graphs1-handouts

graphs1-handouts - ECE 537 Foundations of Computing Module...

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ECE 537 – Foundations of Computing Module 8, Lecture 1: Graph Algorithms – Introduction and Basic Search Methods Professor G.L. Heileman University of New Mexico Fall 2008 c ± 2008 G. L. Heileman Module 8, Lecture 1
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References Cormen et. al., Chapters 22–24 c ± 2008 G. L. Heileman Module 8, Lecture 1
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Overview Background Definitions Representing Problems as Graphs Representing Graphs in Computer Memory Breadth-first and Depth-first Search The Minimum Spanning Tree Problem Shortest Paths Problems c ± 2008 G. L. Heileman Module 8, Lecture 1
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Introduction Graphs are also commonly used to model physical situations involving discrete entities that are related in some fashion. For example, graphs are routinely used to represent and study: Transportation networks Communication networks Parallel computer architectures Precedence relationships between tasks that make up a job Computation Electrical circuits Given a graph that is modeling some physical entity, we will want to infer something about this entity by studying the properties of its corresponding graph. c ± 2008 G. L. Heileman Module 8, Lecture 1
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Introduction We use G = ( V , E ) to denote a graph, where V is the vertex set , and E the edge set containing edges that may exist between the vertices in V . We use | V | and | E | to denote the sizes of the vertex and edge sets, respectively. In an undirected graph , the edges do not have a direction, while in a directed graph they do. Consider the edge ( u , v ) E . This is an edge between vertex u V and vertex v V . If G is undirected, then ( u , v ) = ( v , u ), and we say that u an v are adjacent . If G is directed, then ( u , v ) and ( v , u ) (if it exists in G ) are different edges, and for ( u , v ) we say that u is adjacent to v , and v is adjacent from u . In a weighted graph , weights are assigned to the edges. A weight assignment for G = ( V , E ) is given by the mapping w : { ( u , v ) E } → < . c ± 2008 G. L. Heileman Module 8, Lecture 1
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Graph Problems After the structure of a problem has been cast as a graph, the next step involves restating the physical problem of interest as a graph problem. The use of graphs in this manner leads to geometric interpretations of physical problems that may be easier analyze. Furthermore, if we can pose a physical problem as a graph problem, we may be able to use existing graph-theoretic results to solve the problem. A single graph problem can often be used to model problems in a large number of application areas. This makes the issue of finding efficient algorithms for solving common graph problems extremely important. Let us now consider some common graph problems. c ± 2008 G. L. Heileman Module 8, Lecture 1
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Connectedness and Components One of the most basic questions one may ask about a graph is whether or not it is connected. If a graph is not connected, it may useful to know how many different connected components it has (in the case of directed graphs, we may want to know the number of strongly connected components).
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graphs1-handouts - ECE 537 Foundations of Computing Module...

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