程式碼隨想錄演算法訓練營第63天 | SPFA演算法最佳化+變式

哆啦**發表於2024-08-08
演算法

94.城市間貨物運輸 I
https://kamacoder.com/problempage.php?pid=1152
Bellman_ford 佇列最佳化演算法（又名SPFA）
https://www.programmercarl.com/kamacoder/0094.城市間貨物運輸I-SPFA.html
95.城市間貨物運輸 II
https://kamacoder.com/problempage.php?pid=1153
bellman_ford之判斷負權迴路
https://www.programmercarl.com/kamacoder/0095.城市間貨物運輸II.html
96.城市間貨物運輸 III
https://kamacoder.com/problempage.php?pid=1154
bellman_ford之單源有限最短路
https://www.programmercarl.com/kamacoder/0096.城市間貨物運輸III.html

94. 城市間貨物運輸 I — SPFA

  • 最佳化方式

    • 採用佇列
    • 只按照邊和節點進行遍歷（不需要n-1遍）
    點選檢視程式碼 
    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. 城市間貨物運輸 II - 判斷負權迴路

  • 問題：判斷是否存在負權迴路

  • 在SPFA中

  • 鬆弛n-1次

    • 如果有沒有負權迴路，n-1次鬆弛所有的邊能得到起點到節點的最短路徑；n次以上minDist陣列中的結果不會有改變；
    • 但是如果第n次也改變：說明存在負權迴路；
    點選檢視程式碼 
    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()

  • 佇列最佳化

    • 如果某個點更新超過n-1次，說明存在負權迴路；
    點選檢視程式碼 
    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. 城市間貨物運輸 III

  • 限制條件：在最多經過 k 個城市的條件下，從城市 src 到城市 dst 的最低運輸成本。

  • 分析：

    • 等價於：起點最多經過k + 1 條邊到達終點的最短距離

  • 鬆弛n-1次

    • 限制迴圈k-1次
    • 複製mindist保證每次更新一步
    點選檢視程式碼 
    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()

  • 佇列最佳化

    • 使用visited陣列記錄遍歷過的點
    • 限制遍歷次數k
    點選檢視程式碼 
    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()

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