0]);
            j = sorted[j][1];
        }

        // If cycle is found then the
        // swapping is required
        if (v.size() > 0)
        {

            // Initialize local minimum with
            // 1st element of the vector as
            // it contains the smallest
            // element in the beginning
            int local_minimum = v.get(0), result1 = 0,
                result2 = 0;

            // Stores the cost with using only
            // local minimum value.
            for(int k = 1; k < v.size(); k++)
                result1 += (local_minimum * v.get(k));

            // Stores the cost of using both
            // local minimum and overall minimum
            for(int k = 0; k < v.size(); k++)
                result2 += (overall_minimum * v.get(k));

            // Update the result2
            result2 += (overall_minimum *
                          local_minimum);

            // Store the minimum of the
            // two result to total cost
            total_cost += Math.min(result1, result2);
        }
    }

    // Return the minimum cost
    return total_cost;
}

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

    // Given array arr[]
    int arr[] = { 1, 8, 9, 7, 6 };
    int n = arr.length;

    // Function call
    System.out.print(minCost(arr, n));
}
}

// This code is contributed by offbeat

Python3

# Python3 program for the above approach

# Function returns the minimum cost
# to sort the given array
def minCost(arr, n):

    # Create array of pairs in which
    # 1st element is the array element
    # and 2nd element is index of first
    sortedarr = []

    # Initialize the total cost
    total_cost = 0

    for i in range(n):
        sortedarr.append([arr[i], i])

    # Sort the array with respect to
    # array value
    sortedarr.sort()

    # Initialize the overall minimum
    # which is the 1st element
    overall_minimum = sortedarr[0][0]

    # To keep track of visited elements
    # create a visited array & initialize
    # all elements as not visited
    vis = [False] * n

    # Iterate over every element
    # of the array
    for i in range(n):

        # If the element is visited or
        # in the sorted position, and
        # check for next element
        if vis[i] and sortedarr[i][1] == i:
            continue

        # Create a vector which stores
        # all elements of a cycle
        v = []
        j = i
        size = 0

        # It covers all the elements
        # of a cycle
        while vis[j] == False:
            vis[j] = True
            v.append(sortedarr[j][0])
            j = sortedarr[j][1]
            size += 1

        # If cycle is found then the
        # swapping is required
        if size != 0:

            # Initialize local minimum with
            # 1st element of the vector as
            # it contains the smallest
            # element in the beginning
            local_minimum = v[0]
            result1 = 0
            result2 = 0

            # Stores the cost with using only
            # local minimum value.
            for k in range(1, size):
                result1 += local_minimum * v[k]

            # Stores the cost of using both
            # local minimum and overall minimum
            for k in range(size):
                result2 += overall_minimum * v[k]

            # Update the result2
            result2 += (overall_minimum *
                          local_minimum)

            # Store the minimum of the
            # two result to total cost
            total_cost += min(result1, result2)

    # Return the minimum cost
    return  total_cost

# Driver code

# Given array arr[]
A = [ 1, 8, 9, 7, 6 ]

# Function call
ans = minCost(A, len(A))

print(ans)

# This code is contributed by kumarkashyap

C#

using System;
using System.Collections.Generic;

class GFG
{
  // Function returns the minimum cost
  // to sort the given array
  static int minCost(int[] arr, int n)
  {
    // Create array of pairs in which
    // 1st element is the array element
    // and 2nd element is index of first
    int[][] sorted = new int[n][];

    // Initialize the total cost
    int total_cost = 0;

    for (int i = 0; i < n; i++)
    {
      sorted[i] = new int[2];
      sorted[i][0] = arr[i];
      sorted[i][1] = i;
    }

    // Sort the array with respect to
    // array value
    Array.Sort(sorted, (a, b) => a[0] - b[0]);

    // Initialize the overall minimum
    // which is the 1st element
    int overall_minimum = sorted[0][0];

