Stone Game III

The key idea

Don't track Alice's and Bob's scores separately. Because both play optimally and the roles are symmetric, track a single value per suffix: dp[i] = the best score difference (current mover minus the other) achievable from index i onward. The current mover picks the k (1, 2, or 3 stones) that maximizes their taken sum minus the opponent's best from the rest.

Problem

Alice and Bob continue their games with piles of stones. There are several stones arranged in a row, and each stone has an associated value which is an integer given in the array values.

Alice and Bob take turns, with Alice starting first. On each player's turn, that player can take 1, 2, or 3 stones from the first remaining stones in the row.

The score of each player is the sum of the values of the stones taken. The score of each player is 0 initially.

The objective of the game is to end with the highest score, and the winner is the player with the highest score and there could be a tie. The game continues until all the stones have been taken.

Assume Alice and Bob play optimally.

Return "Alice" if Alice will win, "Bob" if Bob will win, or "Tie" if they will end the game with the same score.

Constraints

1 <= values.length <= 50000
-1000 <= values[i] <= 1000

Examples

Input: values = [1,2,3,7] Output: "Bob"

Input: values = [1,2,3,-9] Output: "Alice"

Input: values = [1,2,3,6] Output: "Tie"

Complexity

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

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