Graphs ORD SFO LAX DFW. Lecture notes adapted from Goodrich and Tomassia. 3/14/18 10:28 AM Graphs 1

Transcription

1 Graphs Lecture otes adapted from Goodrich ad Tomassia 3/14/18 10:28 AM Graphs 1

2 Graph 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 Example: A vertex represets a airport ad stores the airport code A edge represets a flight route betwee two airports PVD LGA HNL MIA 3/14/18 10:28 AM Graphs 2

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 street Directed graph: all edges are directed Weighted edge: has a real umber associated to it e.g. distace betwee cities e.g. badwidth betwee iteret routers Weighted graph: all edges have weights 849 miles PVD PVD PVD 3/14/18 10:28 AM Graphs 3

4 Labeled graphs HNL Labeled graphs: vertices have idetifiers LGA PVD MIA = HNL PVD Note: Geometric layout does t matter - oly coectios matter MIA Ulabeled graph: vertices have o idetifiers LGA = 3/14/18 10:28 AM Graphs 4

5 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 qwest.et att.et cox.et Joh Paul David 3/14/18 10:28 AM Graphs 5

6 Termiology Edpoits of a edge U ad V are the edpoits of a Edges icidet o a vertex a, b, ad d are icidet o V Adjacet vertices Coected by a edge U ad V are adjacet Degree of a vertex Number of icidet edges X has degree 5 Parallel edges h ad i are parallel edges Self-loop j is a self-loop U a c V d W f b e X Y g h i Z j 3/14/18 10:28 AM Graphs 6

7 Termiology (cot.) Path sequece of adjacet vertices Simple path path such that all its vertices are distict Examples P 1 =(V, X, Z) is a simple path P 2 =(U, W, X, Y, W, V) is a path that is ot simple Graph is coected iff For all pair of vertices u ad v, there is a path betwee u ad v U a c V d P 2 W f b e P 1 X g Y h Z 3/14/18 10:28 AM Graphs 7

8 Termiology (cot.) Cycle path that starts ad eds at the same vertex Simple cycle cycle where each vertex is distict Examples C 1 =(V, X, Y, W, U, ) is a simple cycle C 2 =(U, W, X, Y, W, V, ) is a cycle that is ot simple A tree is a coected acyclic graph U a c V d C 2 W f b X e Y C 1 g h Z 3/14/18 10:28 AM Graphs 8

9 Properties Property 1 S v Î V deg(v) = 2 E Why? Property 2 I a udirected graph with o self-loops ad o multiple edges E V ( V - 1)/2 Why? Notatio V E deg(v) umber of vertices umber of edges degree of vertex v Example V = 4 E = 6 deg(v) = 3 3/14/18 10:28 AM Graphs 9

10 Data structure for graphs - Adjacecy lists Graph ca be stored as A dictioary of pairs (key, ifo) where key = vertex idetifier ifo cotais a list (called adj) of adjacet vertices Example: if the dictioary is implemeted as a liked-list vertices LGA LGA DWF LGA 3/14/18 10:28 AM Graphs 10

11 Adjacecy lists - Operatios addvertex(key k): vertices.isert(k, emptylist) addedge(key k, key l): vertices.fid(k).adj.isert(l) vertices.fid(l).adj.isert(k) areadjacet(key k, key l): retur vertices.fid(k).adj.fid(l) 3/14/18 10:28 AM Graphs 11

12 Data structure for graphs - Adjacecy matrix Defie some order o the vertices, for example:,, LGA,, Graph with vertices is stored as x array M of boolea, where M[i][j] = 1 if there is a edge betwee i-th ad j-th vertices 0 otherwise LGA LGA LGA /14/18 10:28 AM Graphs 12

13 Adjacecy matrix - Operatios addedge(i,j): matrix[i][j] = 1 removeedge(i,j): matrix[i][j] = 0 Not very good for isertig/removig vertices: requires shiftig elemets of matrix. Requires space O( 2 ) 3/14/18 10:28 AM Graphs 13

14 Lists vs Matrices Adjacecy lists are better if: You frequetly eed to add/remove vertices The graph has few edges Need to traverse the graph Adjacecy matrices are better if you frequetly eed to w add/remove edges, but NOT vertices w Check for the presece/absece of a edge betwee i,j matrix is small eough to fit i memory 3/14/18 10:28 AM Graphs 14