Near-Optimal Multicriteria Spanner Constructions in Wireless Ad Hoc Networks

Shpungin, Hanan; Segal, Michael

doi:10.1109/tnet.2010.2053381

Cited by 34 publications

(6 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Additional applications of such spanners in distributed systems include network synchronization and computing global functions [Awe85, PU89a, ABP90, ABP91, Pel00]. Light and sparse spanners were also found useful for various data gathering and dissemination tasks in overlay networks [BKR + 02, VWF + 03, KV02], in wireless and sensor networks [vRW04, BSDS04,SS10], for VLSI circuit design [CKR + 91, CKR + 92a, CKR + 92b, SCRS01], for routing [WCT02,PU89a,PU89b,TZ01b], to compute distance oracles and labels [Pel99,TZ01a,RTZ05], and to compute almost shortest paths [Coh93, RZ04, Elk05, EZ06, FKM + 05]. Low degree spanners are also very useful in many of these applications.…”

mentioning

confidence: 99%

The Greedy Spanner is Existentially Optimal

Filtser

Solomon

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive.In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area.Further, we observe that approximate-greedy algorithms are existentially near-optimal as well. Consequently, we provide an O(n log n)-time construction of (1 + )-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is O(n log 2 n) and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al. [SICOMP'02]) in all the involved parameters (up to dependencies on and the dimension).

show abstract

mentioning

confidence: 99%

The Greedy Spanner is Existentially Optimal

Filtser

Solomon

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

show abstract

“…One of the most popular applications of the spanners is to serve as overlay networks in wireless networks [vRW04,BDS04,SS10]. In these applications, we are often interested in solving computational problems in the spanner, such as shortest path [LWF03,GZ05], independent set [Bas01,MM09], dominating sets [MM09,PCAT18], connected dominating set [YWWY13].…”

Section: Algorithmic Implicationsmentioning

confidence: 99%

“…Over more than 30 years, research on spanners has evolved into an independent field of study, and has found numerous applications, such as to VLSI circuit design [CKR + 91, CKR + 92, SCRS01], to distributed computing [Awe85, ABP92, Pel00], to approximation algorithms [RS98, Kle05, Kle06, Got15, BLW17, CFKL20], to wireless and sensor networks [vRW04,BDS04,SS10], to machine learning [GKK17], and to computational biology [RG05]. Spanners enjoy wide applicability since they possess many desirable properties such as low sparsity, implying, e.g., low storage cost; small lightness, implying, e.g., low construction cost; low maximum degree, implying, e.g., small routing tables in routing applications.…”

Section: Introductionmentioning

confidence: 99%

Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

Lê¹,

Than²

2021

Preprint

View full text Add to dashboard Cite

The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [EK21] showed that the greedy spanner in R 2 admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in R 2 has a separator of size O( √ k). Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in R d for any constant d ≥ 3 as an open problem.In this paper, we resolve the problem of Eppstein and Khodabandeh [EK21] by showing that any subgraph of k vertices of the greedy spanner in R d has a separator of size O(k 1−1 d ). We introduce a new technique that gives a simple characterization for any geometric graph to have a sublinear separator that we dub τ -lanky: a geometric graph is τ -lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ -lanky geometric graph of n vertices in R d has a separator of size O(τ n 1−1 d ). We then derive our main result by showing that the greedy spanner is O(1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in R d .Our technique naturally extends to doubling metrics. We use the τ -lanky characterization to show that there exists a (1 + )-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size O(n 1− 1 d ); this result resolves an open problem posed by Abam and Har-Peled [AHP10] a decade ago. We then introduce another simple characterization for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new characterization to show that the greedy spanner of an n-point metric space of doubling dimension d has a separator of size O((n 1− 1 d ) + log ∆) where ∆ is the spread of the metric; the factor log(∆) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree. Finally, we discuss algorithmic implications of our results.

show abstract

“…Various spanner constructions have been analyzed on uniformly distributed point sets [1,8,13,25,27]. Some of these constructions are a t-spanner for fixed t, others are parameterizable with arbitrary t > 1.…”

Section: Introductionmentioning

confidence: 99%

Distribution-Sensitive Construction of the Greedy Spanner

Alewijnse

Bouts

Brink

2014

Algorithms - ESA 2014

View full text Add to dashboard Cite

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take Ω(n 2 ) time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many 'short' edges that can be determined locally and usually quickly. To find the usually few remaining 'long' edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and reallife data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points 'close' to each other, then it has t-paths between all pairs of points. This characterization gives an O(n log 2 n log 2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an O((n +|E|) log 2 n log log n) expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E.

show abstract

Near-Optimal Multicriteria Spanner Constructions in Wireless Ad Hoc Networks

Cited by 34 publications

References 60 publications

The Greedy Spanner is Existentially Optimal

The Greedy Spanner is Existentially Optimal

Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

Distribution-Sensitive Construction of the Greedy Spanner

Contact Info

Product

Resources

About