Lecture 9-10

# Introduction to Algorithms, Second Edition

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The University of Texas at Austin Lectures 9 & 10 Department of Computer Sciences Professor Vijaya Ramachandran Graphs; shortest paths CS357: ALGORITHMS, Spring 2006 1 Graph-theoretic Definitions An undirected graph G = ( V, E ) consists of a finite set of vertices V and a set E of unordered pairs of distinct vertices in V . An edge containing vertices x and y is denoted by ( x, y ) (or equivalently as ( y, x )). A directed graph , also called G = ( V, E ) consists of a finite set of vertices V , and a set of edges E which is a binary relation on V , i.e., E is a subset of V × V . The degree of a vertex v is the number of edges of the form ( v, x ), x V . In a directed graph, this is also called the out-degree of v ; its in-degree is the number of edges of the form ( x, v ), x V . In both an undirected and directed graph, if ( u, v ) is an edge, then v is adjacent to u . A path of length k from vertex u to vertex u 0 in G = ( V, E ) is a sequence of vertices < v 0 , v 1 , · · · , v k > with v 0 = u and v k = u 0 such that each ( v i - 1 , v i ) E , 1 i k . For each v V , < v > represents a path of length 0. A path is simple if all vertices on the path are distinct. We say that v is reachable from u if there is a path (of some length) from u to v . Given a graph G = ( V, E ) (directed or undirected), a graph G 0 = ( V 0 , E 0 ) is a subgraph of G if V 0 V and E 0 E . Given a set of vertices V 1 V , the subgraph of G induced by V 1 is the graph G 1 = ( V 1 , E 1 ) where E 1 contains all edges ( u, u 0 ) E with both u and u 0 in V 1 . Representation of graphs. The running time of graph algorithms is usually measured in terms of the size of the representation used to describe the graph G = ( V, E ). This size is | V | + | E | , if an adjacency-lists representation of G is used, as described in the textbook. Throughout our discussion of graph algorithms, we will use n to denote | V | and m to denote E .

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