Ms.Kox enjoys her job, but she does not like to waste extra time traveling to and from her office. After working for many years, she knows the shortest-distance route to her office on a regular day.

Recently, the city began regular maintenance of various roads. Every day a road gets blocked and no one can use it that day, but all other roads can be used. You are Ms. Kox's new intern and she needs some help. Every day, you need to determine the minimum distance that she has to travel to reach her office.

Input Format

There are N cities numbered 0 to N-1 and M bidirectional roads.  

The first line of the input contains two integers N and M.  
M lines follow, each containing three space-separated integers u , v and w, where u and v are cities connected by a bi-directional road and w is the length of this road. There is at most one road between any two cities and no road connects a city to itself.  
The next line contains two integers S and D. S is the city where Ms. Kox lives and D is the city where her office is located.  
The next line contains an integer Q, the number of queries.  
Q lines follow, each containing two integers u and v, where the road between u and v has been blocked that day.

Constraints

$0<N<200,000$ 

$0<M<200,000$ 

$0<Q<200,000$ 

$0<w<1001$ 

$0\leq S,D<N$

Output Format

Output Q lines, with each line containing the minimum distance Ms.Kox has to travel on that day. If there is no path, print "Infinity".

Sample Input
6 9  
0 1 1  
1 2 1  
2 3 1  
3 4 1  
4 5 1  
2 4 5  
3 5 8  
1 3 3  
0 2 4  
0 5  
9  
0 1  
1 2  
2 3  
3 4  
4 5  
2 4  
3 5  
1 3  
0 2

Sample Output
7  
6  
6  
8  
11  
5  
5  
5  
5

Difficulty: HARD
Tags: Graph algorithms

Starter code:

Solutions:
import math
import copy
from heapq import heappush, heappop
(cities, roads) = map(int, input().split())
graph = {n: [] for n in range(cities)}
edges = []
for road in range(roads):
	(fr, to, dist) = map(int, input().strip().split())
	edges.append((fr, to, dist))
	graph[fr].append((to, dist))
	graph[to].append((fr, dist))
graph_copy = copy.deepcopy(graph)
(start, finish) = map(int, input().strip().split())
number_of_queries = int(input().strip())
queries = []
for q in range(number_of_queries):
	queries.append(tuple(map(int, input().strip().split())))

def min_path_length(g, s, f):
	city_list = [{'visited': False, 'dist': math.inf} for c in range(cities)]
	pq = []
	heappush(pq, (0, s))
	city_list[s]['dist'] = 0
	while pq:
		(current_dist, current_city) = heappop(pq)
		if city_list[current_city]['visited']:
			continue
		city_list[current_city]['visited'] = True
		for neighbor in g[current_city]:
			if not city_list[neighbor[0]]['visited']:
				old_dist = city_list[neighbor[0]]['dist']
				new_dist = current_dist + neighbor[1]
				if new_dist < old_dist:
					heappush(pq, (new_dist, neighbor[0]))
					city_list[neighbor[0]]['dist'] = new_dist
	return [x['dist'] for x in city_list]

def optimal_edges(edges, ds, dt, opt):
	result = {}
	for edge in edges:
		(u, v, dist) = edge
		if ds[u] + dist + dt[v] == opt:
			result[u, v] = dist
		if ds[v] + dist + dt[u] == opt:
			result[v, u] = dist
	return result

def bridges(opt_edges, ds, opt):
	result = {}
	sorted_nodes = sorted([((ds[u], ds[v]), (u, v)) for (u, v) in opt_edges.keys()])
	sorted_nodes = [((0, 0), (start, start)), *sorted_nodes, ((opt, opt), (finish, finish))]
	for i in range(1, len(sorted_nodes) - 1):
		if sorted_nodes[i][0][0] >= sorted_nodes[i - 1][0][1] and sorted_nodes[i][0][1] <= sorted_nodes[i + 1][0][0]:
			result[sorted_nodes[i][1]] = opt_edges[sorted_nodes[i][1]]
	return result

def arrowed_graph(edges, ds):
	a_graph = {n: [] for n in range(cities)}
	for (u, v, dist) in edges:
		if ds[u] + dist == ds[v]:
			a_graph[u].append(v)
		if ds[v] + dist == ds[u]:
			a_graph[v].append(u)
	return a_graph

def islands(ag, bridges):
	isl = [0] * cities
	sorted_bridges = sorted([((ds[u], ds[v]), (u, v)) for (u, v) in bridges.keys()])
	sorted_bridges_ends = [v for ((dsu, dsv), (u, v)) in sorted_bridges]

	def dfs(fr, ag, island_id):
		stack = []
		stack.append(fr)
		while stack:
			current_node = stack.pop()
			isl[current_node] = island_id
			for node in ag[current_node]:
				if isl[node] == 0:
					stack.append(node)
		return isl
	isl_count = len(bridges) + 1
	for i in reversed(sorted_bridges_ends):
		dfs(i, ag, isl_count)
		isl_count -= 1
	dfs(start, ag, isl_count)
	return isl

def bypasses(edges, bridges, islands, ds, dt):
	result = {(u, v): math.inf for (u, v) in bridges.keys()}
	island_to_bridge = {min(islands[u], islands[v]): (u, v) for (u, v) in bridges}
	edges_to_iterate = {(u, v): dist for (u, v, dist) in edges if (u, v) not in bridges.keys() and (v, u) not in bridges.keys()}
	for (u, v) in edges_to_iterate.keys():
		if islands[u] < islands[v]:
			for i in range(islands[u], islands[v]):
				result[island_to_bridge[i]] = min(result[island_to_bridge[i]], ds[u] + edges_to_iterate[u, v] + dt[v])
		if islands[v] < islands[u]:
			for i in range(islands[v], islands[u]):
				result[island_to_bridge[i]] = min(result[island_to_bridge[i]], ds[v] + edges_to_iterate[u, v] + dt[u])
	return result
ds = min_path_length(graph, start, finish)
dt = min_path_length(graph, finish, start)
opt = ds[finish]
opt_edges = optimal_edges(edges, ds, dt, opt)
br = bridges(opt_edges, ds, opt)
ag = arrowed_graph(edges, ds)
isl = islands(ag, br)
bp = bypasses(edges, br, isl, ds, dt)
for (u, v) in queries:
	if (u, v) in br:
		print(bp[u, v] if bp[u, v] < math.inf else 'Infinity')
	elif (v, u) in br:
		print(bp[v, u] if bp[v, u] < math.inf else 'Infinity')
	else:
		print(opt)