<p>You are given an array of integers <code>stones</code> where <code>stones[i]</code> is the weight of the <code>i<sup>th</sup></code> stone.</p>

<p>We are playing a game with the stones. On each turn, we choose any two stones and smash them together. Suppose the stones have weights <code>x</code> and <code>y</code> with <code>x &lt;= y</code>. The result of this smash is:</p>

<ul>
	<li>If <code>x == y</code>, both stones are destroyed, and</li>
	<li>If <code>x != y</code>, the stone of weight <code>x</code> is destroyed, and the stone of weight <code>y</code> has new weight <code>y - x</code>.</li>
</ul>

<p>At the end of the game, there is <strong>at most one</strong> stone left.</p>

<p>Return <em>the smallest possible weight of the left stone</em>. If there are no stones left, return <code>0</code>.</p>

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

<pre>
<strong>Input:</strong> stones = [2,7,4,1,8,1]
<strong>Output:</strong> 1
<strong>Explanation:</strong>
We can combine 2 and 4 to get 2, so the array converts to [2,7,1,8,1] then,
we can combine 7 and 8 to get 1, so the array converts to [2,1,1,1] then,
we can combine 2 and 1 to get 1, so the array converts to [1,1,1] then,
we can combine 1 and 1 to get 0, so the array converts to [1], then that&#39;s the optimal value.
</pre>

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

<pre>
<strong>Input:</strong> stones = [31,26,33,21,40]
<strong>Output:</strong> 5
</pre>

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

<ul>
	<li><code>1 &lt;= stones.length &lt;= 30</code></li>
	<li><code>1 &lt;= stones[i] &lt;= 100</code></li>
</ul>

class Solution {
public:
    int lastStoneWeightII(std::vector<int>& stones) {
        std::vector<std::unordered_map<int, int>> memo(stones.size());
        int stone_sum = std::accumulate(stones.begin(), stones.end(), 0);
        int closest_half = dp(stones, 0, stone_sum / 2, memo);
        return stone_sum - 2*closest_half;
    }
private:
    int dp(const std::vector<int> &stones, int i, int target, std::vector<std::unordered_map<int, int>> &memo) {
        if (target < 0) { // TAKE NOTE OF THE ORDER OF BASE CASES!!!
            return std::numeric_limits<int>::max();
        } else if (i == stones.size() || target == 0) {
            return 0;
        } else if (memo[i].find(target) != memo[i].end()) {
            return memo[i][target];
        }
        int choose_opt = dp(stones, i + 1, target - stones[i], memo);
        int ignore_opt = dp(stones, i + 1, target, memo);
        if (choose_opt == std::numeric_limits<int>::max()) {
            memo[i][target] = ignore_opt;
        } else {
            memo[i][target] = std::max(choose_opt + stones[i], ignore_opt);
        }
        return memo[i][target];
    }
};
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