Classes of graphs which approximate the complete euclidean graph

Keil, J. Mark; Gutwin, Carl

doi:10.1007/bf02187821

Cited by 285 publications

(205 citation statements)

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“…Geometric spanners tend to fall into three categories: (i) Long-known geometric graphs that happen to be spanners, such as Delaunay triangulations; (ii) cone-based constructions, such as Keil's θ-graphs [20]; and (iii) well-separated pair decomposition(WSPD) based constructions introduced by Callaghan and Kosaraju [14]. Note that graphs in the first category have fixed worst-case spanning ratios bounded away from 1.…”

Section: Motivationmentioning

confidence: 99%

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Local Routing in Spanners Based on WSPDs

Bose

Carufel

Dujmović

et al. 2017

Lecture Notes in Computer Science

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The well-separated pair decomposition (WSPD) of the complete Euclidean graph defined on points in Although competitive local-routing strategies exist for various spanners such as Yao-graphs, Θ-graphs, and variants of Delaunay graphs, few local-routing strategies are known for any WSPD-spanner. Our main contribution is a local-routing algorithm with a near-optimal competitive routing ratio of 1 + O(1/s) on a WSPD-spanner.Specifically, using Callahan and Kosaraju's fair split-tree, we show how to build a WSPD-spanner with spanning ratio 1 + 4/s + 4/(s − 2) which is a slight improvement over 1 + 8/(s − 4). We then present a 2-local and a 1-local routing algorithm on this spanner with competitive routing ratios of 1 + 6/(s − 2) + 4/s and 1 + 8/(s − 2) + 4/s + 8/s 2 , respectively. Moreover, we prove that there exists a point set for which our WSPD-spanner has a spanning ratio of at least 1 + 8/s, thereby proving the near-optimality of its spanning ratio and the near-optimality of the routing ratio of both our routing algorithms.ii

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Section: Motivationmentioning

confidence: 99%

“…In most cases, proving tight spanning ratios and routing ratios for graphs in this category is difficult. For example, even the exact spanning ratio of the Delaunay triangulation is unknown, despite over 30 years of study [9,17,20,23].…”

Section: Motivationmentioning

confidence: 99%

Local Routing in Spanners Based on WSPDs

Bose

Carufel

Dujmović

et al. 2017

Lecture Notes in Computer Science

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“…Given V , there may be more than one Delaunay triangulation, but only if V contains four or more co-circular vertices. Del has constant spanning ratio [8]. Del can not be constructed locally, because it may contain arbitrary long edges.…”

Section: Related Workmentioning

confidence: 99%

A novel family of geometric planar graphs for wireless ad hoc networks

Mitton

Simplot-Ryl

et al. 2011

2011 Proceedings IEEE INFOCOM

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Abstract-We propose a radically new family of geometric graphs, i.e., Hypocomb, Reduced Hypocomb and Local Hypocomb. The first two are extracted from a complete graph; the last is extracted from a Unit Disk Graph (UDG). We analytically study their properties including connectivity, planarity and degree bound. All these graphs are connected (provided the original graph is connected) planar. Hypocomb has unbounded degree while Reduced Hypocomb and Local Hypocomb have maximum degree 6 and 8, respectively. To our knowledge, Local Hypocomb is the first strictly-localized, degree-bounded planar graph computed using merely 1-hop neighbor position information. We present a construction algorithm for these graphs and analyze its time complexity. Hypocomb family graphs are promising for wireless ad hoc networking. We report our numerical results on their average degree and their impact on FACE routing [2]. We discuss their potential applications and some open problems.

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“…Our construction will be based on a combination of the spanner based on the well-separated pair decomposition of Callahan and Kosaraju [6,7] (see also Section 2) and ideas that have been used in analyzing the Θ-graph spanner of Clarkson [9] and Keil and Gutwin [15]. As we will show in Section 3, this combination leads to a simple and sufficient condition for a graph being a spanner.…”

Section: Organization Of the Papermentioning

confidence: 99%

“…The property is based on a combination of the WSDP-spanner of [6] and techniques that have been used in the analysis of the Θ-graph spanner of [9,15].…”

Section: A Sucient Condition For Being a Spannermentioning

confidence: 99%

An Optimal Algorithm for Computing Angle-Constrained Spanners

Carmi

Smid

2010

Algorithms and Computation

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Abstract. Let S be a set of n points in R d and let t > 1 be a real number. A graph G = (S, E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0 < θ < π/3, a θ-angle-constrained t-spanner can be computed in O(n log n) time, where t depends only on θ. For values of θ approaching 0, we have t = 1 + O(θ).

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Classes of graphs which approximate the complete euclidean graph

Cited by 285 publications

References 9 publications

Local Routing in Spanners Based on WSPDs

Local Routing in Spanners Based on WSPDs

A novel family of geometric planar graphs for wireless ad hoc networks

An Optimal Algorithm for Computing Angle-Constrained Spanners

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