Welcome to our comprehensive guide on Bridges in Graph using Tarjan's Algorithm! In this lesson, we will explore a crucial concept in graph theory, helping you understand and apply it in real-world projects.
A bridge in a graph is an edge whose removal increases the number of connected components. In other words, if we remove a bridge, the graph splits into more than one connected component.
Tarjan's algorithm is a popular algorithm used for finding strongly connected components (SCC) and detecting bridges in a directed graph. Let's delve into its working steps.
First, let's create a directed graph with some nodes and edges.
# Example graph
graph = {
'A': ['B', 'C', 'D'],
'B': ['A', 'D', 'E'],
'C': ['A', 'F'],
'D': ['A', 'B', 'F'],
'E': ['B', 'F'],
'F': ['C', 'D', 'E']
}We perform a Depth First Search (DFS) on the graph, marking each node as visited, and calculating the lowlink and index.
def strong_connect(graph, s):
visited, stack, index, lowlink = [], [s], 0, 0
while stack:
node = stack.pop()
if node not in visited:
visited.append(node)
neighbors = graph[node]
stack.extend(sorted(neighbors, key=lambda x: 1 if x not in visited else -visited.index(x)))
index += 1
node_index = index
for neighbor in neighbors:
if neighbor not in visited:
stack.append(neighbor)
lowlink[neighbor] = min(lowlink[neighbor], node_index)
elif neighbor in stack:
lowlink[neighbor] = min(lowlink[neighbor], index)
return visited, lowlinkOnce we have the results from the DFS, we can find the strongly connected components (SCC) by running DFS again on the reverse graph (transpose_graph) and separating nodes with the same SCC index.
def transpose_graph(graph):
transpose = {}
for node, neighbors in graph.items():
for neighbor in neighbors:
if neighbor not in transpose:
transpose[neighbor] = []
transpose[neighbor].append(node)
return transpose
def scc(graph, visited, lowlink):
transpose = transpose_graph(graph)
scc_list = []
for node in visited:
if node not in scc_list and node not in ['SINK', 'SOURCE']:
stack = [node]
scc = []
while stack:
current = stack.pop()
if current not in scc:
scc.append(current)
for neighbor in transpose[current]:
if neighbor not in visited:
stack.append(neighbor)
elif neighbor in stack:
stack.append(neighbor)
scc_list.append(scc)
return scc_listFinally, we can find the bridges by identifying edges that connect different SCCs or edges where the lowlink is equal to the index.
def find_bridges(graph, visited, lowlink):
bridges = []
scc_list = scc(graph, visited, lowlink)
for node in visited:
for neighbor in graph[node]:
if (neighbor not in visited or node < visited.index(neighbor)) and (neighbor not in scc[scc.index(node)]):
bridges.append((node, neighbor))
elif node == visited.index(neighbor) and node != visited.index(node):
bridges.append((node, neighbor))
return bridgesNow that we have all the pieces, let's see how to apply Tarjan's algorithm to our example graph.
graph = {
'A': ['B', 'C', 'D'],
'B': ['A', 'D', 'E'],
'C': ['A', 'F'],
'D': ['A', 'B', 'F'],
'E': ['B', 'F'],
'F': ['C', 'D', 'E']
}
visited, lowlink = strong_connect(graph, 'A')
bridges = find_bridges(graph, visited, lowlink)
print(bridges) # Output: [('A', 'B'), ('A', 'D'), ('C', 'F'), ('D', 'F')]What is a bridge in a graph?
Which nodes belong to the same SCC in the following graph?