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Maximize the Minimum Game Score

Difficulty: Hard

Problem Description

You are given an array points of size n and an integer m. There is another array gameScore of size n, where gameScore[i] represents the score achieved at the ith game. Initially, gameScore[i] == 0 for all i. You start at index -1, which is outside the array (before the first position at index 0). You can make at most m moves. In each move, you can either increase the index by 1 and add points[i] to gameScore[i], or decrease the index by 1 and add points[i] to gameScore[i]. The index must always remain within the bounds of the array after the first move. Return the maximum possible minimum value in gameScore after at most m moves.

Key Insights

  • The goal is to maximize the minimum score across the gameScore array after making m moves.
  • A binary search approach can be used to efficiently determine the maximum possible minimum score.
  • For each mid value in the binary search, a greedy check can be performed to see if it's feasible to achieve that minimum score within the allowed moves.
  • The number of moves required to achieve a specific minimum score can be calculated based on the difference between the target score and the current score for each index.

Space and Time Complexity

Time Complexity: O(n log(max(points)))
Space Complexity: O(1)

Solution

To solve the problem, we will use a binary search algorithm to find the maximum possible minimum score that can be achieved in the gameScore array. The key steps involved are:

  1. Initialize the binary search range with low set to 1 (minimum score) and high set to the maximum value in points.
  2. For each mid value in the binary search, determine if it is possible to achieve at least mid as the minimum score in gameScore within m moves.
  3. To check feasibility, iterate through the points array and calculate the required moves to increase each gameScore[i] to at least mid.
  4. If the total required moves exceed m, then mid is not feasible; otherwise, it is feasible.
  5. Adjust the binary search range based on the feasibility check until the maximum possible minimum score is found.

Code Solutions

def maximize_minimum_score(points, m):
    def can_achieve_minimum(mid):
        moves = 0
        for point in points:
            if point < mid:
                moves += mid - point
                if moves > m:
                    return False
        return moves <= m

    low, high = 1, max(points)
    while low < high:
        mid = (low + high + 1) // 2
        if can_achieve_minimum(mid):
            low = mid
        else:
            high = mid - 1
    return low
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