Problem Description

Alice and Bob are playing a game with a row of piles of stones, where each pile has a positive number of stones, given by piles[i]. The goal is to end with the most stones. Alice starts the game and the players alternate turns. Initially, the variable M = 1.

On each turn, the current player can take all the stones from the first X piles, where 1 <= X <= 2M. After choosing the number of piles to take the stones from, the player updates M to be the maximum of M and the number X chosen.

The game ends when all stones have been taken. The objective is to maximize the number of stones Alice can obtain if both players play optimally.

Intuition

To find the solution, we need to consider the game from a dynamic programming perspective, specifically using a minimax approach with memorization to handle the overlapping subproblems.

Since Alice aims to maximize the number of stones she can get, while Bob aims to minimize that same value, we can define a recursive function that returns the maximum number of stones a player can achieve from the current state, defined by the index i of the first available pile and the value of M.

The stopping condition for our recursive function dfs(i, m) is when there are 2M or more piles remaining starting from index i. In this case, the current player can take all remaining stones. Otherwise, the player iterates over all valid choices of X (from 1 to 2M), recursively calling dfs(i + x, max(m, x)) which represents the new state after the opponent has played their turn.

The function then chooses the maximum value from these recursive calls, which will represent the optimal decision for the current player. To optimize and avoid recalculating the same states, we use the @cache decorator from the functools module, which memorizes the results of the function calls with the given arguments.

We also utilize the accumulate function from the itertools module to create a prefix sum array, s, which allows us to quickly calculate the sum of stones from any range of piles during the recursive calls. The final answer for the maximum number of stones Alice can get will be the return value of dfs(0, 1), which is the starting state of the game.

Learn more about Math, Dynamic Programming and Prefix Sum patterns.

Solution Approach

The given Python solution uses a depth-first search (DFS) recursive strategy with memoization. The implementation comprises several key parts:

1. Memoization: The @cache decorator from functools is used to automatically store the results of expensive function calls and return the cached result when the same inputs occur again. This optimization reduces the computational complexity by preventing redundant calculations of subproblems.

2. Depth-First Search (DFS) Recursive Function: The dfs(i, m) function is the heart of the solution. Here, i represents the current index starting from which the stones can be taken, and m is the maximum number of piles that can be chosen based on the previous move.

3. Prefix Sum Array: The accumulate function from the itertools module creates a prefix sum array s. It allows for constant-time computation of the sum of stones in any subarray of piles. The initial 0 is added to the sums to make sure that the sum from the start is easily accessible.

4. Minimax Algorithm: Given that Alice wants to maximize her stones and Bob wants to minimize Alice's stones, the algorithm looks at each possible choice of X (from 1 to 2M) that the players can make. For each choice, it recursively calculates what would be the best score Alice can get if she picks X piles and then the opponent plays optimally after her. The result of this recursive call is subtracted from the total number of stones from i to the end (s[n] - s[i]) to get the sequence of optimal choices for Alice.

5. Base Case: If we can take all remaining piles (m * 2 >= n - i), we simply take all the remaining stones (s[n] - s[i]) without further recursion because this is the maximum we can get from this point.

6. Calculation and Return: In the dfs function, we use a generator expression inside the max function to iterate over all possible X within the 1 to 2M range and calculate the corresponding score Alice would get if the optimal choice for X is made.

By calling dfs(0,1) we start the DFS recursion from the beginning of the piles and with an initial M of 1, as per the game's rules. The return value of this call provides the answer to the problem, which is the maximum number of stones Alice can get if both she and Bob play optimally.

Example Walkthrough

Let's illustrate the solution approach with an example. Suppose the piles array is [2, 7, 9, 4], representing four piles with 2, 7, 9, and 4 stones each.

We apply the depth-first search (DFS) recursive function dfs with memoization to compute the maximum number of stones Alice can obtain.

1. Initial Call: Start with dfs(0, 1) with an index of 0 and M of 1.

2. Prefix Sum Array: First, we construct a prefix sum array s: [0, 2, 9, 18, 22]. This represents the total stones one can get by taking piles up to that index. E.g., taking the first three piles yields s[3] - s[0] which equals 18.

3. Possible Choices: Now, Alice can take from 1 to 2 * M piles. Since M is 1, Alice can take 1 or 2 piles.

4. Alice's Turn:

   • If Alice takes 1 pile, she gets 2 stones. Now Bob will use dfs(1, max(1, 1)) = dfs(1, 1) on the remaining [7, 9, 4] piles.
   • If Alice takes 2 piles (1 to 2M), she gets 9 stones (2 + 7). Now Bob will use dfs(2, max(1, 2)) = dfs(2, 2) on the remaining [9, 4] piles.

5. Bob's Turn: Bob plays optimally to give Alice the minimum number of remaining stones.

   • From dfs(1, 1): Bob follows the same logic and has the option to take 1 or 2 piles. If he takes 1 pile, Alice is left with dfs(2, 1) and if he takes 2 piles, Alice is left with dfs(3, 2).
   • From dfs(2, 2): Bob has the option to take 1, 2, 3, or 4 piles. However, only the options taking 1 or 2 piles are valid as 2M = 4 and there are only two piles left. Both lead to Alice not being able to pick any more piles, as Bob would clear the remaining stones.

6. Subproblem Results: The return values of these recursive calls are cached, preventing redundant calculations. Alice's choice will be the one that maximizes her stones considering optimal plays by Bob.

7. Calculation:

   • dfs(0, 1) is the maximum of:
     • 2 + (Total stones - 2 - dfs(1, 1))
     • 9 + (Total stones - 9 - dfs(2, 2))
   • The recursive calls continue breaking down subsequent turns in the same manner until the game ends.

8. Base Case:

   • When dfs(2, 2) is called, M is 2 and there are only 2 piles left, so Alice can take all the remaining stones, returning s[4] - s[2] = 22 - 18 = 4.
   • When dfs(3, 2) is called, M is 2 and there is only 1 pile left, so Alice can take all the remaining stones, returning s[4] - s[3] = 22 - 18 = 4.

By the end of these calculations, we deduce that Alice's optimal strategy yields the most stones based on the initial call dfs(0, 1). This result is the maximum number of stones Alice can collect if both players play optimally. In this case, Alice's best initial move is to take the first two piles, and she will end up with 9 stones.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n^3). This is because in the function dfs(i, m), we iterate up to 2m - 1 times where m can go up to n/2 in the worst case. For each iteration, we have a recursive call, which can lead to a maximum of n levels of recursion (since i can range from 0 to n - 1), and the update of m in the recursive calls could happen n times as well. Since we are memoizing the results of the recursive calls, each state (i, m) is only computed once. The number of unique states for i is n and for m is also up to n, this results in n^2 unique states. Thus the overall time complexity is O(n * n * n) which simplifies to O(n^3).

The space complexity of the function is O(n^2). The memoization cache dfs(i, m) will store at most n * n states, since both i and m range in [0, n]. The list s that stores the accumulated sums introduces another O(n) space, but since it grows linearly with the input, it doesn't affect the overall O(n^2) space complexity dominated by the memoization cache.

Learn more about how to find time and space complexity quickly using problem constraints.