LC 656. Coin Path 【lock, Hard】
Given an array A
(index starts at 1
) consisting of N integers: A1, A2, ..., AN and an integer B
. The integer B
denotes that from any place (suppose the index is i
) in the array A
, you can jump to any one of the place in the array A
indexed i+1
, i+2
, …, i+B
if this place can be jumped to. Also, if you step on the index i
, you have to pay Ai coins. If Ai is -1, it means you can’t jump to the place indexed i
in the array.
Now, you start from the place indexed 1
in the array A
, and your aim is to reach the place indexed N
using the minimum coins. You need to return the path of indexes (starting from 1 to N) in the array you should take to get to the place indexed N
using minimum coins.
If there are multiple paths with the same cost, return the lexicographically smallest such path.
If it's not possible to reach the place indexed N then you need to return an empty array.
Example 1:
Input: [1,2,4,-1,2], 2
Output: [1,3,5]
Example 2:
Input: [1,2,4,-1,2], 1
Output: []
Note:
- Path Pa1, Pa2, ..., Pan is lexicographically smaller than Pb1, Pb2, ..., Pbm, if and only if at the first
i
where Pai and Pbi differ, Pai < Pbi; when no suchi
exists, thenn
<m
. - A1 >= 0. A2, ..., AN (if exist) will in the range of [-1, 100].
- Length of A is in the range of [1, 1000].
- B is in the range of [1, 100].
参考了lee215的解答:
设dp数组中dp[i]为到第i个位置最小花费,那么dp数组就可以求出来。递推公式为
for i in 1 : len:
dp[i] = min(dp[j] + A[i-1]) for j in range(max(0,j-B),j)
大意就是从当前位置往回找B个位置,并把之前的花费和当前的A相加,求最小值。
而又要返回字典序的最小index。在python中可以用min求数组的最小,就是字典序。
Runtime: 236ms, beats 24.14% 时间复杂度(N*B*N),最后一个N是因为比较数组的时候,数组长度是N,空间复杂度(N*N)
class Solution:
def cheapestJump(self, A, B):
"""
:type A: List[int]
:type B: int
:rtype: List[int]
"""
if not A or A[0] == -1: return 0
dp = [[float('inf')] for _ in A]
dp[0] = [A[0], 1]
for j in range(1, len(A)):
if A[j] == -1: continue
dp[j] = min([dp[i][0] + A[j]] + dp[i][1:] + [j+1] for i in range(max(0,j-B),j))
return dp[-1][1:] if dp[-1][0] != float('inf') else []
看来还能再优化,
这是另一种解法,利用堆的性质,同样把花费和路径都放进堆中,每次取最小的一个花费,加上当前的花费再推进堆中。时间复杂度(N*log(B)*N),优化了选取的步骤,但堆中元素每一次比较花费的时间还是O(N)的。
Runtime:68ms beats: 100%
def cheapestJump(self, A, B): N = len(A)
A = ['dummy'] + A
if A[N] == -1: return []
heap = [(A[N], [N])]
new_path = [N]
for i in range(N-1, 0, -1): # From N-1 sweeping to 1
if A[i] == -1: continue while heap:
cost, path = heapq.heappop(heap)
if path[0] <= i + B: break #当前的index加上B后应该大于之前保存的路径的第一个,这样才能连的上。
else: # exhausted heap without finding the previous path
return [] new_cost = cost + A[i]
new_path = [i] + path heapq.heappush(heap, (new_cost, new_path))
heapq.heappush(heap, (cost, path))
return new_path
这题如果用C++,JAVA来做,没有python的min能比较数组或者tuple的性质就麻烦一点。
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