Reading. Graph Introduction. What are graphs? Graphs. Motivation for Graphs. Varieties. Reading. CSE 373 Data Structures Lecture 18

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1 Reading Graph Introduction Reading Section 9.1 S 373 ata Structures Lecture 18 11/25/02 Graph Introduction - Lecture 18 2 What are graphs? Yes, this is a graph. Graphs Graphs are composed of Nodes (vertices) dges node ut we are interested in a different kind of graph 11/25/02 Graph Introduction - Lecture 18 3 edge 11/25/02 Graph Introduction - Lecture 18 4 Varieties Nodes Labeled or unlabeled dges irected or undirected Labeled or unlabeled 11/25/02 Graph Introduction - Lecture 18 5 Motivation for Graphs 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 + 2 outgoing edges inomial trees/-trees: nodes with 1 incoming edge + multiple outgoing edges Up-trees: nodes with multiple incoming edges + 1 outgoing edge node Value Next 11/25/02 Graph Introduction - Lecture 18 6 d a g b 10 node Value Next

2 Motivation for Graphs What is common among these data structures? How can you generalize them? onsider data structures for representing the following problems S ourse Prerequisites at UW Nodes = courses irected edge = prerequisite /25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture 18 8 Representing a Maze Representing lectrical ircuits attery Switch Nodes = rooms dge = door or passage Nodes = battery, switch, resistor, etc. dges = connections Resistor 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture Program statements Precedence x1=q+y*z x2=y*z-q x1 + Naive: q * y*z calculated twice y z * x2 - q S 1 S 2 S 3 S 4 S 5 S 6 a=0; b=1; c=a+1 d=b+a; e=d+1; e=c+d; 6 5 Nodes = symbols/operators dges = relationships common subexpression eliminated: x1 + q * y x2 - z q Which statements must execute before S 6? S 1, S 2, S 3, S 4 Nodes = statements dges = precedence requirements /25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture

3 Information Transmission in a omputer Network Traffic low on Highways Seoul Tokyo 30 Seattle New York UW Nodes = cities dges = # vehicles on connecting highway Sydney L.. Nodes = computers dges = transmission rates 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture Soap Opera Relationships Victor Michelle shley Wayne rad Peter Trisha 11/25/02 Graph Introduction - Lecture Six egrees of Separation from Kevin acon Kevin acon pollo 13 Tom Hanks Where s my Oscar? Gary Sinise orest Gump Robin Wright Wallace Shawn The Princess ride Rosanna rquette ary lwes fter Hours Toy Story heech Marin esperately Seeking Susan Laurie Metcalf Six egrees of Separation from Kevin acon Niche overlaps Kevin acon pollo 13 pollo 13 Gary Sinise Rosanna rquette fter Hours heech Marin Raccoon Hawk Owl Tom Hanks orest Gump The Princess ride ary Toy Story Laurie Robin lwes Metcalf Wright 11/25/02 Graph Introduction - Lecture The Princess ride Wallace Shawn esperately Seeking Susan Opossum row Squirrel Shrew Woodpecker Mouse 11/25/02 Graph Introduction - Lecture

4 Graph efinition 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 11/25/02 Graph Introduction - Lecture Graph xample Here is a directed graph G = (V, ) ach edge is a pair (v 1, v 2 ), where v 1, v 2 are vertices in V V = {,,,,, } = {(,), (,), (,), (,), (,), (,)} 11/25/02 Graph Introduction - Lecture irected vs Undirected Graphs If the order of edge pairs (v 1, v 2 ) matters, the graph is directed (also called a digraph): (v 1, v 2 ) (v 2, v 1 ) v 1 v 2 If the order of edge pairs (v 1, v 2 ) does not matter, the graph is called an undirected graph: in this case, (v 1, v 2 ) = (v 2, v 1 ) v 1 v 2 Undirected Terminology 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) 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture Undirected Terminology (,) is incident to and to egree = 3 is adjacent to and is adjacent to 11/25/02 Graph Introduction - Lecture egree = 0 irected Terminology 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 11/25/02 Graph Introduction - Lecture

5 irected Terminology Handshaking Theorem In-degree = 2 Out-degree = 1 adjacent to and adjacent from In-degree = 0 Out-degree = 1 Let G=(V,) be an undirected graph with =e edges Then 2edeg(v) vv 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 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture 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 11/25/02 Graph Introduction - Lecture djacency Matrix if (v, w) is in M(v, w) = 0 otherwise Space = V 2 11/25/02 Graph Introduction - Lecture djacency Matrix for a igraph if (v, w) is in M(v, w) = 0 otherwise Space = V 2 11/25/02 Graph Introduction - Lecture djacency List or each v in V, L(v) = list of w such that (v, w) is in a b 11/25/02 Graph Introduction - Lecture Space = a V + 2 b 5

6 djacency List for a igraph or each v in V, L(v) = list of w such that (v, w) is in a b Space = a V + b 11/25/02 Graph Introduction - Lecture ipartite simple graph is bipartite if: its vertex set V can be partitioned into two disjoint non-empty sets such that every edge in the graph connects a vertex in one set to a vertex in the other set which also means that no edge connects a vertex in one set to another vertex in the same set no triangular or other odd length cycles 11/25/02 Graph Introduction - Lecture g ipartite examples a b c ipartite example - not a b {a b d} {c e f g} f a e b d f e d c g f d a says that b and f should be in S 2, but b says a and f should be in S 1. TILT! e c 11/25/02 Graph Introduction - Lecture ipartite Graph pplication lassroom scheduling Nodes are lassrooms and lasses dge between a classroom and class if the class will fit in the classroom and has the right technology. Matching Problem ind an assignment of classes to classrooms so that every class fits and has the right technology. classes classrooms classes classrooms 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture

7 Steps in Solving the Problem bstract the problem as a graph problem. ind an algorithm for solving the graph problem. esign data structures and algorithms to implement the graph solution. Write code lternating Path Let G = (U,V,) be a bipartite graph where (u,v) in only if u in U and v in V. partial matching M is subset of such that if (u,v) and (u,v ) in M then either (u = u and v = v ) or (u u or v v ) n alternating path is x 1,x 2,,x 2n such that (x i,x i+1 ) in M if i is odd (x i,x i+1 ) in M if i is even x 1 and x 2n are not matched in the partial matching 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture Partial Matching lternating Path M -M x 3 x 1 x 2 x4 M -M x 5 x 6 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture Matching lgorithm One step in the lgorithm set M to be the empty set initially repeat find an alternating path x 1,x 2,,x 2n // (x i,x i+1 ) in M if i is odd and (x i,x i+1 ) in M if i is even neither x 1 nor x 2n matched // delete (x i,x i+1 ) from M if i is even add (x i,x i+1 ) to M if i is odd until no alternating path can be found if M has every vertex of U then M is a matching if M does not have some vertex then there is complete matching, but M is a maximum size matching x3 x 2 x 1 x4 x 5 x 6 x3 x 2 x 1 x4 x 5 x 6 11/25/02 Graph Introduction - Lecture /25/02 Graph Introduction - Lecture

8 Maximum Matching Prove that M is maximum size if and only if there is no alternating path. esign data structures algorithms to find alternating paths or determine they don t exist. Goal: fast data structures and algorithms 11/25/02 Graph Introduction - Lecture