Problem Description
Alice and Bob play a game with piles of stones. There are an even number of piles arranged in a row, and each pile has a positive integer number of stones. The objective of the game is to end with the most stones. The total number of stones across all the piles is odd, so there are no ties. Alice and Bob take turns, with Alice starting first. Each turn, a player takes the entire pile of stones either from the beginning or from the end of the row. This continues until there are no more piles left, at which point the person with the most stones wins. Assuming Alice and Bob play optimally, return true if Alice wins the game, or false if Bob wins.
Key Insights
- The game involves strategic decision-making where players must consider not only their immediate gain but also the future implications of their choices.
- Since the total number of stones is odd and both players play optimally, Alice, starting first, can always secure a winning strategy.
- The problem can be approached using dynamic programming, where we calculate the maximum stones a player can collect given a range of piles.
Space and Time Complexity
Time Complexity: O(n^2)
Space Complexity: O(n)
Solution
The solution involves using dynamic programming to keep track of the maximum stones that can be collected by Alice and Bob. We define a DP table where dp[i][j] represents the maximum number of stones a player can collect from the piles between indices i and j. The optimal move for a player is to choose either the first or the last pile, and the decision will also affect the remaining piles. The formula used to update the DP table takes into account the best choice between the two options (taking from the start or the end).