<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:
        """
        i_i__i__i_i__i____i_____i
        """
        if stones[0] != 0 or stones[1] != 1:
            return False

        rocks = {pos: False for pos in stones}
        memo = {}
        def helper(dist, k):
            if (dist, k) in memo:
                return memo[(dist, k)]
            if dist not in rocks:
                return False
            if dist == stones[-1]:
                rocks[dist] = True
            for i in range(-1, 2):
                jump = k+i
                next_dist = dist+jump
                if jump > 0 and next_dist in rocks:
                    if helper(next_dist, jump):
                        return True
            memo[(dist, k)] = False
            return False
        helper(stones[1], 1)
        return rocks[stones[-1]]
            







        # -------------------
        # bfs
        # q = collections.deque()
        # rocks = set(stones)
        # q.append((1,1))
        # while q:
        #     dist, k = q.popleft()
        #     if dist == stones[-1]:
        #         return True
        #     for i in range(-1, 2):
        #         jump = k+i
        #         next_dist = dist+jump
        #         if next_dist in rocks and jump > 0:
        #             q.append((next_dist, jump))
        # return False
        # ------------------
        #failed bfs
        # q = collections.deque()

        # rocks = {pos: i for pos, i in enumerate(stones)}
        # q.append((1,1))
        # while q:
        #     dist, k = q.popleft()
    #         if rocks[dist] == len(stones)-1: -> fails, idk why
    #             return True
    #         for i in range(-1, 2):
    #             if k + i > 0 and dist + k + i in rocks:
    #                 q.append((dist+k+i, k+i))
        # return False
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