Minimize cost to empty given array where cost of removing an element is its absolute difference with Time instant

Given an array arr[] consisting of N integers, the task is to find the minimum cost to remove all elements from the array such that the cost of removing any element is the absolute difference between the current time instant T (initially 1) and the array element arr[i] i.e., abs(T – arr[i]) where T.

Examples:

Input: arr[] = {3, 6, 4, 2}
Output: 0
Explanation:
T = 1: No removal
T = 2: Remove arr[3]. Cost = |2 – 2| = 0
T = 3: Remove arr[0]. Cost = |3 – 3| = 0
T = 4: Remove arr[2]. Cost = |4 – 4| = 0
T = 5: No removal.
T = 6: Remove arr[1]. Cost = |0| + |6 – 6| = 0
Therefore, the total cost = 0

Input: arr[] = {4, 2, 4, 4, 5, 2}
Output: 4

Naive Approach: The idea is to use recursion to solve the problem. At each instant of time, two possibilities exists, either to remove any element or not. Therefore, to minimize the cost, sort the array. Then, starting from index 0 and time T = 1, solve the problem using following recurrence relation: 

minCost(index, time) = min(minCost(index + 1, T + 1) + abs(time – a[index]), minCost(index, T + 1))
where, Base Case: If current index exceeds the current size of the array.

Time Complexity: O(2^N)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is to use Dynamic Programming as there are overlapping subproblems and overlapping substructure to the above recurrence relation. Follow the steps below to solve the problem:

• Initialize a 2-D array, cost[][] of size N*2N with some large value where cost[i][j] denotes the minimum cost to delete all the elements up to the i^th index from the given array using j amount of time.
• Also, initialize cost[0][0] with 0 and variable, prev with 0 where prev will store the minimum of all previous cost values of the previous index.
• Traverse the given array arr[] using a variable i and then, for each i, iterate in the range [1, 2N] using variable j:
  • If the value of (prev + abs(j – arr[i – 1]) is less than cost[i][j], then update cost[i][j] to this value.
  • If cost[i – 1][j] is less than prev, then update prev to this value.
• After the above steps, print the minimum cost as cost[N][j].

Below is the implementation of the above approach: 

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;
#define INF 10000

// Function to find the minimum cost
// to delete all array elements
void minCost(int arr[], int n)
{

    // Sort the input array
    sort(arr, arr + n);

    // Store the maximum time to delete
    // the array in the worst case
    int m = 2 * n;

    // Store the result in cost[][] table
    int cost[n + 1][m + 1];

    // Initialize the table cost[][]
    for (int i = 0; i <= n; i++) {
        for (int j = 0; j <= m; j++) {
            cost[i][j] = INF;
        }
    }

    // Base Case
    cost[0][0] = 0;

    // Store the minimum of all cost
    // values of the previous index
    int prev = 0;

    // Iterate from range [1, n]
    // using variable i
    for (int i = 1; i <= n; i++) {

        // Update prev
        prev = cost[i - 1][0];

        // Iterate from range [1, m]
        // using variable j
        for (int j = 1; j <= m; j++) {

            // Update cost[i][j]
            cost[i][j] = min(cost[i][j],
                             prev
                                 + abs(j - arr[i - 1]));

            // Update the prev
            prev = min(prev, cost[i - 1][j]);
        }
    }

    // Store the minimum cost to
    // delete all elements
    int minCost = INF;

    // Find the minimum of all values
    // of cost[n][j]
    for (int j = 1; j <= m; j++) {
        minCost = min(minCost, cost[n][j]);
    }

    // Print minimum cost
    cout << minCost;
}

// Driver Code
int main()
{
    int arr[] = { 4, 2, 4, 4, 5, 2 };
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function Call
    minCost(arr, N);

    return 0;
}

Java

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

class GFG{

static int INF = 10000;

// Function to find the minimum cost
// to delete all array elements
static void minCost(int arr[], int n)
{

    // Sort the input array
    Arrays.sort(arr);

    // Store the maximum time to delete
    // the array in the worst case
    int m = 2 * n;

    // Store the result in cost[][] table
    int cost[][] = new int[n + 1][m + 1];

    // Initialize the table cost[][]
    for(int i = 0; i <= n; i++)
    {
        for(int j = 0; j <= m; j++)
        {
            cost[i][j] = INF;
        }
    }

    // Base Case
    cost[0][0] = 0;

    // Store the minimum of all cost
    // values of the previous index
    int prev = 0;

    // Iterate from range [1, n]
    // using variable i
    for(int i = 1; i <= n; i++)
    {

        // Update prev
        prev = cost[i - 1][0];

        // Iterate from range [1, m]
        // using variable j
        for(int j = 1; j <= m; j++)
        {

            // Update cost[i][j]
            cost[i][j] = Math.min(cost[i][j],
                                  prev + Math.abs(
                                     j - arr[i - 1]));

            // Update the prev
            prev = Math.min(prev, cost[i - 1][j]);
        }
    }

    // Store the minimum cost to
    // delete all elements
    int minCost = INF;

    // Find the minimum of all values
    // of cost[n][j]
    for(int j = 1; j <= m; j++)
    {
        minCost = Math.min(minCost, cost[n][j]);
    }