    // To keep track of visited elements
    // create a visited array & initialize
    // all elements as not visited
    bool[] vis = new bool[n];

    // Iterate over every element
    // of the array
    for (int i = 0; i < n; i++)
    {
      // If the element is visited or
      // in the sorted position, and
      // check for next element
      if (vis[i] && sorted[i][1] == i)
        continue;

      // Create a vector which stores
      // all elements of a cycle
      List<int> v = new List<int>();
      int j = i;

      // It covers all the elements
      // of a cycle
      while (!vis[j])
      {
        vis[j] = true;
        v.Add(sorted[j][0]);
        j = sorted[j][1];
      }

      // If cycle is found then the
      // swapping is required
      if (v.Count > 0)
      {
        // Initialize local minimum with
        // 1st element of the vector as
        // it contains the smallest
        // element in the beginning
        int local_minimum = v[0], result1 = 0,
        result2 = 0;

        // Stores the cost with using only
        // local minimum value.
        for (int k = 1; k < v.Count; k++)
          result1 += (local_minimum * v[k]);

        // Stores the cost of using both
        // local minimum and overall minimum
        for (int k = 0; k < v.Count; k++)
          result2 += (overall_minimum * v[k]);

        // Update the result2
        result2 += (overall_minimum *
                    local_minimum);

        // Store the minimum of the
        // two result to total cost
        total_cost += Math.Min(result1, result2);
      }
    }

    // Return the minimum cost
    return total_cost;
  }
  // Driver code
  public static void Main(String[] args)
  {
    // Given array arr[]
    int[] arr = {1, 8, 9, 7, 6};
    var n = arr.Length;
    // Function call
    Console.Write(GFG.minCost(arr, n));
  }
}

Javascript

<script>

// JavaScript program for the above approach

// Function returns the minimum cost
// to sort the given array
function minCost(arr, n){

    // Create array of pairs in which
    // 1st element is the array element
    // and 2nd element is index of first
    let sortedarr = []

    // Initialize the total cost
    let total_cost = 0

    for(let i = 0; i < n; i++){
        sortedarr.push([arr[i], i])
   }

    // Sort the array with respect to
    // array value
    sortedarr.sort()

    // Initialize the overall minimum
    // which is the 1st element
    let overall_minimum = sortedarr[0][0]

    // To keep track of visited elements
    // create a visited array & initialize
    // all elements as not visited
    let vis = new Array(n).fill(false)

    // Iterate over every element
    // of the array
    for(let i = 0; i < n; i++){

        // If the element is visited or
        // in the sorted position, and
        // check for next element
        if(vis[i] && sortedarr[i][1] == i)
            continue

        // Create a vector which stores
        // all elements of a cycle
        let v = []
        let j = i
        let size = 0

        // It covers all the elements
        // of a cycle
        while(vis[j] == false){
            vis[j] = true
            v.push(sortedarr[j][0])
            j = sortedarr[j][1]
            size += 1
      }

        // If cycle is found then the
        // swapping is required
        if(size != 0){

            // Initialize local minimum with
            // 1st element of the vector as
            // it contains the smallest
            // element in the beginning
            let local_minimum = v[0]
            let result1 = 0
            let result2 = 0

            // Stores the cost with using only
            // local minimum value.
            for(let k = 1; k < size; k++)
                result1 += local_minimum * v[k]

            // Stores the cost of using both
            // local minimum and overall minimum
            for(let k = 0; k < size; k++)
                result2 += overall_minimum * v[k]

            // Update the result2
            result2 += (overall_minimum *
                        local_minimum)

            // Store the minimum of the
            // two result to total cost
            total_cost += Math.min(result1, result2)
      }
   }

    // Return the minimum cost
    return total_cost
}

// Driver code

// Given array arr[]
A = [ 1, 8, 9, 7, 6 ]

// Function call
ans = minCost(A, A.length)
document.write(ans,"</br>")

// This code is contributed by shinjanpatra

</script>

Output:

36

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