In Stone Game VII, Alice and Bob take turns removing stones to maximize or minimize the score difference. The optimal strategy uses state transition dynamic programming to compute scores based on subarrays of remaining stones. By considering every possible left or right removal and caching results, the solution efficiently determines the maximum achievable score difference for Alice against Bob's minimizing strategy.
Problem Statement
Alice and Bob play a game with a row of n stones. On each turn, a player removes either the leftmost or rightmost stone and earns points equal to the sum of the remaining stones. The game ends when no stones remain, and the player with the higher score wins.
Alice aims to maximize the score difference while Bob tries to minimize it. Implement a function that, given the initial array of stone values, returns the maximum difference Alice can achieve assuming both players play optimally.
Examples
Example 1
Input: stones = [5,3,1,4,2]
Output: 6
- Alice removes 2 and gets 5 + 3 + 1 + 4 = 13 points. Alice = 13, Bob = 0, stones = [5,3,1,4].
- Bob removes 5 and gets 3 + 1 + 4 = 8 points. Alice = 13, Bob = 8, stones = [3,1,4].
- Alice removes 3 and gets 1 + 4 = 5 points. Alice = 18, Bob = 8, stones = [1,4].
- Bob removes 1 and gets 4 points. Alice = 18, Bob = 12, stones = [4].
- Alice removes 4 and gets 0 points. Alice = 18, Bob = 12, stones = []. The score difference is 18 - 12 = 6.
Example 2
Input: stones = [7,90,5,1,100,10,10,2]
Output: 122
Example details omitted.
Constraints
- n == stones.length
- 2 <= n <= 1000
- 1 <= stones[i] <= 1000
Solution Approach
Precompute Prefix Sums
Calculate prefix sums of the stones array to quickly compute the sum of any subarray in constant time. This enables efficient computation of points earned after removing a stone on either end.
Dynamic Programming State Definition
Define dp[i][j] as the maximum score difference the current player can achieve for the subarray stones[i..j]. Transition using dp[i+1][j] and dp[i][j-1] combined with the remaining sum after removing left or right stones.
Iterative DP Filling
Fill the DP table for increasing subarray lengths, ensuring smaller subproblems are solved first. The final result is dp[0][n-1], representing the maximum difference Alice can achieve for the entire array.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(n^2) due to filling an n x n DP table and computing transitions in constant time using prefix sums. Space complexity is O(n^2) for the DP table, reducible to O(n) with optimization since only adjacent subarray results are needed at each step.
What Interviewers Usually Probe
- Candidate immediately considers DP subarray state transitions.
- Candidate tries a naive greedy approach and needs redirection to subarray sums.
- Candidate correctly identifies that prefix sums optimize repeated sum computations.
Common Pitfalls or Variants
Common pitfalls
- Forgetting to subtract the removed stone from the remaining sum when computing points.
- Incorrectly defining dp state leading to miscounted score differences.
- Overlooking base cases for single-stone subarrays.
Follow-up variants
- Allow players to remove up to k stones from either end per turn.
- Modify scoring so removed stone values are added instead of remaining sums.
- Extend to three players with cyclic turns and maximize score difference for first player.
How GhostInterview Helps
- GhostInterview guides through DP subarray setup and prefix sum optimization.
- It highlights score difference calculation mistakes specific to this pattern.
- It walks through example arrays like [5,3,1,4,2] to validate state transitions and final output.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the best approach for Stone Game VII?
Use state transition dynamic programming with prefix sums to compute maximum score difference efficiently.
Can Stone Game VII be solved using recursion alone?
Recursion without memoization is too slow; memoization or iterative DP is required for n up to 1000.
How are points calculated when removing a stone?
Points equal the sum of the remaining stones after removing either the leftmost or rightmost stone.
Why use prefix sums in Stone Game VII?
Prefix sums allow O(1) calculation of remaining subarray sums, avoiding repeated O(n) summations per move.
What common mistakes occur with state transition DP in this problem?
Misdefining dp[i][j] or failing to account for opponent's minimizing strategy often leads to incorrect maximum differences.
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