08. graph traversal
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This document discusses graph traversal algorithms. It analyzes the running time of breadth-first search (BFS) on an input graph G, showing that BFS runs in O(V+E) time, where V is the number of vertices and E is the number of edges. It also defines the four edge types that can exist in a depth-first search (DFS) forest produced by DFS on a graph G: tree edges, back edges, forward edges, and cross edges.
08. graph traversal
- 1. Design & Analysis of Algorithm Graph Traversal Algorithms Kamalesh Karmakar Assistant Professor Department of C.S.E. MeghnadSahaInstitute of Technology Kolkata.
- 2. Design & Analysis of Algorithm Graph Traversal Algorithms
- 3. Design & Analysis of Algorithm Graph Traversal Algorithms
- 4. Design & Analysis of Algorithm Graph Traversal Algorithms
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- 6. Design & Analysis of Algorithm Beforeprovingthevariouspropertiesofbreadth-firstsearch,wetakeonthesomewhateasierjobofanalyzingitsrunningtimeonaninputgraphG=(V,E).Weuseaggregateanalysis,aswesawinSection17.1.Afterinitialization,novertexiseverwhitened,andthusthetestinline13ensuresthateachvertexisenqueuedatmostonce,andhencedequeuedatmostonce.TheoperationsofenqueuinganddequeuingtakeO(1)time,sothetotaltimedevotedtoqueueoperationsisO(V).Becausetheadjacencylistofeachvertexisscannedonlywhenthevertexisdequeued,eachadjacencylistisscannedatmostonce.Sincethesumofthelengthsofalltheadjacencylistsis(E),thetotaltimespentinscanningadjacencylistsisO(E).TheoverheadforinitializationisO(V),andthusthetotalrunningtimeofBFSisO(V+E).Thus,breadth-firstsearchrunsintimelinearinthesizeoftheadjacency-listrepresentationofG. Graph Traversal Algorithms
- 7. Design & Analysis of Algorithm Graph Traversal Algorithms
- 8. Design & Analysis of Algorithm Graph Traversal Algorithms
- 9. Design & Analysis of Algorithm Graph Traversal Algorithms
- 10. Design & Analysis of Algorithm Graph Traversal Algorithms
- 11. Design & Analysis of Algorithm G. Graph Traversal Algorithms
- 12. Design & Analysis of Algorithm Wecandefinefouredgetypesintermsofthedepth-firstforestGπproducedbyadepth-firstsearchonG. 1.Treeedgesareedgesinthedepth-firstforestGπ.Edge(u,v)isatreeedgeifvwasfirstdiscoveredbyexploringedge(u,v). 2.Backedgesarethoseedges(u,v)connectingavertexutoanancestorvinadepth-firsttree.Self-loops,whichmayoccurindirectedgraphs,areconsideredtobebackedges. 3.Forwardedgesarethosenon-treeedges(u,v)connectingavertexutoadescendantvinadepth-firsttree. 4.Crossedgesareallotheredges.Theycangobetweenverticesinthesamedepth-firsttree,aslongasonevertexisnotanancestoroftheother,ortheycangobetweenverticesindifferentdepth-firsttrees. Graph Traversal Algorithms