algoadvance

You are given an array of integers nums. Two players take turns, picking either the first or last element from the array, removing it, and gaining the number of points equal to the value of the element they picked. The game ends when there are no more elements to pick.

Given an array of scores, predict whether Player 1 is the winner. If Player 1 wins, return True. If Player 1 loses or the game ends in a draw, return False.

Player 1 always starts first, and both players try to maximize their score.

Clarifying Questions

  1. What are the constraints on the length of the array nums?
    • The length of the array will be between 1 and 20.
  2. Are all elements in the array integers?
    • Yes, all elements are integers, and they can be positive, negative, or zero.
  3. What should be the behavior in case of a draw?
    • If the game ends in a draw, the function should return False.

Strategy

The problem can be approached using a recursive dynamic programming algorithm. We need to keep track of the scores that both players can achieve. To do this, we use memoization to avoid recalculating results for the same subproblems.

Here’s a breakdown of the approach:

  1. Use a recursive function score_diff(l, r) that returns the maximum score difference between Player 1 and Player 2 when they play optimally on the subarray nums[l:r+1].
  2. Initialize a memoization dictionary to store already computed results of subproblems.
  3. The base case is when l == r, which means only one element is left.
  4. If it is Player 1’s turn, they will choose the maximum score they can get by picking either the left or right end.
  5. If it is Player 2’s turn, they will also choose in a similar manner but to minimize Player 1’s advantage.
  6. The difference calculated will be the final score difference.

Code

def PredictTheWinner(nums):
    def score_diff(l, r):
        if l == r:
            return nums[l]
        if (l, r) in memo:
            return memo[(l, r)]
        
        pick_left = nums[l] - score_diff(l + 1, r)
        pick_right = nums[r] - score_diff(l, r - 1)
        
        memo[(l, r)] = max(pick_left, pick_right)
        return memo[(l, r)]

    n = len(nums)
    memo = {}
    return score_diff(0, n - 1) >= 0

# Example usage:
nums = [1, 5, 233, 7]
print(PredictTheWinner(nums))  # Output: True

Time Complexity

The time complexity of this approach is O(n^2) because:

The space complexity is O(n^2) due to the storage required for memoization.

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