Geometric Spanner Networks

Narasimhan, Giri; Smid, Michiel

doi:10.1017/cbo9780511546884

Cited by 385 publications

(490 citation statements)

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“…Since then, spanners have received a lot of attention; see the survey papers [10,13,22] and the books [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown that for any set V of n points in the Euclidean space R d and for any fixed t > 1, the greedy spanner has O(n) edges, maximum degree O(1), and total weight O(wt(M ST (V ))), where wt(M ST (V )) is the weight of a minimum spanning tree of V ; see [8,18]. Thus, in R d , the naïve implementation of the greedy algorithm runs in near-cubic time.…”

Section: Introductionmentioning

confidence: 99%

“…Due to the high time complexity of the greedy algorithm, researchers have proposed algorithms for computing other types of sparse t-spanners, see [18]. For Euclidean space R d , there are several algorithms that construct t-spanners with O(n) edges in O(n log n) time.…”

Section: Introductionmentioning

confidence: 99%

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Computing the Greedy Spanner in Near-Quadratic Time

Bose

Carmi

Farshi

et al. 2008

Algorithm Theory – SWAT 2008

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The greedy algorithm produces high-quality spanners and, therefore, is used in several applications. However, even for points in d-dimensional Euclidean space, the greedy algorithm has near-cubic running time. In this paper, we present an algorithm that computes the greedy spanner for a set of n points in a metric space with bounded doubling dimension in O(n 2 log n) time. Since computing the greedy spanner has an Ω(n 2 ) lower bound, the time complexity of our algorithm is optimal within a logarithmic factor.

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“…Since then, spanners have received a lot of attention; see the survey papers [10,13,22] and the books [18,19].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computing the Greedy Spanner in Near-Quadratic Time

Bose

Carmi

Farshi

et al. 2008

Algorithm Theory – SWAT 2008

Self Cite

View full text Add to dashboard Cite

show abstract

“…Euclidean spanners are being subject of ongoing intensive research since the mid-eighties. See the recent book by Narasimhan and Smid [19] for an excellent survey on this subject. Euclidean single-sink spanners were studied by Arya et al [2]; see also [19], Chapter 4.2.…”

Section: Rtvmentioning

confidence: 99%

“…See the recent book by Narasimhan and Smid [19] for an excellent survey on this subject. Euclidean single-sink spanners were studied by Arya et al [2]; see also [19], Chapter 4.2. Lukovszki [16,15] devised fault-tolerant constructions of single-sink spanners.…”

Section: Rtvmentioning

confidence: 99%

Narrow-Shallow-Low-Light Trees with and without Steiner Points

Elkin

Solomon

2009

Lecture Notes in Computer Science

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Abstract. We show that for every set S of n points in the plane and a designated point rt ∈ S, there exists a tree T that has small maximum degree, depth and weight. Moreover, for every point v ∈ S, the distance between rt and v in T is within a factor of (1 + ) close to their Euclidean distance rt, v . We call these trees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction achieves optimal (up to constant factors) tradeoffs between all parameters of NSLLTs. Our construction extends to point sets in R d , for an arbitrarily large constant d. The running time of our construction is O(n · log n). We also study this problem in general metric spaces, and show that NSLLTs with small maximum degree, depth and weight can always be constructed if one is willing to compromise the root-distortion. On the other hand, we show that the increased root-distortion is inevitable, even if the point set S resides in a Euclidean space of dimension Θ(log n). On the bright side, we show that if one is allowed to use Steiner points then it is possible to achieve root-distortion (1 + ) together with small maximum degree, depth and weight for general metric spaces. Finally, we establish some lower bounds on the power of Steiner points in the context of Euclidean spanning trees and spanners.

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Spanners in randomly weighted graphs: Euclidean case

Frieze

Pegden

2023

Journal of Graph Theory

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Given a connected graph G=(V,E) $G=(V,E)$ and a length function ℓ:E→double-struckR $\ell :E\to {\mathbb{R}}$ we let dv,w ${d}_{v,w}$ denote the shortest distance between vertex v $v$ and vertex w $w$. A t $t$‐spanner is a subset Efalse′⊆E $E^{\prime} \subseteq E$ such that if dv,w′ ${d}_{v,w}^{^{\prime} }$ denotes shortest distances in the subgraph Gfalse′=(V,Efalse′) $G^{\prime} =(V,E^{\prime} )$ then dv,w′≤tdv,w ${d}_{v,w}^{^{\prime} }\le t{d}_{v,w}$ for all v,w∈V $v,w\in V$. We study the size of spanners in the following scenario: we consider a random embedding Xp ${{\mathscr{X}}}_{p}$ of Gn,p ${G}_{n,p}$ into the unit square with Euclidean edge lengths. For ϵ>0 $\epsilon \gt 0$ constant, we prove the existence w.h.p. of (1+ϵ) $(1+\epsilon )$‐spanners for Xp ${{\mathscr{X}}}_{p}$ that have Oϵ(n) ${O}_{\epsilon }(n)$ edges. These spanners can be constructed in Oϵ(n2logn) ${O}_{\epsilon }({n}^{2}\mathrm{log}n)$ time. (We will use Oϵ ${O}_{\epsilon }$ to indicate that the hidden constant depends on ε $\varepsilon $). There are constraints on p $p$ preventing it going to zero too quickly.

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Geometric Spanner Networks

Cited by 385 publications

References 1 publication

Computing the Greedy Spanner in Near-Quadratic Time

Computing the Greedy Spanner in Near-Quadratic Time

Narrow-Shallow-Low-Light Trees with and without Steiner Points

Spanners in randomly weighted graphs: Euclidean case

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