<p>A frog is crossing a river. The river is divided into some number of units, and at each unit, there may or may not exist a stone. The frog can jump on a stone, but it must not jump into the water.</p>

<p>Given a list of <code>stones</code>&nbsp;positions (in units) in sorted <strong>ascending order</strong>, determine if the frog can cross the river by landing on the last stone. Initially, the frog is on the first stone and assumes the first jump must be <code>1</code> unit.</p>

<p>If the frog&#39;s last jump was <code>k</code> units, its next jump must be either <code>k - 1</code>, <code>k</code>, or <code>k + 1</code> units. The frog can only jump in the forward direction.</p>

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

<pre>
<strong>Input:</strong> stones = [0,1,3,5,6,8,12,17]
<strong>Output:</strong> true
<strong>Explanation:</strong> The frog can jump to the last stone by jumping 1 unit to the 2nd stone, then 2 units to the 3rd stone, then 2 units to the 4th stone, then 3 units to the 6th stone, 4 units to the 7th stone, and 5 units to the 8th stone.
</pre>

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

<pre>
<strong>Input:</strong> stones = [0,1,2,3,4,8,9,11]
<strong>Output:</strong> false
<strong>Explanation:</strong> There is no way to jump to the last stone as the gap between the 5th and 6th stone is too large.
</pre>

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

<ul>
	<li><code>2 &lt;= stones.length &lt;= 2000</code></li>
	<li><code>0 &lt;= stones[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>stones[0] == 0</code></li>
	<li><code>stones</code>&nbsp;is sorted in a strictly increasing order.</li>
</ul>

class Solution {
public:
    bool helper(const vector<int>& stones, const int pos, const int lastjump, std::unordered_map<int, vector<short>>& cache) const {
        if(lastjump == 0) {
            return false;
        }
        else if(cache.count(pos) != 0 && cache.find(pos)->second[lastjump] != -1) {
            return cache.find(pos)->second[lastjump];
        }
        auto& insert = cache.emplace(std::make_pair(pos, std::vector<short>(stones.size(), -1))).first->second;
        const auto found = std::lower_bound(stones.cbegin(), stones.cend(), pos);
        if(found == stones.cend() || *found != pos) {
            insert[lastjump] = 0;
            return false;
        }
        else if(found == std::prev(stones.cend())) {
            insert[lastjump] = 1;
            return true;
        }
        const auto res = helper(stones, pos+lastjump+1, lastjump+1, cache) ||
        helper(stones, pos+lastjump, lastjump, cache) ||
        helper(stones, pos+lastjump-1, lastjump-1, cache);
        insert[lastjump] = res ? 1 : 0;
        return res;
    }

    bool canCross(const vector<int>& stones) const {
        if(stones[1] != 1) {
            return false;
        }
        else if(stones.size() == 1) {
            return true;
        }
        const auto maxJump = static_cast<int>(stones.size());
        std::unordered_map<int, std::vector<short>> cache;
        auto lastjump = 1;
        auto pos = 1;
        return helper(stones, pos, lastjump, cache);
    }
};
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