Reading. Graph Terminology. Graphs. What are graphs? Varieties. Motivation for Graphs. Reading. CSE 373 Data Structures. Graphs are composed of.

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1 Reading Graph Reading Goodrich and Tamassia, hapter 1 S 7 ata Structures What are graphs? Graphs Yes, this is a graph. Graphs are composed of Nodes (vertices) dges (arcs) node ut we are interested in a different kind of graph edge Varieties Motivation for Graphs Nodes Labeled or unlabeled dges irected or undirected Labeled or unlabeled onsider the data structures we have looked at so far Linked list: nodes with 1 incoming edge + 1 outgoing edge inary trees/heaps: nodes with 1 incoming edge + outgoing edges -trees: nodes with 1 incoming edge + multiple outgoing edges node Value Next 10 node Value Next

2 Motivation for Graphs How can you generalize these data structures? onsider data structures for representing the following problems 10 S ourse Prerequisites at UW Nodes = courses irected edge = prerequisite Representing a Maze Representing lectrical ircuits attery Switch S S Nodes = rooms dge = door or passage Nodes = battery, switch, resistor, etc. dges = connections Resistor 9 10 Program statements Precedence x1=q+y*z x=y*z-q x1 + Naive: q * y*z calculated twice y z * x - q S 1 S S S S S 6 a=0; b=1; c=a+1 d=b+a; e=d+1; e=c+d; 6 Nodes = symbols/operators dges = relationships common subexpression eliminated: x1 + q * y x - z q Which statements must execute before S 6? S 1, S, S, S Nodes = statements dges = precedence requirements

3 Information Transmission in a omputer Network Traffic low on Highways Seoul 6 Tokyo Seattle 18 UW New York Nodes = cities dges = # vehicles on connecting highway Sydney L.. Nodes = computers dges = transmission rates 1 1 Graph efinition Graph xample graph is simply a collection of nodes plus edges Linked lists, trees, and heaps are all special cases of graphs The nodes are known as vertices (node = vertex ) ormal efinition: graph G is a pair (V, ) where V is a set of vertices or nodes is a set of edges that connect vertices 1 Here is a directed graph G = (V, ) ach edge is a pair (v 1, v ), where v 1, v are vertices in V V = {,,,,, } = {(,), (,), (,), (,), (,), (,)} 16 irected vs Undirected Graphs If the order of edge pairs (v 1, v ) matters, the graph is directed (also called a digraph): (v 1, v ) (v, v 1 ) v 1 v If the order of edge pairs (v 1, v ) does not matter, the graph is called an undirected graph: in this case, (v 1, v ) = (v, v 1 ) v 1 v Undirected Two vertices u and v are adjacent in an undirected graph G if {u,v} is an edge in G edge e = {u,v} is incident with vertex u and vertex v The degree of a vertex in an undirected graph is the number of edges incident with it a self-loop counts twice (both ends count) denoted with deg(v) 17 18

4 Undirected (,) is incident to and to egree = is adjacent to and is adjacent to Self-loop egree = 0 irected Vertex u is adjacent to vertex v in a directed graph G if (u,v) is an edge in G vertex u is the initial vertex of (u,v) Vertex v is adjacent from vertex u vertex v is the terminal (or end) vertex of (u,v) egree in-degree is the number of edges with the vertex as the terminal vertex out-degree is the number of edges with the vertex as the initial vertex 19 0 irected Handshaking Theorem In-degree = Out-degree = 1 adjacent to and adjacent from In-degree = 0 Out-degree = 0 Let G=(V,) be an undirected graph with =e edges. Then e= v V deg(v) dd up the degrees of all vertices. very edge contributes +1 to the degree of each of the two vertices it is incident with number of edges is exactly half the sum of deg(v) the sum of the deg(v) values must be even 1 Graph Representations Space and time are analyzed in terms of: Number of vertices = V and Number of edges = There are at least two ways of representing graphs: The adjacency matrix representation The adjacency list representation M(v, w) = djacency Matrix 1 if (v, w) is in 0 otherwise Space = V

5 M(v, w) = djacency Matrix for a igraph 1 if (v, w) is in 0 otherwise Space = V djacency List or each v in V, L(v) = list of w such that (v, w) is in a b list of neighbors Space = a V + b 6 or each v in V, L(v) = list of w such that (v, w) is in a b djacency List for a igraph Space = a V + b 7