Lower Bounds on the Dilation of Plane Spanners

Dumitrescu, Adrian; Ghosh, Anirban

doi:10.1142/s0218195916500059

Cited by 18 publications

(8 citation statements)

References 39 publications

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“…In the geometric setting, where the vertices are embedded in a metric space, spanners have been studied in [3,10,12,13,26,28] and many other papers. In particular, plane geometric spanners were studied in [7,8,17,16]. The reader is also referred to the surveys [8,20,30] and the monograph [33] dedicated to this subject.…”

Section: :3mentioning

confidence: 99%

Sparse hop spanners for unit disk graphs

Dumitrescu

Ghosh

Tóth

2022

Computational Geometry

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A unit disk graph G on a given set of points P in the plane is a geometric graph where an edge exists between two points p, q ∈ P if and only if |pq| ≤ 1. A subgraph G of G is a k-hop spanner if and only if for every edge pq ∈ G, the topological shortest path between p, q in G has at most k edges. We obtain the following results for unit disk graphs. I Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from 9n to 5.5n. II Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges. III Every n-vertex unit disk graph has a 2-hop spanner with O(n log n) edges. This is the first nontrivial construction of 2-hop spanners. IV For every sufficiently large n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. V For every point set on a circle, there exists a plane 4-hop spanner. As such, this provides a tight bound for points on a circle. VI The maximum degree of k-hop spanners cannot be bounded from above by a function of k.

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Section: :3mentioning

confidence: 99%

Sparse hop spanners for unit disk graphs

Dumitrescu

Ghosh

Tóth

2022

Computational Geometry

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“…In the geometric setting, where the vertices are embedded in a metric space, spanners have been studied in [3,8,10,11,22,24] and many other papers. In particular, plane geometric spanners were studied in [5,6,14,15]. The reader is also referred to the surveys [6,17,25] and the monograph [27] dedicated to this subject.…”

Section: Related Work Peleg and Schäfermentioning

confidence: 99%

Sparse Hop Spanners for Unit Disk Graphs

Dumitrescu¹,

Ghosh²,

Tóth³

2020

Preprint

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A unit disk graph G on a given set of points P in the plane is a geometric graph where an edge exists between two points p, q ∈ P if and only if |pq| ≤ 1. A subgraph G of G is a k-hop spanner if and only if for every edge pq ∈ G, the topological shortest path between p, q in G has at most k edges. We obtain the following results for unit disk graphs.(i) Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (WADS 2019) and improve the upper bound on the number of edges from 9n to 5.5n. (ii) Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges. (iii) Every n-vertex unit disk graph has a 2-hop spanner with O(n 3/2 ) edges. This is the first construction of a 2-hop spanner with a subquadratic number of edges. (iv) For every sufficiently large n, there exists a set P of n points such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. (v) For every point set on a circle, there exists a plane 4-hop spanner. As such, this provides a tight bound for points on a circle. (vi) The maximum degree of k-hop spanners cannot be bounded above by a function of k.

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“…In two special cases, Bose et al [5] showed that reducing the degree to 3 is possible. In terms of lower bounds, Dumitrescu and Ghosh [19] showed that there exist point sets that require a spanning ratio of at least 1.4308. They also strengthened this bound to 2.1755 for spanners of degree 4 and 2.7321 for spanners of degree 3.…”

Section: Introductionmentioning

confidence: 99%

Local Routing in Sparse and Lightweight Geometric Graphs

et al. 2022

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Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no O(1)-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin (SIAM J Comput 33(4):937–951, 2004) showed that there exists an online routing algorithm that is O(1)-competitive. However, a Delaunay triangulation can have $$\varOmega (n)$$ Ω ( n ) vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree. We show a simple construction, given a set V of n points in the Euclidean plane, of a planar geometric graph on V that has small weight (within a constant factor of the weight of a minimum spanning tree on V), constant degree, and that admits a local routing strategy that is O(1)-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an O(1)-competitive routing strategy.

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Lower Bounds on the Dilation of Plane Spanners

Cited by 18 publications

References 39 publications

Sparse hop spanners for unit disk graphs

Sparse hop spanners for unit disk graphs

Sparse Hop Spanners for Unit Disk Graphs

Local Routing in Sparse and Lightweight Geometric Graphs

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