There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

Levcopoulos, Christos; Lingas, Andrzej

doi:10.1007/bf01758846

Cited by 48 publications

(38 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similarly, Czumaj and Lingas [7] showed approximation schemes for minimum-cost multiconnectivity problems in geometric graphs. The problem of constructing geometric spanners has received considerable attention from a theoretical perspective; see [1,3,4,5,8,9,10,17,20,21,23,24,33,36], the surveys [12,16,34], and the book by Narasimhan and Smid [28]. Note that considerable research has also been done in the construction of spanners for general graphs; see, for example, the book by Peleg [31] or the recent work by Elkin and Peleg [11] and Thorup and Zwick [35].…”

mentioning

confidence: 99%

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Farshi¹,

Giannopoulos²,

Gudmundsson³

2008

SIAM J. Comput.

View full text Add to dashboard Cite

Abstract. Given a Euclidean graph G in R d with n vertices and m edges, we consider the problem of adding an edge to G such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in O(nm + n 2 log n) time, resulting in a trivial O(n 3 m + n 4 log n)-time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in O(n 4 ) time using O(n 2 ) space. To reduce the running time we consider several approximation algorithms.Key words. computational geometry, approximation algorithms, geometric networks AMS subject classifications. 65D18, 68U05, 68Q25 DOI. 10.1137/050635675 1. Introduction. Consider a set V of n points in R d . A network on V can be modeled as an undirected graph G with vertex set V of size n and an edge set E of size m where every edge (u, v) has a positive weight w (u, v). A Euclidean network is a geometric network where the weight of the edge (u, v) is equal to the Euclidean distance |uv| between its two endpoints u and v.For two vertices u, v in a weighted graph G we use δ G (u, v) to denote a shortest path between u and v in G, and the length of the path is denoted by d G (u, v). Consider a weighted graph G = (V, E) and a graph G = (V, E ) on the same vertex set but with edge set E ⊆ E. We say that G is a t-spanner of G if for each pair of vertices (u, v). The minimum t such that G is a t-spanner for V is called the stretch factor, or dilation, of G.We say that a Euclidean network G = (V, E) is a t-spanner if G = (V, E) is a t-spanner of the complete network on V . In other words, for any two points p, q ∈ V the graph distance in G is at most t times the Euclidean distance between the two points.Complete graphs represent ideal communication networks, but they are expensive to build; sparse spanners represent low-cost alternatives. The weight of the spanner network is a measure of its sparseness; other sparseness measures include the number of edges, the maximum degree, and the number of Steiner points. Spanners for complete Euclidean graphs as well as for arbitrary weighted graphs find applications in robotics, network topology design, distributed systems, design of parallel machines, and many other areas. Recently spanners found interesting practical applications

show abstract

mentioning

confidence: 99%

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Farshi¹,

Giannopoulos²,

Gudmundsson³

2008

SIAM J. Comput.

View full text Add to dashboard Cite

show abstract

“…The constant hidden in the O(1) is fairly large. When considering plane graphs specifically, Levcopoulos and Lingas [46] showed that, for any given real number r > 2, the Delaunay triangulation can be used to construct a plane graph that is a t-spanner for t = (r − 1)4π √ 3/9 and whose total weight is at most 1 + 2/(r − 2) times the weight of a minimum spanning tree. Subsequently, Kanj et al [40] showed how this method can be generalized to build bounded degree plane spanners.…”

Section: Low Weight Plane Spannersmentioning

confidence: 99%

On plane geometric spanners: A survey and open problems

Bose

Smid

2013

Computational Geometry

View full text Add to dashboard Cite

show abstract

“…The concept of graph spanners was first invented first by [26] in a geometric context. To the best of our knowledge the spanner problem on general graphs was first invented indirectly by Peleg and Upfal [30] in their work on small routing tables.…”

Section: The Basic K-spanner Problem and Previous Workmentioning

confidence: 99%

Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-Spanner

Dinitz

Kortsarz

Raz

2012

Automata, Languages, and Programming

View full text Add to dashboard Cite

We study the well-known Label Cover problem under the additional requirement that problem instances have large girth. We show that if the girth is some k, the problem is roughly 2 log 1− n/k hard to approximate for all constant > 0. A similar theorem was claimed by Elkin and Peleg [ICALP 2000], but their proof was later found to have a fundamental error. We use the new proof to show inapproximability for the basic k-spanner problem, which is both the simplest problem in graph spanners and one of the few for which super-logarithmic hardness was not known. Assuming N P ⊆ BP T IM E(2 polylog(n) ), we show that for every k ≥ 3 and every constant > 0 it is hard to approximate the basic k-spanner problem within a factor better than 2 (log 1− n)/k (for large enough n). A similar hardness for basic k-spanner was claimed by Elkin and Peleg [ICALP 2000], but the error in their analysis of Label Cover made this proof fail as well. Thus for the problem of Label Cover with large girth we give the first non-trivial lower bound. For the basic k-spanner problem we improve the previous best lower bound of Ω(log n)/k from [24]. Our main technique is subsampling the edges of 2-query PCPs, which allows us to reduce the degree of a PCP to be essentially equal to the soundness desired. This turns out to be enough to essentially guarantee large girth.

show abstract

There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

Cited by 48 publications

References 7 publications

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

On plane geometric spanners: A survey and open problems

Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-Spanner

Contact Info

Product

Resources

About