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© 2004 Goodrich, Tamassia Breadth-First Search1 CB A E D L0L0 L1L1 F L2L2

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© 2004 Goodrich, Tamassia Breadth-First Search2 Breadth-first search (BFS) is a general technique for traversing a graph A BFS traversal of a graph G Visits all the vertices and edges of G Determines whether G is connected Computes the connected components of G Computes a spanning forest of G BFS on a graph with n vertices and m edges takes O(n m ) time BFS can be further extended to solve other graph problems Find and report a path with the minimum number of edges between two given vertices Find a simple cycle, if there is one

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© 2004 Goodrich, Tamassia Breadth-First Search3 BFS Algorithm The algorithm uses a mechanism for setting and getting “labels” of vertices and edges Algorithm BFS(G, s) L 0 new empty sequence L 0.insertLast(s) setLabel(s, VISITED ) i 0 while L i.isEmpty() L i 1 new empty sequence for all v L i.elements() for all e G.incidentEdges(v) if getLabel(e) UNEXPLORED w opposite(v,e) if getLabel(w) UNEXPLORED setLabel(e, DISCOVERY ) setLabel(w, VISITED ) L i 1.insertLast(w) else setLabel(e, CROSS ) i i 1 Algorithm BFS(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u G.vertices() setLabel(u, UNEXPLORED ) for all e G.edges() setLabel(e, UNEXPLORED ) for all v G.vertices() if getLabel(v) UNEXPLORED BFS(G, v)

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© 2004 Goodrich, Tamassia Breadth-First Search4 Example C B A E D discovery edge cross edge A visited vertex A unexplored vertex unexplored edge L0L0 L1L1 F CB A E D L0L0 L1L1 F CB A E D L0L0 L1L1 F

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© 2004 Goodrich, Tamassia Breadth-First Search5 Example (cont.) CB A E D L0L0 L1L1 F CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2

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© 2004 Goodrich, Tamassia Breadth-First Search6 Example (cont.) CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2

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© 2004 Goodrich, Tamassia Breadth-First Search7 Properties Notation G s : connected component of s Property 1 BFS(G, s) visits all the vertices and edges of G s Property 2 The discovery edges labeled by BFS(G, s) form a spanning tree T s of G s Property 3 For each vertex v in L i The path of T s from s to v has i edges Every path from s to v in G s has at least i edges CB A E D L0L0 L1L1 F L2L2 CB A E D F

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© 2004 Goodrich, Tamassia Breadth-First Search8 Analysis Setting/getting a vertex/edge label takes O(1) time Each vertex is labeled twice once as UNEXPLORED once as VISITED Each edge is labeled twice once as UNEXPLORED once as DISCOVERY or CROSS Each vertex is inserted once into a sequence L i Method incidentEdges is called once for each vertex BFS runs in O(n m) time provided the graph is represented by the adjacency list structure Recall that v deg(v) 2m

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© 2004 Goodrich, Tamassia Breadth-First Search9 Applications Using the template method pattern, we can specialize the BFS traversal of a graph G to solve the following problems in O(n m) time Compute the connected components of G Compute a spanning forest of G Find 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

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© 2004 Goodrich, Tamassia Breadth-First Search10 DFS vs. BFS CB A E D L0L0 L1L1 F L2L2 CB A E D F DFSBFS ApplicationsDFSBFS Spanning forest, connected components, paths, cycles Shortest paths Biconnected components

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© 2004 Goodrich, Tamassia Breadth-First Search11 DFS vs. BFS (cont.) Back edge (v,w) w is an ancestor of v in the tree of discovery edges Cross edge (v,w) w is in the same level as v or in the next level in the tree of discovery edges CB A E D L0L0 L1L1 F L2L2 CB A E D F DFSBFS

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