Find Strongly Connected Components in a Directed Graph
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Overview
Finding strongly connected components in a directed graph is a form of clustering heuristics: strongly connected components represent clusters where the objects represented by the vertices are clustered in some way.
Strongly connected components of a directed graph can be computed with Kosaraju's Two-Pass algorithm, which consists of two passes of depth-first search. In consequence, the running time is O(n + m). The key idea of the algorithm is that for directed graphs, starting a depth-first search from the "right" nodes discovers strongly connected components, while starting from the "wrong" nodes may discover the entire graph, which is not useful. The first pass of the algorithm computes the right order in which to use the nodes as start nodes in the second depth-first search pass.
Kosaraju's Two-Pass Algorithm
The algorithm has three steps:
1. Reverse the Arcs. Reverse all of the arcs of the given graph. Let Grev = G with all arcs reversed.
- Note that reversing the arcs helps with reasoning about the problem. A practical implementation can simply run DFS on the original graph G but going backwards, in the opposite direction of the arcs, instead of creating a copy of the graph where the arcs are effectively reversed. The recursive depth-first search implementation can be used as is, with just a minimal modification: when looking for edges to explore, use edges incident to the current node instead of edges incident from the current node. A pseudocode implementation will be provided below.
2. First Pass DFS. Do the first depth-first search pass on the reversed graph Grev, from node n to node 1. This pass computes an ordering of nodes that allows the second pass to discover the strongly connected components. The pass labels each node with its "finishing time". Let f(v) = "finishing time" of each v ∈ V. Once f(v) for each node is computed at the end of the first pass, the nodes are relabeled with their finishing time.
First_Pass(graph G)
global variable t = 0 # Counts the total number of nodes processed so far in the reverse DFS routine.
# Used by the reverse DFS routine to compute the "finishing time".
# Assume nodes labelled 1 to n
for i = n down to 1
if i not yet explored
Reverse_DFS(G, i)
The Reverse_DFS() is similar to the standard recursive DFS implementation, with the exception that it uses the edges incident to the current nodes instead of the edges incident from the current node, and also computes the finishing time for each node:
Reverse_DFS(graph G, start vertex s)
mark s as explored
for every edge incident to s (s ← v):
if v unexplored:
Reverse_DFS(G, v)
# ⚠️ We are currently assuming nodes labeled from 1 to n. For zero-based node labels, set the finishing time first, then increment t
t++ # Increment the finishing time, in the first pass
set f(s) = t # Set the finishing time on the current node
3. Second Pass DFS. Do the second depth-first search pass again, on the original graph G, with the arcs "restored" in the original direction. The graph vertices must be processed in the decreasing order of the "finishing time" computed during the first pass, so the entire graph must be relabeled and re-ordered using the "finishing time" as vertex labels. This re-labeling should be performed in linear time. The pseudocode of a possible O(n + m) relabelling algorithm that uses additional memory is presented below. Note that by processing the graph we reset the "explored" states on the nodes. The re-labeled graph nodes are all "unexplored".
Relabel_With_Finishing_Time(graph G)
# Allocate a new adj2 adjacency list representation for the entire graph G
# adj is the current representation carrying the old labeling
adj2 = new Vertex[adj.size]
for each o in adj # For each old-labeled vertex
adj2[o.finishingTime] = new Vertex()
# Rewrite the adjacency list of the newly create vertex using the finishing time of the old ajacencies
for i in o.adjacencyList
adj2[o.finishingTime].addHead(adj[i].finishingTime)
# Swap adj with adj2
adj.clear()
for v in adj2
adj.add(v)
Once the graph is re-labeled, the second pass runs depth-first search recursively on unexplored nodes, and discovers the strongly connected components one by one. During this second pass the algorithm labels each node with its "leader" ID, which is the node we start the recursive depth-first search on. The idea is all nodes in the same strongly connected component will be labeled with the same leader node.
Second_Pass(graph G)
global variable leader = NULL # Keeps track of the most recent vertex from which the DFS was initiated (the leader).
# Assume nodes re-labelled 1 to n using finishing times
for i = n down to 1
if i not yet explored
leader = i
DFS(G, i)
DFS() is almost identical to the standard recursive DFS implementation, with the exception that it annotates each node it processes with its leader, maintained by Second_Pass(graph G):
DFS(graph G, start vertex s)
mark s as explored # Once the node is marked explored, it is explored for the rest of Second_Pass() loop
set leader(s) = leader # Mark the node with its "leader" in the second pass
for every edge incident from s (s → v):
if v unexplored:
DFS(G, v)
At the end of the second loop, each node is annotate with its leader.
The state of the graph allows to linearly scan the list of node, identify distinct leaders and compute the SCCs.