Stone Game VI

Problem Description

Alice and Bob take turns playing a game, with Alice starting first. There are n stones in a pile, and on each player's turn, they can remove a stone from the pile and receive points based on the stone's value. Alice and Bob may value the stones differently. Given two integer arrays, aliceValues and bobValues, determine the result of the game: return 1 if Alice wins, -1 if Bob wins, or 0 if the game results in a draw.

Key Insights

• Both players play optimally and are aware of the other's values for the stones.
• The strategy for both players is to maximize their own score while minimizing the opponent's potential score.
• The stones should be prioritized based on the combined value both players assign to them, as this influences the strategy for optimal plays.

Space and Time Complexity

Time Complexity: O(n log n) - sorting the stones based on their combined values requires O(n log n) time. Space Complexity: O(n) - storing the combined values and indices of the stones.

Solution

To solve the problem, we can follow these steps:

1. Create a list of tuples where each tuple contains the combined value of a stone for both players (aliceValues[i] + bobValues[i]) and its index.
2. Sort this list in descending order based on the combined values. This allows players to prioritize stones that have the highest total value.
3. Iterate through the sorted list, assigning points to Alice and Bob based on whose turn it is (Alice takes the first stone, Bob the second, and so on).
4. At the end of the iteration, compare the scores of Alice and Bob to determine the winner.

Code Solutions