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. 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 all of the arcs of the given graph. Let Grev = G with all arcs revered.
2. Do the first depth-first search pass on the reversed graph Grev. The practical implementation should run DFS on the original graph G but going in the opposite direction of the arcs, backwards. This pass computes an ordering of nodes that allows the second pass, while starting in the given order, to discover the strongly connected components. 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.
3. Do the second depth-first search pass again, on the original graph G. The "right" order is to go through the vertices in the decreasing order of the "finishing time" computed during the first pass. In this pass we use the order computed by the first pass - remember that the nodes have been relabeled according to their computed finishing time - and we discover the strongly connected components one by one. During this second pass we will label each node with its "leader". The idea is all nodes in the same strongly connected component will be labeled with the same leader node.
DFS_Loop(graph G)
global variable t = 0 # Counts the total number of nodes processed so far.
# Used in the first DFS pass to compute the "finishing time".
global variable s = NULL # Keeps track of the most recent vertex from which the DFS
# was initiated (the leader). Used in the second pass.
Assume nodes labelled 1 to n.
for i = n down to 1
if i not yet explored
s = i
DFS(G, i)
DFS(graph G, node i)
mark i as explored # Once the node is marked explored, it is explored for the rest of DFS_Loop
set leader(i) = s # Mark the node with its "leader" in the second pass
for ech arc (i, j) ∈ G
if j not yet explored
DFS(G, j)
t ++ # Increment the finishing time, in the first pass
set f(i) = t # Finishing time of i, in the first pass