Sccd and topological sorting
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The document discusses strongly connected component decomposition (SCCD) which uses depth-first search (DFS) to separate a directed graph into subsets of mutually reachable vertices. It describes running DFS on the original graph and its transpose to find these subsets in Θ(V+E) time, then provides an example applying the three step process of running DFS on the graph and transpose, finding two strongly connected components.
Sccd and topological sorting
- 1. Strongly Connected Component Decomposition (SCCD) Dr. AMIT KUMAR @JUET
- 2. Strongly Connected Component Decomposition (SCCD) • The third useful application of DFS is to perform a strongly connected component decomposition • Given a directed graph G(V,E), the strongly connected component decomposition determi nes sets of vertices Ci ∈ V such that For every pair u,v ∈ Ci ⇒ u ↝ v and v ↝ u. • In otherwords, it separates the graph into subsets of vertices that are mutually reachable from each other. Dr. AMIT KUMAR @JUET
- 3. Strongly Connected Component Decomposition (SCCD) • Define the transpose of a graph • i.e. it is the original graph with all edges reversed. The transpose graph can be created in O(V+E) time if the graph is represented as an adjacency list. • It can then be shown that u and v are mutually reachable from each other in G if and only if u and v are mutually reachable from each other in GT. Dr. AMIT KUMAR @JUET
- 4. Strongly Connected Component Decomposition (SCCD) Thus the procedure for finding strongly connected components is as follows: 1. Run DFS on G to find u.f's ⇒ O(V+E) 2. Compute GT ⇒ O(V+E) 3. Run DFS on GT considering the vertices in decreasing order of u.f's from step 1 ⇒ O(V+E) The resulting depth-first trees from step 3 are the strongly connected components of G. Furthermore, the running time of SCCD is Θ(V+E). Dr. AMIT KUMAR @JUET
- 5. Strongly Connected Component Decomposition (SCCD) Example Consider the following directed graph Dr. AMIT KUMAR @JUET
- 6. Strongly Connected Component Decomposition (SCCD) Example Step 1 - Run DFS on G Running DFS on G (starting at vertex 1 and taking the vertices in numerical order) gives Thus the vertices in decreasing order of u.f is {1,4,2,5,3}. Dr. AMIT KUMAR @JUET
- 7. Strongly Connected Component Decomposition (SCCD) Example Step 2 - Compute GT Dr. AMIT KUMAR @JUET
- 8. Strongly Connected Component Decomposition (SCCD) Example Step 3 - Run DFS on GT Running DFS on GT taking the vertices in the order {1,4,2,5,3} Dr. AMIT KUMAR @JUET
- 9. Strongly Connected Component Decomposition (SCCD) Example The depth-first trees from step 3 are Hence the strongly connected components are {1,2,3,4} and {5}. Dr. AMIT KUMAR @JUET
- 10. Topological Sorting • A second application of DFS is to create a directed ordering of nodes, e.g. task dependencies. This can be done based on an important theorem that can be shown using DFS • A directed graph is acyclic (i.e. a DAG) if and only if a DFS produces no back edges. Thus DFS can be used to test whether or not a graph has cycles on O(V+E) time. Dr. AMIT KUMAR @JUET
- 11. Topological Sorting • Given a directed acyclic graph G (i.e. DAG, which can be determined via DFS), running DFS on G and sorting the vertices by decreasing finishing times produces a sequential ordering of the vertices. • This can be seen since for any edge (u,v) in the depth-first tree, u must appear before v in the sorted list (since G is a DAG we know there are no back edges). Dr. AMIT KUMAR @JUET
- 12. Topological Sorting • For example, If the vertices represent a set of tasks with edges representing dependencies between tasks, a topological sort can be used to find a critical path. Since DFS runs in Θ(V + E), topological sort runs in the same time Dr. AMIT KUMAR @JUET