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.

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

5. Design & Analysis of Algorithm
Graph Traversal Algorithms

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