11. Clustered Representations: Clusters, Covers and Partitions

Abstract

Previous chapter Next chapter Discrete Mathematics and Applications Distributed Computing: A Locality-Sensitive Approach11. Clustered Representations: Clusters, Covers and Partitionspp.123 - 133Chapter DOI:https://doi.org/10.1137/1.9780898719772.ch11PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt We now temporarily put aside distributed network algorithms and turn to discussing LP-graph representations. We begin this part by focusing on the first type of LP-representations, namely, cluster-based representations. Later, in Chapters 15-18 we turn to skeletal representations, and finally, in Chapter 19 we discuss also a third type of network representation, based on vertex labelings. This chapter introduces the basic concepts of locality-preserving clustered representations and presents the terminology and notation used throughout the discussion of those representations and their properties. 11.1 The graph model We consider an arbitrary weighted graph G= (V,E,ω) , where V is the set of vertices, E ⊆ V×V is the set of edges and ω:E→ R+ is a weight function, assigning a nonnegative weight ω(e) to every edge e ∈ E. The weights are assumed to satisfy the triangle inequality. Usually, this causes no loss of generality, since an edge whose weight exceeds that of some alternate path connecting its two endpoints can be replaced by that path for many practical purposes. We may occasionally restrict ourselves to unweighted graphs by assuming that ω (e) =1 for every edge e ∈ E. Such a graph is denoted G= (V,E) . We usually denote n= ∣V∣ . When discussing an unspecified graph G′ , we may refer to its vertex and edge sets by V( G′ ) and E( G′ ) , respectively. 11.2 Clusters, covers and partitions 11.2.1 Clusters Clusters are naturally the most basic concept in our discussion of locality-preserving clustered representations of graphs. A cluster is essentially a collection of vertices in the graph. However, we are also interested in the topological connections among the vertices in the cluster. Given a set of vertices S ⊆ V , let G(S) denote the subgraph induced by S in G, namely, G (S) = (S, E′ ) , where E′ consists of all the edges of G whose endpoints both belong to S. In what follows we may sometimes interchange S with its induced subgraph G(S). Previous chapter Next chapter RelatedDetails Published:2000ISBN:978-0-89871-464-7eISBN:978-0-89871-977-2 https://doi.org/10.1137/1.9780898719772Book Series Name:Discrete Mathematics and ApplicationsBook Code:DT05Book Pages:xvi + 324Key words:electronic data processing, distributed processing, distributed network algorithms