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L18cs2110fa08-6up

# L18cs2110fa08-6up - Announcements Prelim 2 Tuesday Nov 18...

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Graphs Lecture 18 CS2110 – Fall 2008 2 Announcements ! Prelim 2 " Tuesday, Nov 18, 7:30-9pm " Uris Auditorium ! Exam conflicts " Email Kelly Patwell ASAP ! Old exams are available for review on the course website 3 These are not Graphs 17.50 35.63 53.75 71.88 90.00 1st Qtr2nd Qtr3rd Qtr4th Qtr East West North ...not the kind we mean, anyway 4 These are Graphs K 5 K 3,3 = 5 Applications of Graphs ! Communication networks ! Routing and shortest path problems ! Commodity distribution (flow) ! Traffic control ! Resource allocation ! Geometric modeling ! ... 6 Graph Definitions ! A directed graph (or digraph ) is a pair (V, E) where " V is a set " E is a set of ordered pairs (u,v) where u,v ! V # Usually require u " v (i.e., no self-loops) ! An element of V is called a vertex (pl. vertices ) or node ! An element of E is called an edge or arc ! |V| = size of V, often denoted n ! |E| = size of E, often denoted m

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Example Directed Graph (Digraph) V = { a,b,c,d,e,f } E = { (a,b), (a,c), (a,e), (b,c), (b,d), (b,e), (c,d), (c,f), (d,e), (d,f), (e,f) } |V| = 6, |E| = 11 b a c d e f 8 Example Undirected Graph An undirected graph is just like a directed graph, except the edges are unordered pairs ( sets ) {u,v} Example: b a c e d f V = { a,b,c,d,e,f } E = { {a,b}, {a,c}, {a,e}, {b,c}, {b,d}, {b,e}, {c,d}, {c,f}, {d,e}, {d,f }, {e,f } } 9 Some Graph Terminology ! Vertices u and v are called the source and sink of the directed edge (u,v), respectively ! Vertices u and v are called the endpoints of (u,v) ! Two vertices are adjacent if they are connected by an edge ! The outdegree of a vertex u in a directed graph is the number of edges for which u is the source ! The indegree of a vertex v in a directed graph is the number of edges for which v is the sink ! The degree of a vertex u in an undirected graph is the number of edges of which u is an endpoint b a c e d f b a c d e f 10 More Graph Terminology ! A path is a sequence v 0 ,v 1 ,v 2 ,...,v p of vertices such that (v i ,v i+1 ) ! E, 0 # i # p – 1 ! The length of a path is its number of edges " In this example, the length is 5 ! A path is simple if it does not repeat any vertices ! A cycle is a path v 0 ,v 1 ,v 2 ,...,v p such that v 0 = v p ! A cycle is simple if it does not repeat any vertices except the first and last ! A graph is acyclic if it has no cycles ! A directed acyclic graph is called a dag v 0 v 5 b a c d e f 11 Is This a Dag? ! Intuition: " If it’s a dag, there must be a vertex with indegree zero – why? ! This idea leads to an algorithm " A digraph is a dag if and only if we can iteratively delete indegree-0 vertices until the graph disappears b a c d e f 12 Is This a Dag? ! Intuition: " If it’s a dag, there must be a vertex with indegree zero – why? ! This idea leads to an algorithm
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L18cs2110fa08-6up - Announcements Prelim 2 Tuesday Nov 18...

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