Problem Description
You are given an integer array nums. Two players are playing a game with this array: player 1 and player 2. Player 1 and player 2 take turns, with player 1 starting first. Both players start the game with a score of 0. At each turn, the player takes one of the numbers from either end of the array (i.e., nums[0] or nums[nums.length - 1]) which reduces the size of the array by 1. The player adds the chosen number to their score. The game ends when there are no more elements in the array. Return true if Player 1 can win the game. If the scores of both players are equal, then player 1 is still the winner, and you should also return true. You may assume that both players are playing optimally.
Key Insights
- Players can only pick from the ends of the array.
- Both players are playing optimally, meaning they will always choose the option that maximizes their score while minimizing the opponent's potential score.
- The problem can be approached using dynamic programming to store intermediate results and avoid redundant calculations.
Space and Time Complexity
Time Complexity: O(n^2)
Space Complexity: O(n^2)
Solution
To solve this problem, we can use dynamic programming. We will create a 2D array dp where dp[i][j] represents the maximum score difference that the current player can achieve over the opponent when playing optimally on the subarray from index i to j. The player can choose either the left end (nums[i]) or the right end (nums[j]), and we update the dp table based on these choices. The base case occurs when only one element is left, where the player takes that element. The result for the entire game will be the value at dp[0][n-1], which indicates the score difference for the whole array.