<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:
    def canCross(self, stones: List[int]) -> bool:
        stone_to_id = {stone: i for i, stone in enumerate(stones)}
        cache = {}

        def dfs(i, step):
            if (i, step) in cache:
                return cache[(i, step)]
            if i == len(stones) - 1:
                return True
            ret = False
            for n_step in [step - 1, step, step + 1]:
                n_stone = stones[i] + n_step
                if n_step > 0 and n_stone in stone_to_id:
                    n_stone_id = stone_to_id[n_stone]
                    ret |= dfs(n_stone_id, n_step)
            cache[(i, step)] = ret
            return ret
        return dfs(0, 0)


        # stone_to_id = {stone: i for i, stone in enumerate(stones)}
        # cache = {}

        # def dfs(index, step):
        #     if (index, step) in cache:
        #         return cache[(index, step)]

        #     if index == len(stones) - 1:
        #         return True
        #     if index > len(stones) - 1:
        #         cache[(index, step)] = False
        #         return False
            
        #     ret = False
        #     for next_step in [step - 1, step, step + 1]:
        #         if next_step > 0 and stones[index] + next_step in stone_to_id:
        #             ret |= dfs(stone_to_id[stones[index] + next_step], next_step)

        #     cache[(index, step)] = ret
        #     return ret

        # return dfs(0, 0)
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