# your code goes here
def articulation_points_and_bridges(n, roads):
def dfs(u, parent):
nonlocal time, articulation_points, bridges
visited[u] = True
discovery[u] = low[u] = time
time += 1
children = 0
for v in graph[u]:
if not visited[v]:
children += 1
parent[v] = u
dfs(v, parent)
low[u] = min(low[u], low[v])
if low[v] > discovery[u]:
bridges.append((u, v))
if parent[u] == -1 and children > 1:
articulation_points.add(u)
if parent[u] != -1 and low[v] >= discovery[u]:
articulation_points.add(u)
elif v != parent[u]:
low[u] = min(low[u], discovery[v])
graph = [[] for _ in range(n + 1)]
for road in roads:
u, v = road
graph[u].append(v)
graph[v].append(u)
visited = [False] * (n + 1)
discovery = [float('inf')] * (n + 1)
low = [float('inf')] * (n + 1)
parent = [-1] * (n + 1)
time = 0
articulation_points = set()
bridges = []
for i in range(1, n + 1):
if not visited[i]:
dfs(i, parent)
return articulation_points, bridges
# Hidden Test Cases
hidden_test_cases = [
# Easy Cases
(1, []), # Single intersection with no roads
(5, [[1, 2], [2, 3], [3, 4], [4, 5]]), # Linear road network with no articulation points or bridges
(4, [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]), # Complete graph with multiple articulation points and bridges
# Medium Cases
(4, [[1, 2], [2, 3], [3, 4], [1, 3]]), # Grid-like road network with multiple articulation points but no bridges
(5, [[1, 2], [1, 3], [2, 3], [3, 4], [4, 5], [2, 5]]), # Random road network with varying degrees of connectivity
(5, [[1, 2], [2, 3], [3, 4], [4, 5], [5, 1], [2, 4], [3, 5]]), # Road network with loops and cycles
(4, [[1, 2], [1, 2], [2, 3], [2, 3], [3, 4], [4, 4]]), # Road network with duplicate edges and self-loops
# Hard Cases
(8, [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 2], [8, 3]]), # Large-scale road network with thousands of intersections and roads
(7, [[1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [3, 5], [4, 5], [4, 6], [5, 6], [5, 7]]), # Highly interconnected road network with numerous articulation points and bridges
(8, [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [4, 8], [3, 7]]) # Road network with complex topology, including multiple disconnected components
]
# Function Calls for Hidden Test Cases
for i, (n, roads) in enumerate(hidden_test_cases, 1):
print(f"Hidden Test Case {i}:")
print("Input:", n)
for road in roads:
print(*road)
articulation_points, bridges = articulation_points_and_bridges(n, roads)
print("Output:")
print("Articulation Points:", ", ".join(map(str, articulation_points)))
print("Bridges:", ", ".join(map(str, bridges)))
print()
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