<p>Alice and Bob continue their games with piles of stones. There are a number of piles <strong>arranged in a row</strong>, and each pile has a positive integer number of stones <code>piles[i]</code>. The objective of the game is to end with the most stones.</p>

<p>Alice and Bob take turns, with Alice starting first.</p>

<p>On each player&#39;s turn, that player can take <strong>all the stones</strong> in the <strong>first</strong> <code>X</code> remaining piles, where <code>1 &lt;= X &lt;= 2M</code>. Then, we set <code>M = max(M, X)</code>. Initially, M = 1.</p>

<p>The game continues until all the stones have been taken.</p>

<p>Assuming Alice and Bob play optimally, return the maximum number of stones Alice can get.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">piles = [2,7,9,4,4]</span></p>

<p><strong>Output:</strong> <span class="example-io">10</span></p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>If Alice takes one pile at the beginning, Bob takes two piles, then Alice takes 2 piles again. Alice can get <code>2 + 4 + 4 = 10</code> stones in total.</li>
	<li>If Alice takes two piles at the beginning, then Bob can take all three piles left. In this case, Alice get <code>2 + 7 = 9</code> stones in total.</li>
</ul>

<p>So we return 10 since it&#39;s larger.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">piles = [1,2,3,4,5,100]</span></p>

<p><strong>Output:</strong> <span class="example-io">104</span></p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= piles.length &lt;= 100</code></li>
	<li><code>1 &lt;= piles[i]&nbsp;&lt;= 10<sup>4</sup></code></li>
</ul>

#include <vector>

class Solution {
public:
    int helper(int pos, int limit) {
        if (pos < n_ && pos + limit >= n_) {
            return dp_[pos][limit] = cumulated_[pos];
        }

        if (pos >= n_) {
            return 0;
        }

        if (dp_[pos][limit] != -1) {
            return dp_[pos][limit];
        }

        int best = 0;
        for (int step = 1; step <= 2 * limit; ++step) {
            int forward = helper(pos + step, std::max(limit, step));
            best = std::max(best, cumulated_[pos] - forward);
        }
        return dp_[pos][limit] = best;
    }

    int stoneGameII(std::vector<int>& piles) {
        n_ = static_cast<int>(piles.size());
        piles_ = piles;
        
        cumulated_ = std::vector<int>(n_ + 1, 0);
        for (int i = n_ - 1; i >= 0; --i) {
            cumulated_[i] = piles[i] + cumulated_[i + 1];
        }

        dp_ = std::vector<std::vector<int>>(n_, std::vector<int>(2 * n_, -1));

        return helper(0, 1);
    }

private:
    int n_;
    std::vector<int> piles_;
    std::vector<int> cumulated_;
    std::vector<std::vector<int>> dp_; // [position][limit]
};
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