# Time:  O(n) ~ O(n^2) 
# Space: O(n)

# A frog is crossing a river. The river is divided into x 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.
#
# Given a list of stones' positions (in units) in sorted ascending order,
# determine if the frog is able to cross the river by landing on the last stone.
# Initially, the frog is on the first stone and assume the first jump must be 1 unit.
#
# If the frog has just jumped k units, then its next jump must be
# either k - 1, k, or k + 1 units. Note that the frog can only jump in the forward direction.
#
# Note:
#
# The number of stones is >= 2 and is < 1,100.
# Each stone's position will be a non-negative integer < 231.
# The first stone's position is always 0.
# Example 1:
#
# [0,1,3,5,6,8,12,17]
#
# There are a total of 8 stones.
# The first stone at the 0th unit, second stone at the 1st unit,
# third stone at the 3rd unit, and so on...
# The last stone at the 17th unit.
#
# Return true. 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.
# Example 2:
#
# [0,1,2,3,4,8,9,11]
#
# Return false. There is no way to jump to the last stone as 
# the gap between the 5th and 6th stone is too large.

# DP with hash table
class Solution(object):
    def canCross(self, stones):
        """
        :type stones: List[int]
        :rtype: bool
        """
        if stones[1] != 1:
            return False

        last_jump_units = {s: set() for s in stones}
        last_jump_units[1].add(1)
        for s in stones[:-1]:
            for j in last_jump_units[s]:
                for k in (j-1, j, j+1):
                    if k > 0 and s+k in last_jump_units:
                        last_jump_units[s+k].add(k)
        return bool(last_jump_units[stones[-1]])
