CSES Solutions - Stick Lengths

There are N sticks with some lengths given by arr[]. Your task is to modify the sticks so that each stick has the same length. You can either lengthen or shorten each stick. Both operations cost X where X is the difference between the new and original length. What is the minimum total cost?

Examples:

Input: N = 5, arr[] = {2, 3, 1, 5, 2}
Output: 5
Explanation: We can make all the sticks of length = 2. The total cost = abs(2 - 2) + abs(3 - 2) + abs(1 - 2) + abs(5 - 2) + abs(2 -2) = 0 + 1 + 1 + 3 + 0= 5.

Input: N = 5, arr[] = {1, 1, 1, 1, 1}
Output: 0
Explanation: All the sticks already have the same length. The total cost = 0.

Approach: To solve the problem, follow the below idea:

Think geometrically. Assume that array elements are co-ordinates on X axis. The problem reduces to finding another co-ordinate such that the sum of distances between this choice and other co-ordinate is minimized.

Observe that: If number of coordinates are odd then Y = middle element. If even then Y is any number in between middle 2 co-ordinates. Say Input = [a, b, c, d]. Output is any number between b and c including both. Hence the cost is sum which can be computed easily now that we have chosen Y. sum|(Y-arr[i])| for all i.

Step-by-step algorithm:

• Sort the array arr[] in ascending order.
• Find the median as the middle element of the array arr[].
• Iterate over the array and find the absolute difference between arr[i] and the median.
• Sum of all these differences will be our answer.

Below is the implementation of the algorithm:

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

// method to calculate the minimum total cost
long long calculateStickLength(int* arr, int N)
{
    // sort the array to find the median
    sort(arr, arr + N);
    int median = arr[N / 2];
    long long ans = 0;
    // Find the cost of individual stick
    for (int i = 0; i < N; i++) {
        ans += abs(median - arr[i]);
    }
    return ans;
}

int main()
{

    // Sample Input
    int N = 5;
    int arr[] = { 2, 3, 1, 5, 2 };
    cout << calculateStickLength(arr, N) << "\n";
    return 0;
}

import java.util.*;

public class StickLength {
    // Method to calculate the minimum total cost
    static long calculateStickLength(int[] arr, int N)
    {
        // Sort the array to find the median
        Arrays.sort(arr);
        int median = arr[N / 2];
        long ans = 0;

        // Find the cost of individual stick
        for (int i = 0; i < N; i++) {
            ans += Math.abs(median - arr[i]);
        }
        return ans;
    }

    public static void main(String[] args)
    {
        // Sample Input
        int N = 5;
        int[] arr = { 2, 3, 1, 5, 2 };
        System.out.println(calculateStickLength(arr, N));
    }
}

// This code is contributed by akshitaguprzj3

using System;

class Program
{
    // Method to calculate the minimum total cost
    static long CalculateStickLength(int[] arr)
    {
        // Sort the array to find the median
        Array.Sort(arr);
        int median = arr[arr.Length / 2];
        long ans = 0;
        // Find the cost of individual stick
        foreach (int num in arr)
        {
            ans += Math.Abs(median - num);
        }
        return ans;
    }

    static void Main(string[] args)
    {
        // Sample Input
        int[] arr = { 2, 3, 1, 5, 2 };
        Console.WriteLine(CalculateStickLength(arr));
    }
}

// Method to calculate the minimum total cost
function calculateStickLength(arr) {
    // Sort the array to find the median
    arr.sort((a, b) => a - b);
    const median = arr[Math.floor(arr.length / 2)];
    let ans = 0;

    // Find the cost of individual stick
    for (let i = 0; i < arr.length; i++) {
        ans += Math.abs(median - arr[i]);
    }

    return ans;
}

// Sample Input
const N = 5;
const arr = [2, 3, 1, 5, 2];
console.log(calculateStickLength(arr));

def calculate_stick_length(arr):
    # Sort the array to find the median
    arr.sort()
    median = arr[len(arr) // 2]
    ans = 0

    # Find the cost of individual stick
    for length in arr:
        ans += abs(median - length)
    return ans

# Driver code
if __name__ == "__main__":
    # Sample Input
    arr = [2, 3, 1, 5, 2]
    print(calculate_stick_length(arr))

5

Time Complexity: O(N* logN), where N is the number of sticks.
Auxiliary Space: O(1)