Find the sum of elements after X deletions and Y sign inversions

Two Players A and B are engaged in a game with an array of positive integers arr[] of size N. The game starts with Player A deleting at max X elements from the array arr[]. Then, Player B can select at max Y elements and multiply them with -1. Player A wants to maximize the sum of arr[] after both the players have made their move whereas Player B wants to minimize the sum. The task is to determine the sum of the elements in the array arr[] if both the players have moved optimally.

Examples:

Input: N = 4, arr[] = {3, 1, 2, 4}, X = 1, Y = 1
Output: 2
Explanation: It is optimal for Player A to not delete any element. Player B will then multiply 4 with -1. So, the sum of elements of arr[] = 3 + 1 + 2 – 4 = 2.

Input: N = 8, arr[] = {5, 5, 3, 3, 3, 2, 9, 9}, X = 5, Y = 3
Output: -5
Explanation: It is optimal for Player A to delete {9, 9}. Player B will then multiply {5, 5, 3} with -1. So, the sum of elements of arr[] = -5 – 5 – 3 + 3 + 3 + 2 = -5.

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

It is optimal for Player B to negate the Y largest elements of the array. Player A will always try to delete the largest elements in the array to prevent Player B to negate them.

To solve the problem, we can sort the array and iterate over i (0 ≤ i ≤ X) where i is the number of elements Player A deletes. For each i, we know that Player A will delete the i largest elements of the array, and Player B will then negate the Y largest remaining elements. So, the sum at the end can be calculated quickly with prefix sums.

Step-by-step algorithm:

• Sort the given array arr[] and modify arr[] to store the prefix sums.
• Iterate i from 0 to X, where i is the number of elements deleted by Player A.
• Find the maximum sum in each of the cases.
• Return the maximum sum among all i from 0 to X.

Below is the implementation of approach:

Java

import java.util.*;

public class Main {
    static final int MOD = 1000000007;

    // Function to solve the problem
    static void solve(int N, ArrayList<Integer> arr, int X, int Y) {

        // Sort the array in non-decreasing order
        Collections.sort(arr, Collections.reverseOrder());

        // Calculate prefix sums
        for (int i = 1; i < N; i++) {
            arr.set(i, arr.get(i) + arr.get(i - 1));
        }

        // Initialize answer
        int ans = -1_000_000_000;

        // Iterate over possible values of i
        for (int i = 0; i <= X; i++) {
            if (i == 0) {
                int curr = arr.get(N - 1)
                        - 2 * (arr.get(Math.min(i + Y - 1, N - 1)));
                ans = Math.max(ans, curr);
            } else {
                int curr = arr.get(N - 1)
                        - 2 * arr.get(Math.min(i + Y - 1, N - 1))
                        + arr.get(i - 1);
                ans = Math.max(ans, curr);
            }
        }

        // Output the answer
        System.out.println(ans);
    }

    // Main function
    public static void main(String[] args) {
        // Input variables: N, X, Y
        int N = 8, X = 5, Y = 3;

        // ArrayList to store array elements
        ArrayList<Integer> arr = new ArrayList<>(Arrays.asList(5, 5, 3, 3, 3, 2, 9, 9));
        solve(N, arr, X, Y);
    }
}

// This code is contributed by Ayush Mishra

C#

using System;
using System.Collections.Generic;
using System.Linq;

public class MainClass
{
    const int MOD = 1000000007;

    // Function to solve the problem
    static void Solve(int N, List<int> arr, int X, int Y)
    {
        // Sort the array in non-decreasing order
        arr.Sort((a, b) => b.CompareTo(a));

        // Calculate prefix sums
        for (int i = 1; i < N; i++)
        {
            arr[i] += arr[i - 1];
        }

        // Initialize answer
        int ans = -1_000_000_000;

        // Iterate over possible values of i
        for (int i = 0; i <= X; i++)
        {
            if (i == 0)
            {
                int curr = arr[N - 1] - 2 * (arr[Math.Min(i + Y - 1, N - 1)]);
                ans = Math.Max(ans, curr);
            }
            else
            {
                int curr = arr[N - 1] - 2 * arr[Math.Min(i + Y - 1, N - 1)] + arr[i - 1];
                ans = Math.Max(ans, curr);
            }
        }

