Optimal strategy for a Game with modifications

Problem Statement: Consider a row of n coins of values v1 . . . vn, where n is even. We play a game against an opponent by alternating turns. In each turn, a player performs the following operation K times.

The player selects either the first or last coin from the row, removes it from the row permanently, and receives the value of the coin.

Determine the maximum possible amount of money the user can definitely win if the user moves first.

Note: The opponent is as clever as the user.

Examples:



Input : array = {10, 15, 20, 9, 2, 5}, k=2
Output :32
Explanation:
Lets say, user has initially picked 10 and 15.
The value of coins which the user has is 25 and
{20, 9, 2, 5} are remaining in the array.
In the second round, the opponent picks 20 and 9 making his value 29.
In the third round, the user picks 2 and 5 which makes his total value as 32.

Input: array = {10, 15, 20, 9, 2}, k=1
Output: 32

Approach:
A recursive solution needs to be formed and the values of sub problems needs to be stored to compute the result.
Taking an example to arrive at the recursive solution;

arr = {10, 15, 20, 9, 2, 5}, k=2
So if the user selects 10, 15 in the first turn then 20, 9, 2, 5 are left in the array.
But if the user selects 10, 5; then 15, 20, 9, 2 are left in the array.
Lastly, if the user selects 5, 2; then 10, 15, 20, 9 are left in the array.

So at any iteration after selecting k elements, a continuous subarray of length n-k is remaining for next computation.

So recursive solution can be formed where :

S(l, r) = (sum(l, r) – sum(l+i, l+i+n-k-1))+(sum(l+i, l+i+n-k-1) – S(l+i, l+i+n-k-1))
where l+i+n-k-1<=r

Sum of chosen elements Sc=(sum(l, r) – sum(l+i, l+i+n-k-1))
Now the opponent will perform the next turn so
Sum of Elements chosen in next steps = Total sum of present array from l to r –
Sum of elements chosen by opponent in next steps which is equal to

Nc=(sum(l+i, l+i+n-k-1) - S(l+i, l+i+n-k-1)).
S(l, r) = Sc + Nc
where,
Nc=(sum(l+i, l+i+n-k-1) - S(l+i, l+i+n-k-1))
Sc=(sum(l, r) - sum(l+i, l+i+n-k-1))

Below is the implementation of the above approach:

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// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
#define ll long long int
  
// Function to return sum of subarray from l to r
ll sum(int arr[], int l, int r)
{
    // calculate sum by a loop from l to r
    ll s = 0;
    for (int i = l; i <= r; i++) {
        s += arr[i];
    }
    return s;
}
  
// dp to store the values of sub problems
ll dp[101][101][101] = { 0 };
  
ll solve(int arr[], int l, int r, int k)
{
    // if length of the array is less than k
    // return the sum
    if (r - l + 1 <= k)
        return sum(arr, l, r);
  
    // if the value is previously calculated
    if (dp[l][r][k])
        return dp[l][r][k];
  
    // else calculate the value
    ll sum_ = sum(arr, l, r);
    ll len_r = (r - l + 1) - k;
    ll len = (r - l + 1);
    ll ans = 0;
  
    // select all the sub array of length len_r
    for (int i = 0; i < len - len_r + 1; i++) {
        // get the sum of that sub array
        ll sum_sub = sum(arr, i + l, i + l + len_r - 1);
  
        // check if it is the maximum or not
        ans = max(ans, (sum_ - sum_sub) + (sum_sub - 
                  solve(arr, i + l, i + l + len_r - 1, k)));
    }
  
    // store it in the table
    dp[l][r][k] = ans;
  
    return ans;
}
  
// Driver code
int main()
{
    int arr[] = { 10, 15, 20, 9, 2, 5 }, k = 2;
    int n = sizeof(arr) / sizeof(int);
    memset(dp, 0, sizeof(dp));
  
    cout << solve(arr, 0, n - 1, k);
  
    return 0;
}
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// Java implementation of the above approach 
class GFG
{
      
    // Function to return sum of subarray from l to r 
    static int sum(int arr[], int l, int r) 
    
        // calculate sum by a loop from l to r 
        int s = 0
        for (int i = l; i <= r; i++)
        
            s += arr[i]; 
        
        return s; 
    
      
    // dp to store the values of sub problems 
    static int dp[][][] = new int[101][101][101] ; 
      
    static int solve(int arr[], int l, int r, int k) 
    
        // if length of the array is less than k 
        // return the sum 
        if (r - l + 1 <= k) 
            return sum(arr, l, r); 
      