    // Print minimum cost
    System.out.print(minCost);
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 4, 2, 4, 4, 5, 2 };
    int N = arr.length;

    // Function Call
    minCost(arr, N);
}
}

// This code is contributed by sanjoy_62

Python3

# Python3 program for the above approach

INF = 10000

# Function to find the minimum cost
# to delete all array elements
def minCost(arr, n):

    # Sort the input array
    arr = sorted(arr)

    # Store the maximum time to delete
    # the array in the worst case
    m = 2 * n

    # Store the result in cost[][] table
    cost = [[INF for i in range(m + 1)] for i in range(n + 1)]

    # Base Case
    cost[0][0] = 0

    # Store the minimum of all cost
    # values of the previous index
    prev = 0

    # Iterate from range [1, n]
    # using variable i
    for i in range(1, n + 1):

        # Update prev
        prev = cost[i - 1][0]

        # Iterate from range [1, m]
        # using variable j
        for j in range(1, m + 1):

            # Update cost[i][j]
            cost[i][j] = min(cost[i][j], prev + abs(j - arr[i - 1]))

            # Update the prev
            prev = min(prev, cost[i - 1][j])

    # Store the minimum cost to
    # delete all elements
    minCost = INF

    # Find the minimum of all values
    # of cost[n][j]
    for j in range(1, m + 1):
        minCost = min(minCost, cost[n][j])

    # Print minimum cost
    print(minCost)

# Driver Code
if __name__ == '__main__':
    arr=[4, 2, 4, 4, 5, 2]
    N = len(arr)

    # Function Call
    minCost(arr, N)

# This code is contributed by mohit kumar 29

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

static int INF = 10000;

// Function to find the minimum cost
// to delete all array elements
static void minCost(int[] arr, int n)
{

    // Sort the input array
    Array.Sort(arr);

    // Store the maximum time to delete
    // the array in the worst case
    int m = 2 * n;

    // Store the result in cost[][] table
    int[,] cost = new int[n + 1, m + 1];

    // Initialize the table cost[][]
    for(int i = 0; i <= n; i++)
    {
        for(int j = 0; j <= m; j++)
        {
            cost[i, j] = INF;
        }
    }

    // Base Case
    cost[0, 0] = 0;

    // Store the minimum of all cost
    // values of the previous index
    int prev = 0;

    // Iterate from range [1, n]
    // using variable i
    for(int i = 1; i <= n; i++)
    {

        // Update prev
        prev = cost[i - 1, 0];

        // Iterate from range [1, m]
        // using variable j
        for(int j = 1; j <= m; j++)
        {

            // Update cost[i][j]
            cost[i, j] = Math.Min(cost[i, j],
                                  prev + Math.Abs(
                                     j - arr[i - 1]));

            // Update the prev
            prev = Math.Min(prev, cost[i - 1, j]);
        }
    }

    // Store the minimum cost to
    // delete all elements
    int minCost = INF;

    // Find the minimum of all values
    // of cost[n][j]
    for(int j = 1; j <= m; j++)
    {
        minCost = Math.Min(minCost, cost[n, j]);
    }

    // Print minimum cost
    Console.Write(minCost);
}

// Driver Code
public static void Main()
{
    int[] arr = { 4, 2, 4, 4, 5, 2 };
    int N = arr.Length;

    // Function Call
    minCost(arr, N);
}
}

// This code is contributed by susmitakundugoaldanga

Javascript

<script>

// JavaScript program for above approach

let INF = 10000;

// Function to find the minimum cost
// to delete all array elements
function minCost(arr, n)
{

    // Sort the input array
    arr.sort();

    // Store the maximum time to delete
    // the array in the worst case
    let m = 2 * n;

    // Store the result in cost[][] table
    let cost = new Array(n + 1);

    // Loop to create 2D array using 1D array
    for (var i = 0; i < cost.length; i++) {
        cost[i] = new Array(2);
    }

    // Initialize the table cost[][]
    for(let i = 0; i <= n; i++)
    {
        for(let j = 0; j <= m; j++)
        {
            cost[i][j] = INF;
        }
    }

    // Base Case
    cost[0][0] = 0;

    // Store the minimum of all cost
    // values of the previous index
    let prev = 0;

    // Iterate from range [1, n]
    // using variable i
    for(let i = 1; i <= n; i++)
    {

        // Update prev
        prev = cost[i - 1][0];

        // Iterate from range [1, m]
        // using variable j
        for(let j = 1; j <= m; j++)
        {

            // Update cost[i][j]
            cost[i][j] = Math.min(cost[i][j],
                                  prev + Math.abs(
                                     j - arr[i - 1]));

            // Update the prev
            prev = Math.min(prev, cost[i - 1][j]);
        }
    }

    // Store the minimum cost to
    // delete all elements
    let minCost = INF;

    // Find the minimum of all values
    // of cost[n][j]
    for(let j = 1; j <= m; j++)
    {
        minCost = Math.min(minCost, cost[n][j]);
    }

    // Print minimum cost
    document.write(minCost);
}

// Driver Code
     let arr = [ 4, 2, 4, 4, 5, 2 ];
    let N = arr.length;

    // Function Call
    minCost(arr, N);

// This code is contributed by avijitmondal1998.
</script>

Output:

4

Time Complexity: O(N²)
Auxiliary Space: O(N²)