Breadth-First Search L 1 L Goodrich, Tamassia, Goldwasser Breadth-First Search 1

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1 readth-irst Search 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 1

2 readth-irst Search readth-first search (S) is a general technique for traversing a graph S traversal of a graph G Visits all the vertices and edges of G etermines whether G is connected omputes the connected components of G omputes a spanning forest of G S on a graph with n vertices and m edges takes O(n + m ) time S can be further extended to solve other graph problems ind and report a path with the minimum number of edges between two given vertices ind a simple cycle, if there is one 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 2

3 S lgorithm The algorithm uses a mechanism for setting and getting labels of vertices and edges lgorithm S(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR S(G, v) lgorithm S(G, s) new empty sequence.addlast(s) setlabel(s, VISIT) i 0 while L i.ismpty() L i +1 new empty sequence for all v L i.elements() for all e G.incidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) setlabel(w, VISIT) L i +1.addLast(w) else setlabel(e, ROSS) i i Goodrich, Tamassia, Goldwasser readth-irst Search 3

4 Python Implementation 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 4

5 xample unexplored vertex visited vertex unexplored edge discovery edge cross edge 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 5

6 xample (cont.) 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 6

7 xample (cont.) 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 7

8 Properties Notation G s : connected component of s Property 1 S(G, s) visits all the vertices and edges of G s Property 2 The discovery edges labeled by S(G, s) form a spanning tree T s of G s Property 3 or each vertex v in L i The path of T s from s to v has i edges very path from s to v in G s has at least i edges 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 8

9 nalysis Setting/getting a vertex/edge label takes O(1) time ach vertex is labeled twice once as UNXPLOR once as VISIT ach edge is labeled twice once as UNXPLOR once as ISOVRY or ROSS ach vertex is inserted once into a sequence L i Method incidentdges is called once for each vertex S runs in O(n + m) time provided the graph is represented by the adjacency list structure Recall that S v deg(v) = 2m 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 9

10 pplications Using the template method pattern, we can specialize the S traversal of a graph G to solve the following problems in O(n + m) time ompute the connected components of G ompute a spanning forest of G ind a simple cycle in G, or report that G is a forest Given two vertices of G, find a path in G between them with the minimum number of edges, or report that no such path exists 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 10

11 S vs. S pplications S S Spanning forest, connected components, paths, cycles Shortest paths iconnected components 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 11 S S

12 S vs. S (cont.) ack edge (v,w) w is an ancestor of v in the tree of discovery edges ross edge (v,w) w is in the same level as v or in the next level S S 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 12

DFS on Directed Graphs BOS. Outline and Reading ( 6.4) Digraphs. Reachability ( 6.4.1) Directed Acyclic Graphs (DAG s) ( 6.4.4)

DFS on Directed Graphs BOS. Outline and Reading ( 6.4) Digraphs. Reachability ( 6.4.1) Directed Acyclic Graphs (DAG s) ( 6.4.4) S on irected Graphs OS OR JK SO LX W MI irected Graphs S 1.3 1 Outline and Reading ( 6.4) Reachability ( 6.4.1) irected S Strong connectivity irected cyclic Graphs (G s) ( 6.4.4) Topological Sorting irected