代码随想录算法训练营第63天 | SPFA算法优化+变式

94.城市间货物运输 I
https://kamacoder.com/problempage.php?pid=1152
Bellman_ford 队列优化算法（又名SPFA）
https://www.programmercarl.com/kamacoder/0094.城市间货物运输I-SPFA.html
95.城市间货物运输 II
https://kamacoder.com/problempage.php?pid=1153
bellman_ford之判断负权回路
https://www.programmercarl.com/kamacoder/0095.城市间货物运输II.html
96.城市间货物运输 III
https://kamacoder.com/problempage.php?pid=1154
bellman_ford之单源有限最短路
https://www.programmercarl.com/kamacoder/0096.城市间货物运输III.html

94. 城市间货物运输 I — SPFA

优化方式

采用队列
只按照边和节点进行遍历（不需要n-1遍）

点击查看代码

from collections import deque
class edge:
	def __init__(self,t,v):
		self.to = t
		self.val = v

def spfa(n,m,edges,start,end):
	isqueue = [False]*(n+1)
	mindist = [float("inf")]*(n+1)
	mindist[start] = 0
	que = deque([start])

	while que:
		curr = que.popleft()
		isqueue[curr] = False
		if len(edges[curr])>0:
			for ed in edges[curr]:
				to = ed.to
				val = ed.val
				if mindist[to]>mindist[curr]+val:
					mindist[to] = mindist[curr]+val
					if not isqueue[to]:
						que.append(to)
						isqueue[to] = True

	if mindist[end]==float("inf"):
		return "unconnected"
	else:
		return mindist[end]


def main():
	n,m = map(int,input().split())
	edges = [[] for _ in range(n+1)]
	# print(edges)
	for _ in range(m):
		s,t,v = map(int,input().split())
		edges[s].append(edge(t,v))
	res = spfa(n,m,edges,1,n)
	print(res)

if __name__ == '__main__':
	main()

95. 城市间货物运输 II - 判断负权回路

问题：判断是否存在负权回路
在SPFA中

松弛n-1次

如果有没有负权回路，n-1次松弛所有的边能得到起点到节点的最短路径；n次以上minDist数组中的结果不会有改变；
但是如果第n次也改变：说明存在负权回路；

点击查看代码

def bellman(n,m,edges,start,end):
	mindist = [float("inf")]*(n+1)
	mindist[start] = 0
	count = [0]*(n+1)

	for i in range(n):
		update = False
		for s,t,v in edges:
			if mindist[s]!=float("inf") and mindist[t]>mindist[s]+v:
				mindist[t] = mindist[s]+v
				update = True
		if not update:
			break

	for s,t,v in edges:
		if mindist[s]!=float("inf") and mindist[t]>mindist[s]+v:
			return "circle"

	if mindist[end]==float("inf"):
		return "unconnected"
	else:
		return mindist[end]

def main():
	n,m = map(int,input().split())
	grid = [[float("inf")]*(n+1) for _ in range(n+1)]
	# print(edges)
	edges = []
	for _ in range(m):
		s,t,v = map(int,input().split())
		edges.append((s,t,v))
	res = bellman(n,m,edges,1,n)
	print(res)
if __name__ == '__main__':
	main()

队列优化

如果某个点更新超过n-1次，说明存在负权回路；

点击查看代码

from collections import deque
class edge:
	def __init__(self,t,v):
		self.to = t
		self.val = v

def spfa(n,m,edges,start,end):
	isqueue = [False]*(n+1)
	mindist = [float("inf")]*(n+1)
	mindist[start] = 0
	que = deque([start])
	count = [0]*(n+1)

	while que:
		curr = que.popleft()
		isqueue[curr] = False
		if len(edges[curr])>0:
			for ed in edges[curr]:
				to = ed.to
				val = ed.val
				if mindist[to]>mindist[curr]+val:
					mindist[to] = mindist[curr]+val
					count[to]+=1
					if count[to]==n:
						return "circle"
					if not isqueue[to]:
						que.append(to)
						isqueue[to] = True

	if mindist[end]==float("inf"):
		return "unconnected"
	else:
		return mindist[end]

def main():
	n,m = map(int,input().split())
	edges = [[] for _ in range(n+1)]
	# print(edges)
	for _ in range(m):
		s,t,v = map(int,input().split())
		edges[s].append(edge(t,v))
	res = spfa(n,m,edges,1,n)
	print(res)
if __name__ == '__main__':
	main()

96. 城市间货物运输 III

限制条件：在最多经过 k 个城市的条件下，从城市 src 到城市 dst 的最低运输成本。
分析：
- 等价于：起点最多经过k + 1 条边到达终点的最短距离

松弛n-1次

限制循环k-1次
复制mindist保证每次更新一步

点击查看代码

from collections import deque

class edge:
	def __init__(self,t,v):
		self.to = t
		self.val = v
def spfa(n,m,edges,start,end,k):
	mindist = [float("inf")]*(n+1)
	mindist[start]=0
	isqueue = [False]*(n+1)
	queue = deque([start])
	k = k+1
	while k and queue:
		k -= 1
		mindist_copy = mindist[:]
		visited = [False]*(n+1)
		que_size = len(queue)
		for _ in range(que_size):
			curr = queue.popleft()
			if len(edges[curr])>0:
				for ed in edges[curr]:
					from_ = curr
					to = ed.to
					val = ed.val
					if mindist[to]>mindist_copy[curr]+val:
						mindist[to] = mindist_copy[curr]+val
						if not visited[to]:
							queue.append(to)
							visited[to] = True
		# print(mindist)
	if mindist[end]!=float("inf"):
		return mindist[end]
	else:
		return "unconnected"
def main():
	n,m = map(int,input().split())
	edges = [[] for _ in range(n+1)]
	# print(edges)
	for _ in range(m):
		s,t,v = map(int,input().split())
		edges[s].append(edge(t,v))
	src,dst,k = map(int,input().split())
	res = spfa(n,m,edges,src,dst,k)
	print(res)
if __name__ == '__main__':
	main()

队列优化

使用visited数组记录遍历过的点
限制遍历次数k

点击查看代码

def bellman(n, m, edges, start, end, k):
	mindist = [float("inf")] * (n + 1)
	mindist[start] = 0
	for i in range(1, k + 2):
		mindist_copy = mindist[:]
		update = False
		for s, t, v in edges:
			if mindist_copy[s] != float("inf") and mindist[t] > mindist_copy[s] + v:
				mindist[t] = mindist_copy[s] + v
				update = True
		if not update:
			break
	if mindist[end] == float("inf"):
		return "unreachable"
	else:
		return mindist[end]
def main():
	n, m = map(int, input().split())
	edges = []
	for _ in range(m):
		s, t, v = map(int, input().split())
		edges.append((s, t, v))
	src, dst, k = map(int, input().split())
	res = bellman(n, m, edges, src, dst, k)
	print(res)
if __name__ == '__main__':
	main()

posted @ 2024-08-08 12:04 哆啦** 阅读(2) 评论(0) 编辑收藏举报

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代码随想录算法训练营第63天 | SPFA算法优化+变式

94. 城市间货物运输 I — SPFA

95. 城市间货物运输 II - 判断负权回路

96. 城市间货物运输 III

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