Overview of Graph Theory,Ad Hoc Networking Instructor: Carlos Pomalaza-Rez Fall 2003 University of Oulu, Finland,Some applications of Graph Theory,Models for communications and electrical networks Models for computer architectures Network optimization models for operations analysis, including scheduling and job assignment Analysis of Finite State Machines Parsing and code optimization in compilers,Application to Ad Hoc Networking,Networks can be represented by graphs The mobile nodes are vertices The communication links are edges,Routing protocols often use shortest path algorithms This lecture is background material to the routing algorithms,Vertices,Edges,Elementary Concepts,A graph G(V,E) is two sets of object Vertices (or nodes) , set V Edges, set E A graph is represented with dots or circles (vertices) joined by lines (edges) The magnitude of graph G is characterized by number of vertices |V| (called the order of G) and number of edges |E| (size of G) The running time of algorithms are measured in terms of the order and size,Graphs Networks,Directed Graph,An edge e E of a directed graph is represented as an ordered pair (u,v), where u, v V. Here u is the initial vertex and v is the terminal vertex. Also assume here that u v,2,4,3,1,V = 1, 2, 3, 4, | V | = 4 E = (1,2), (2,3), (2,4), (4,1), (4,2), | E |=5,Undirected Graph,2,4,3,1,V = 1, 2, 3, 4, | V | = 4 E = (1,2), (2,3), (2,4), (4,1), | E |=4,An edge e E of an undirected graph is represented as an unordered pair (u,v)=(v,u), where u, v V. Also assume that u v,Degree of a Vertex,Degree of a vertex in an undirected graph is the number of edges incident on it. In a directed graph, the out degree of a vertex is the number of edges leaving it and the in degree is the number of edges entering it,2,4,3,1,2,4,3,1,The degree of vertex 2 is 3,The in degree of vertex 2 is 2 and the in degree of vertex 4 is 1,Weighted Graph,A weighted graph is a graph for which each edge has an associated weight, usually given by a weight function w: E R,2,4,3,1,2,4,3,1,1.2,2.1,0.2,0.5,4,8,6,2,9,Walks and Paths,3,2,3,4,1,1,V5,V4,V3,V2,V1,V6,4,1,A walk is an sequence of nodes (v1, v2,., vL) such that (v1, v2), (v1, v2),., (v1, v2) E, e.g. (V2, V3,V6, V5,V3),A cycle is an walk (v1, v2,., vL) where v1=vL with no other nodes repeated and L3, e.g. (V1, V2,V5, V4,V1),A simple path is a walk with no repeated nodes, e.g. (V1, V4,V5, V2,V3),A graph is called cyclic if it contains a cycle; otherwise it is called acyclic,Complete Graphs,A,D,C,B,4 nodes and (4*3)/2 edges V nodes and V*(V-1)/2 edges,C,A,B,3 nodes and 3*2 edges V nodes and V*(V-1) edges,A complete graph is an undirected/directed graph in which every pair of vertices is adjacent. If (u, v ) is an edge in a graph G, we say that vertex v is adjacent to vertex u.,Connected Graphs,A,D,E,F,B,C,A,B,C,D,An undirected graph is connected if you can get from any node to any other by following a sequence of edges OR any two nodes are connected by a path,A directed graph is strongly connected if there is a directed path from any node to any other node,A graph is sparse if | E | | V | A graph is dense if | E | | V |2,Bipartite Graph,A bipartite graph is an undirected graph G = (V,E) in which V can be partitioned into 2 sets V1 and V2 such that ( u,v) E implies either u V1 and v V2 OR v V1 and u V2.,u1,u2,u3,u4,v1,v2,v3,V1,V2,An example of bipartite graph application to telecommunication problems can be found in, C.A. Pomalaza-Rez, “A Note on Efficient SS/TDMA Assignment Algorithms,” IEEE Transactions on Communications, September 1988, pp. 1078-1082. For another example of bipartite graph applications see the slides in the Addendum section,Trees,A,B,D,F,C,E,Let G = (V, E ) be an undirected graph. The following statements are equivalent, G is a tree Any two vertices in G are connected by unique simple path G is connected, but if any edge is removed from E, the resulting graph is disconnected G is connected, and | E | = | V | -1 G is acyclic, and | E | = | V | -1 G is acyclic, but if any edge is added to E, the resulting graph contains a cycle,Spanning Tree,A tree (T ) is said to span G = (V,E) if T = (V,E) and E E,V5,V4,V3,V2,V1,V6,V5,V4,V3,V2,V1,V6,V5,V4,V3,V2,V1,V6,For the graph shown on the right two possible spanning trees are shown below,For a given graph there are usually several possible spanning trees,Minimum Spanning Tree,1,3,8,2,6,7,4,5,5,23,10,21,14,16,6,18,9,7,11,8,1,3,8,2,6,7,4,5,5,6,4,9,7,11,8,G = (V, E),T = (V, F),w(T) = 50,24,4,Given connected graph G with real-valued edge weights ce, a Minimum Spanning Tree (MST) is a spanning tree of G whose sum of edge weights is minimized,Cayleys Theorem (1889) There are nn-2 spanning trees of a complete graph Kn n = |V|, m = |E| Cant solve MST by brute force (because of