        // if the value is previously calculated 
        if (dp[l][r][k] != 0
            return dp[l][r][k]; 
      
        // else calculate the value 
        int sum_ = sum(arr, l, r); 
        int len_r = (r - l + 1) - k; 
        int len = (r - l + 1); 
        int ans = 0
      
        // select all the sub array of length len_r 
        for (int i = 0; i < len - len_r + 1; i++) 
        
            // get the sum of that sub array 
            int sum_sub = sum(arr, i + l, i + l + len_r - 1); 
      
            // check if it is the maximum or not 
            ans = Math.max(ans, (sum_ - sum_sub) + (sum_sub - 
                    solve(arr, i + l, i + l + len_r - 1, k))); 
        
      
        // store it in the table 
        dp[l][r][k] = ans; 
      
        return ans; 
    
      
    // Driver code 
    public static void main (String[] args) 
    {
      
        int arr[] = { 10, 15, 20, 9, 2, 5 }, k = 2
        int n = arr.length; 
  
        System.out.println(solve(arr, 0, n - 1, k)); 
      
    }
}
  
// This code is contributed by AnkitRai01
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# Python3 implementation of the above approach 
import numpy as np
  
# Function to return sum of subarray from l to r 
def Sum(arr, l, r) : 
  
    # calculate sum by a loop from l to r 
    s = 0
    for i in range(l, r + 1) :
        s += arr[i]; 
  
    return s; 
  
# dp to store the values of sub problems 
dp = np.zeros((101, 101, 101)); 
  
def solve(arr, l, r, k) :
  
    # if length of the array is less than k 
    # return the sum 
    if (r - l + 1 <= k) :
        return Sum(arr, l, r); 
  
    # if the value is previously calculated 
    if (dp[l][r][k]) :
        return dp[l][r][k]; 
  
    # else calculate the value 
    sum_ = Sum(arr, l, r); 
    len_r = (r - l + 1) - k; 
    length = (r - l + 1); 
    ans = 0
  
    # select all the sub array of length len_r 
    for i in range(length - len_r + 1) :
          
        # get the sum of that sub array 
        sum_sub = Sum(arr, i + l, i + l + len_r - 1); 
  
        # check if it is the maximum or not 
        ans = max(ans, (sum_ - sum_sub) + (sum_sub -
                            solve(arr, i + l, i + l + len_r - 1, k))); 
  
    # store it in the table 
    dp[l][r][k] = ans; 
  
    return ans; 
  
  
# Driver code 
if __name__ == "__main__"
  
    arr = [ 10, 15, 20, 9, 2, 5 ]; k = 2
      
    n = len(arr); 
  
    print(solve(arr, 0, n - 1, k)); 
  
# This code is contributed by AnkitRai01
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// C# implementation of the above approach 
using System;
  
class GFG
{
      
    // Function to return sum of subarray from l to r 
    static int sum(int []arr, int l, int r) 
    
        // calculate sum by a loop from l to r 
        int s = 0; 
        for (int i = l; i <= r; i++)
        
            s += arr[i]; 
        
        return s; 
    
      
    // dp to store the values of sub problems 
    static int [,,]dp = new int[101, 101, 101] ; 
      
    static int solve(int []arr, int l, int r, int k) 
    
        // if length of the array is less than k 
        // return the sum 
        if (r - l + 1 <= k) 
            return sum(arr, l, r); 
      
        // if the value is previously calculated 
        if (dp[l, r, k] != 0) 
            return dp[l, r, k]; 
      
        // else calculate the value 
        int sum_ = sum(arr, l, r); 
        int len_r = (r - l + 1) - k; 
        int len = (r - l + 1); 
        int ans = 0; 
      
        // select all the sub array of length len_r 
        for (int i = 0; i < len - len_r + 1; i++) 
        
            // get the sum of that sub array 
            int sum_sub = sum(arr, i + l, i + l + len_r - 1); 
      
            // check if it is the maximum or not 
            ans = Math.Max(ans, (sum_ - sum_sub) + (sum_sub - 
                    solve(arr, i + l, i + l + len_r - 1, k))); 
        
      
        // store it in the table 
        dp[l, r, k] = ans; 
      
        return ans; 
    
      
    // Driver code 
    public static void Main () 
    {
        int []arr = { 10, 15, 20, 9, 2, 5 };
        int k = 2; 
        int n = arr.Length; 
  
        Console.WriteLine(solve(arr, 0, n - 1, k)); 
    }
}
  
// This code is contributed by AnkitRai01
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Output:
32

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Third year Department of Information Technology Jadavpur University

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