What are graphs? (Take 1)

Transcription

1 Lecture 22: Let s Get Graphic Graph lgorithms Today s genda: What is a graph? Some graphs that you already know efinitions and Properties Implementing Graphs Topological Sort overed in hapter 9 of the textbook Some slides based on: S 326 by S. Wolfman, What are graphs? (Take 1) Yes, this is a graph. ut we are interested in a different kind of graph 2

2 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 d a g b 10 node Value Next Motivation for Graphs What is common among these data structures? How can you generalize them? onsider data structures for representing the following problems 4

3 ourse Prerequisites for S at UW Nodes = courses irected edge = prerequisite Representing the loor Plan of a House Nodes = rooms dge=doororpassage 6

4 Representing lectrical ircuits attery Switch Nodes = battery, switch, resistor, etc. dges = connections Resistor 7 Representing xpressions in ompilers x1=q+y*z x2=y*z-q Naive: x1 + q * * x2 - q y*z calculated twice y z Nodes = symbols/operators dges = relationships common subexpression eliminated: x1 + q * y x2 - z q 8

5 Information Transmission in a omputer Network Seoul 56 Tokyo Seattle New York Sydney L.. Nodes = computers dges = transmission rates 9 Traffic low on Highways UW Nodes = cities dges = # vehicles on connecting highway 10

6 Soap Opera Relationships Victor Michelle shley Wayne rad Trisha Peter 11 Six egrees of Separation from Kevin acon Kevin acon pollo 13 Tom Hanks Where s my Oscar? Gary Sinise orest Gump Robin Wright The Princess ride fter Hours Rosanna rquette heech Marin esperately Seeking Susan ased on: S 326, 2000 Wallace Shawn ary lwes Toy Story Laurie Metcalf 12

7 Six egrees of Separation from Kevin acon Kevin acon pollo 13 pollo 13 orest Gump Gary Sinise Rosanna rquette fter Hours heech Marin Tom Hanks Robin Wright The Princess ride The Princess ride Wallace Shawn ary lwes esperately Seeking Susan Toy Story Laurie Metcalf 13 Graphs: 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 14

8 Graph xample Here is a graph G =(V, ) ach edge is a pair (v 1, v 2 ), where v 1, v 2 are vertices in V V = {,,,,, } = {(,), (,), (,), (,), (,), (,)} 15 irected versus 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 16

9 Graph Representations Space and time are measured in terms of: Number of vertices = V and Number of edges = There are two ways of representing graphs: The adjacency matrix representation The adjacency list representation 17 Graph Representation: djacency Matrix The adjacency matrix representation: Space =? 1 if(v, w)isin M(v, w) = 0 otherwise

10 djacency Matrix for a igraph M(v, w) = 1 if(v, w)isin 0 otherwise Space = V Graph Representation: djacency List The adjacency list representation: or each v in V, L(v)=listofwsuch that (v, w)isin Space =? 20

11 Graph Representation: djacency List Space = a V + 2 b a b 21 djacency List for a igraph Space =? a b igraph djacency List 22

12 djacency List for a igraph Space = a V +b a b igraph djacency List 23 Graph lgorithm #1: Topological Sort Graph of course prerequisites Problem: indanorderin which all these courses can 378 be taken. 421 xample: 142! 143! ! 370! 321! 341! 322! 326! 421! 401 To take a course, all its prerequisites must be taken first 24

13 Topological Sort Topological sorting problem: givendigraphg =(V, ), find a linear ordering of its vertices such that: for any edge (v, w)in, v precedes w in the ordering On-board example: Topo-Sort this digraph 25 Topological Sort Topological sorting problem: given digraph G =(V, ), find a linear ordering of its vertices such that: for any edge (v, w)in, v precedes w in the ordering ny linear ordering in which all the arrows go to the right is a valid solution 26

14 Topological Sort Topological sorting problem: given digraph G =(V, ), find a linear ordering of its vertices such that: for any edge (v, w)in, v precedes w in the ordering Not a valid topological sort! 27 Topological Sort lgorithm #1 Step 1: Identify vertices that have no incoming edges The in-degree of these vertices is zero 28

15 Topological Sort lgorithm #1 Step 1: Identify vertices that have no incoming edges Ifno such vertices, graph has cycle(s) (cyclic graph) Topological sort not possible Halt. xample of a cyclic graph 29 Topological Sort lgorithm #1 Step 1: Identify vertices that have no incoming edges Select one such vertex Select 30

16 Topological Sort lgorithm #1 Step 2: elete this vertex of in-degree 0 and all its outgoing edges from the graph. Place it in the output. 31 Topological Sort lgorithm #1 Repeat Step 1 andstep2until graph is empty Select 32

17 Topological Sort lgorithm #1 Repeat Step 1 andstep2until graph is empty Select 33 Topological Sort lgorithm #1 Repeat Step 1 andstep2until graph is empty Select 34

18 Topological Sort lgorithm #1 Repeat Step 1 andstep2until graph is empty inal Result: 35 Topological Sort lgorithm #1: nalysis or input graph G = (V,), Run Time =? reak down into total time to:! ind a vertex with in-degree 0! Remove its edges! Place vertex in output ssume adjacency list representation 36

19 Topological Sort lgorithm #1: nalysis alculate and store In-egree of all vertices in an array! ind vertex with in-degree 0: Search this array! Remove its edges: Update this array In-egree array Topological Sort lgorithm #1: nalysis or input graph G = (V,), Run Time =? reak down into total time to:! ind vertices with in-degree 0: V vertices, each takes O( V ) to search In-egree array = O( V 2 )! Remove edges: edges! Place vertices in output: V vertices Total time = O( V 2 + ) an we do better than quadratic time? an you think of a faster way to find vertices with in-degree 0? 38

20 Next lass: aster Topological Sort and inding shortest ways to get to your classrooms To o: Read and enjoy chapter 9 Have a great weekend! 39