        // Output the answer
        Console.WriteLine(ans);
    }

    // Main function
    public static void Main(string[] args)
    {
        // Input variables: N, X, Y
        int N = 8, X = 5, Y = 3;

        // List to store array elements
        List<int> arr = new List<int> { 5, 5, 3, 3, 3, 2, 9, 9 };
        Solve(N, arr, X, Y);
    }
}

Javascript

// Function to solve the problem
function solve(N, arr, X, Y) {
    // Sort the array in non-decreasing order
    arr.sort((a, b) => b - a);

    // Calculate prefix sums
    for (let i = 1; i < N; i++) {
        arr[i] += arr[i - 1];
    }

    // Initialize answer
    let ans = -1e9;

    // Iterate over possible values of i
    for (let i = 0; i <= X; i++) {
        if (i === 0) {
            let curr = arr[N - 1] - 2 * (arr[Math.min(i + Y - 1, N - 1)]);
            ans = Math.max(ans, curr);
        } else {
            let curr =
                arr[N - 1] -
                2 * arr[Math.min(i + Y - 1, N - 1)] +
                (i > 0 ? arr[i - 1] : 0);
            ans = Math.max(ans, curr);
        }
    }

    // Output the answer
    console.log(ans);
}

// Main function
function main() {
    // Input variables: N, X, Y
    const N = 8,
        X = 5,
        Y = 3;

    // Array to store elements
    const arr = [5, 5, 3, 3, 3, 2, 9, 9];
    solve(N, arr, X, Y);
}

// Calling the main function
main();

// This code is contributed by Ayush Mishra

C++14

#include <bits/stdc++.h>
using namespace std;
#define MOD 1000000007

// Function to solve the problem
void solve(int N, vector<int>& arr, int X, int Y)
{

    // Sort the array in non-decreasing order
    sort(arr.begin(), arr.end(), greater<int>());

    // Calculate prefix sums
    for (int i = 1; i < N; i++) {
        arr[i] += arr[i - 1];
    }

    // Initialize answer
    int ans = -1e9;

    // Iterate over possible values of i
    for (int i = 0; i <= X; i++) {
        if (i == 0) {
            int curr = arr[N - 1]
                    - 2 * (arr[min(i + Y - 1, N - 1)]);
            ans = max(ans, curr);
        }
        else {
            int curr = arr[N - 1]
                    - 2 * arr[min(i + Y - 1, N - 1)]
                    + arr[i - 1];
            ans = max(ans, curr);
        }
    }

    // Output the answer
    cout << ans << "\n";
}

// Main function
int main()
{
    // Input variables: N, X, Y
    int N = 8, X = 5, Y = 3;

    // Vector to store array elements
    vector<int> arr = { 5, 5, 3, 3, 3, 2, 9, 9};
    solve(N, arr, X, Y);
}

Python3

def solve(N, arr, X, Y):
    # Sort the array in non-decreasing order
    arr.sort(reverse=True)

    # Calculate prefix sums
    for i in range(1, N):
        arr[i] += arr[i - 1]

    # Initialize answer
    ans = -1_000_000_000

    # Iterate over possible values of i
    for i in range(X + 1):
        if i == 0:
            curr = arr[N - 1] - 2 * (arr[min(i + Y - 1, N - 1)])
            ans = max(ans, curr)
        else:
            curr = arr[N - 1] - 2 * arr[min(i + Y - 1, N - 1)] + arr[i - 1]
            ans = max(ans, curr)

    # Output the answer
    print(ans)

# Main function
if __name__ == "__main__":
    # Input variables: N, X, Y
    N, X, Y = 8, 5, 3

    # List to store array elements
    arr = [5, 5, 3, 3, 3, 2, 9, 9]
    solve(N, arr, X, Y)

#This code is contributed by rohit singh

Output

-5

Time Complexity: O(NlogN + X), where N is the size of input array arr[] and X is the number of elements Player A can delete.
Auxiliary Space: O(1)