Graph Terminology and Representations

Transcription

1 Presetatio for use with the textbook, Algorithm Desig ad Applicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 2015 Graph Termiology ad Represetatios 2015 Goodrich ad Tamassia Graphs 1

2 Graphs A graph is a pair (V, E), where V is a set of odes, called vertices E is a collectio of pairs of vertices, called edges Vertices ad edges are positios ad store elemets Example: A vertex represets a airport ad stores the three-letter airport code A edge represets a flight route betwee two airports ad stores the mileage of the route HNL 2555 SFO 337 LAX ORD DFW 849 LGA PVD MIA 2015 Goodrich ad Tamassia Graphs

3 Edge Types Directed edge ordered pair of vertices (u,v) first vertex u is the origi secod vertex v is the destiatio e.g., a flight Udirected edge uordered pair of vertices (u,v) e.g., a flight route Directed graph all the edges are directed e.g., route etwork Udirected graph all the edges are udirected e.g., flight etwork ORD ORD flight AA miles PVD PVD 2015 Goodrich ad Tamassia Graphs 3

4 Applicatios Electroic circuits Prited circuit board Itegrated circuit Trasportatio etworks Highway etwork Flight etwork Computer etworks Local area etwork Iteret Web Databases Etity-relatioship diagram cslab1a cslab1b math.brow.edu cs.brow.edu brow.edu west.et att.et cox.et Joh Paul David 2015 Goodrich ad Tamassia Graphs 4

5 Termiology Ed vertices (or edpoits) of a edge U ad V are the edpoits of a Edges icidet o a vertex a, d, ad b are icidet o V Adjacet vertices U ad V are adjacet Degree of a vertex X has degree 5 Parallel edges h ad i are parallel edges Self-loop j is a self-loop U a c V W d f b e X Y g h i Z j 2015 Goodrich ad Tamassia Graphs 5

6 Termiology (cot.) Path seuece of alteratig vertices ad edges begis with a vertex eds with a vertex each edge is preceded ad followed by its edpoits Simple path path such that all its vertices ad edges are distict Examples P 1 =(V,b,X,h,Z) is a simple path P 2 =(U,c,W,e,X,g,Y,f,W,d,V) is a path that is ot simple U a c V d P 2 W f b e P 1 X g Y h Z 2015 Goodrich ad Tamassia Graphs 6

7 Termiology (cot.) Cycle circular seuece of alteratig vertices ad edges each edge is preceded ad followed by its edpoits Simple cycle cycle such that all its vertices ad edges are distict Examples C 1 =(V,b,X,g,Y,f,W,c,U,a, ) is a simple cycle C 2 =(U,c,W,e,X,g,Y,f,W,d,V,a, ) is a cycle that is ot simple U a c V d C 2 W f b X e Y C 1 g h Z 2015 Goodrich ad Tamassia Graphs 7

8 Properties Property 1 Σ v deg(v) = 2m Proof: each edge is couted twice Property 2 I a udirected graph with o self-loops ad o multiple edges m ( - 1)/2 Proof: each vertex has degree at most ( - 1) Notatio umber of vertices m umber of edges deg(v) degree of vertex v Example = 4 m = 6 deg(v) = 3 What is the boud for a directed graph? 2015 Goodrich ad Tamassia Graphs 8

9 Vertices ad Edges A graph is a collectio of vertices ad edges. A Vertex is ca be a abstract ulabeled object or it ca be labeled (e.g., with a iteger umber or a airport code) or it ca store other objects A Edge ca likewise be a abstract ulabeled object or it ca be labeled (e.g., a flight umber, travel distace, cost), or it ca also store other objects Goodrich ad Tamassia Graphs 9

10 Graph Operatios 2015 Goodrich ad Tamassia Graphs 10

11 Graph Operatios, Cotiued 2015 Goodrich ad Tamassia Graphs 11

12 Edge List Structure Vertex object elemet referece to positio i vertex seuece Edge object elemet origi vertex object destiatio vertex object referece to positio i edge seuece Vertex seuece seuece of vertex objects Edge seuece seuece of edge objects 2015 Goodrich ad Tamassia Graphs 12

13 Adjacecy List Structure Icidece seuece for each vertex seuece of refereces to edge objects of icidet edges Augmeted edge objects refereces to associated positios i icidece seueces of ed vertices 2015 Goodrich ad Tamassia Graphs 13

14 Adjacecy Matrix Structure Edge list structure Augmeted vertex objects Iteger key (idex) associated with vertex 2D-array adjacecy array Referece to edge object for adjacet vertices Null for o oadjacet vertices The old fashioed versio just has 0 for o edge ad 1 for edge 2015 Goodrich ad Tamassia Graphs 14

15 Performace (All bouds are big-oh ruig times, except for Space ) vertices, m edges o parallel edges o self-loops Edge List Adjacecy List Adjacecy Matrix Space + m + m 2 icidetedges(v) m deg(v) areadjacet (v, w) m mi(deg(v), deg(w)) 1 isertvertex(o) isertedge(v, w, o) removevertex(v) m deg(v) 2 removeedge(e) Goodrich ad Tamassia Graphs 15