(148064384) bfs
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Breadth-first search (BFS) is an algorithm for traversing or searching trees or graphs. It begins at a root node and explores all neighboring nodes at the present depth prior to moving on to the nodes at the next depth level. The key properties of BFS are that it visits all vertices and edges, computes connected components, and finds the shortest path between any two vertices in terms of the number of edges. BFS runs in O(n+m) time on a graph with n vertices and m edges.
(148064384) bfs
- 1. Breadth-First Searc hL L 1 0 B L 2 © 2004 G drich, Tamassia A Breadth-First Search C E D F 1
- 2. Breadth-First Searc (§ 12.3.3) Breadth-first search h BF on a graph ith n w (BFS) is a general nique for traversing te ch a graph ABF traversal o a graph G S f v rtices and edges of G e „ Determines whether G is „ Visits all the connected „ Computes the connected components of G „ Computes a spanning forest of G © 2004 G drich, Tamassia Breadth-First Search vertices and m edges S ta s O(n + m ) time ke s BF can be further extended to solve other S graph problems Find and report a p t with the minimum number of edges a h between two given vertices „ Find a simple cycle, if there is one „ 2
- 3. BF Algorithm The algorithm uses a S mechanism for setting and getting “labels” of vertices of and edges Algorithm BFS(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u ∈ G.ertce() setLabel(u, UNEXPLORED) for all e ∈ G.edges() ∈ v i s setLabel(e, UNEXPLORED) for all v ∈ G.vertices() ∈ if getLabel(v) = UNEXPLORED BFS(G, v) © 2004 G drich, Tamassia Algorithm BFS(G, s) L L0 ← new empty L0.insertL ) bel(s, VISITED sequence se 0 be(s tLa i← ast(s) l while ,new empty sequence ← Lii.isEmpty() for ¬L all v ∈∈ i +1 Breadth-First Search for all e ∈ G.incidentEdges(v) if Li.elements getLabel(e) = ← opposite(v,e) () w← UNEXPLO if getLabel(w) = setLabel(e, DISCOVERY) RED setLabel(w, VISITED) w, UNEXPLO Li RED Li else setLabel(e, CROSS +1.insertLas ) e, i ← i +1 i t(w) 3
- 4. Example unexplored vertex visited vertex unexplored edge scovery edge di cross edge A A L L 1 0 G L 1 B B L C drich, Tamassia 0 D F A C E A E © 2004 L L 1 0 F A B C E Breadth-First Search D D F 4
- 5. Example (cont.) L L 0 B 1 L A 0 C E L L 1 0 2 © 2004 G L F drich, Tamassia C 2 D L E L C E B 1 A B L D L A A 0 B 1 L F 2 Breadth-First Search D F C E D F 5
- 6. Example (cont.) L L L 1 0 B C L L E L 1 0 B L 2 © 2004 G D L 1 0 A B C D A 2 L L A drich, Tamassia L F C E 2 E F D F Breadth-First Search 6
- 7. Properties Notation G s: Property B connecte all the vertices and 1 BFS(G, s) visits d edges Property of Gs 2 The discovery la eled by BFS(G, s) form a spanning tree edges L0 b of G Ts Property 3 s ach L B F r e vertex v in „ The p h of Ts s to v has i 1 to o edges Li L 2 at from Every path from s to v in „ le ii edges a © 2004 G drich, Tamassia st component of s inGs a has t Breadth-First Search A C E D F A C E D F 7
- 8. Analysi s Setting/getting a vertex/edge Each vertex is labeled twice label takes O(1) time O a UNEXPLORE aD s VISITED Each edge is s UNEXPLOREtwice labeled „ once aD once a DISCOVERYoor s CROSS Each vertex is inserted once into a sequen s r is called Method incidentEdges for each vertex „ once „ once „ once BFS the Li O + m) time providedce graph i O the adjacency list structure ( + represented by runs „ © 2004 G n that Σv n Recall deg(v) = 2m drich, Tamassia Breadth-First Search i s 8
- 9. Applications Using the template method pattern, we can specialize the BFS traversal of a G to solve graphthe 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 G number edges, or them wi the mi nimum t o report that no such path exists h © 2004 G drich, Tamassia f Breadth-First Search 9
- 10. DF vs. BF S S Applications Spanning forest, connected components, paths, cycles Shortest paths DF S √ √ L A C E © 2004 DF S G drich, Tamassia D F L 1 0 A B L 2 Breadth-First Search √ √ Biconnected components B BF S C E BF S D F 10
- 11. DF vs. BF (cont.) S edge (v,w) S Back s edge ,w) Cr swi i the, same level as os (v ancestor of v in wi a „ „ next vo n thew level in i s the tree of discovery r n edges the tree of discovery sn edges L A B C E © 2004 G DF S drich, Tamassia D F L 1 0 B L 2 Breadth-First Search A C E BF S D F 11