Minimum Levels to Gain More Points

Problem Description

You are given a binary array possible of length n. Alice and Bob are playing a game that consists of n levels. Some of the levels in the game are impossible to clear while others can always be cleared. A player gains 1 point on clearing a level and loses 1 point if the player fails to clear it. At the start of the game, Alice will play some levels in the given order starting from the 0th level, after which Bob will play for the rest of the levels. Alice wants to know the minimum number of levels she should play to gain more points than Bob, if both players play optimally to maximize their points. Return the minimum number of levels Alice should play to gain more points. If this is not possible, return -1. Note that each player must play at least 1 level.

Key Insights

• Alice needs to play levels such that her total points exceed Bob's points.
• Alice and Bob's scores depend on the levels they play, specifically the number of clearable levels.
• The game structure is sequential with Alice playing first and Bob playing the rest.
• If no levels are clearable, it's impossible for Alice to gain more points than Bob.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we will use a greedy approach by iterating through the levels and calculating Alice's potential score based on the number of levels she plays. At each level, we will compute both Alice's and Bob's scores considering that Bob plays optimally to maximize his score after Alice's choice. Specifically, we will:

1. Count the total number of clearable levels.
2. Use a loop to check how many levels Alice must play to ensure her score exceeds Bob's.
3. If at any point Alice's score exceeds Bob's, we note the number of levels she has played.
4. If it's impossible for Alice to ever gain more points, we return -1.

Code Solutions