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Minimum Levels to Gain More Points

Number: 3355

Difficulty: Medium

Paid? No

Companies: IBM

Problem Description

Alice and Bob are competing in a game consisting of n levels. Each level is represented in a binary array "possible" where a value of 1 means the level is clearable (gains 1 point when cleared) and 0 means it is impossible to clear (loses 1 point when attempted). Alice plays the levels in order starting from the 0th index and Bob plays the remaining levels. Both players aim to maximize their final score, and each must play at least one level. The goal is to determine the minimum number of levels Alice needs to play so that her score becomes strictly greater than Bob’s score. If this cannot be achieved, return -1.

Key Insights

  • Use a prefix sum to track Alice’s performance as she plays levels sequentially.
  • Calculate the score for any segment of levels using the formula: score = 2*(number of possible levels) - (number of levels played).
  • Precompute the total score for all levels so that Bob's score can be derived by subtracting Alice's score.
  • Iterate from k = 1 to n - 1 ensuring that Bob plays at least one level, and check if Alice’s score exceeds Bob’s score.
  • Return the smallest k that satisfies the condition, or -1 if none exists.

Space and Time Complexity

Time Complexity: O(n) – A single pass is used to calculate prefix sums and check the score condition. Space Complexity: O(1) – Only a few extra variables are used for counting; otherwise, operations are done in-place.

Solution

We approach the problem by simulating the levels Alice plays one-by-one and calculating her cumulative score. The score for a segment is derived from the difference between the number of levels that are clearable and those that are impossible. With the total score (computed from the entire array), we can subtract Alice’s score to obtain Bob's score. The iteration checks for the first split where Alice’s score becomes greater than Bob’s score.

Code Solutions

def min_levels_to_gain_more_points(possible):
    n = len(possible)
    # Total points if a player plays all levels
    total_possible = sum(possible)
    total_score = 2 * total_possible - n  # score = 2*(# possible) - n
    prefix_possible = 0
    # Iterate through possible split points ensuring Bob plays at least one level
    for k in range(1, n):
        prefix_possible += possible[k - 1]  # update count for Alice's possible levels
        alice_score = 2 * prefix_possible - k  # score for first k levels
        bob_score = total_score - alice_score   # remaining levels score for Bob
        if alice_score > bob_score:
            return k
    return -1

# Example usage:
print(min_levels_to_gain_more_points([1, 0, 1, 0]))  # Expected output: 1
print(min_levels_to_gain_more_points([1, 1, 1, 1, 1]))  # Expected output: 3
print(min_levels_to_gain_more_points([0, 0]))          # Expected output: -1
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