Sparse hop spanners for unit disk graphs

Dumitrescu, Adrian; Ghosh, Anirban; Tóth, Csaba D.

doi:10.1016/j.comgeo.2021.101808

Cited by 3 publications

(3 citation statements)

References 37 publications

(52 reference statements)

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“…In-browser visualizations of some of the algorithms (those based on the L 2 -Delaunay triangulation) have been recently presented in [3]. In related works, the construction of plane hop spanners (where the number of hops in shortest paths is of interest) for unit disk graphs has been considered in [6,19,26].…”

Section: Referencementioning

confidence: 99%

Bounded-degree plane geometric spanners in practice

Anderson¹,

Ghosh²,

Graham³

et al. 2022

Preprint

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The construction of bounded-degree plane geometric spanners has been a focus of interest since 2002 when Bose, Gudmundsson, and Smid proposed the first algorithm to construct such spanners. To date, eleven algorithms have been designed with various trade-offs in degree and stretch factor. We have implemented these sophisticated algorithms in C++ using the CGAL library and experimented with them using large synthetic and real-world pointsets. Our experiments have revealed their practical behavior and real-world efficacy. We share the implementations via GitHub 1 for broader uses and future research.We present a simple practical algorithm, named AppxStretchFactor, that can estimate stretch factors (obtains lower bounds on the exact stretch factors) of geometric spanners -a challenging problem for which no practical algorithm is known yet. In our experiments with bounded-degree plane geometric spanners, we find that AppxStretchFactor estimates stretch factors almost precisely. Further, it gives linear runtime performance in practice for the pointset distributions considered in this work, making it much faster than the naive Dijkstra-based algorithm for calculating stretch factors.

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Section: Referencementioning

confidence: 99%

Bounded-degree plane geometric spanners in practice

Anderson¹,

Ghosh²,

Graham³

et al. 2022

Preprint

Self Cite

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“…A unit disk is a closed disk of unit diameter in R 2 ; two unit disks intersect if and only if their centers are at distance at most 1 apart. Previous work [10,20,29] reduced the problem to a bipartite setting, where we need to find a 2-hop spanner between two cliques on opposite sides of a line. For finite sets A, B ⊂ R 2 , let U (A, B) denote the unit disk graph on A ∪ B, and let G(A, B) denote the bipartite subgraph of U (A, B) of all edges between A and B.…”

Section: Two-hop Spanners For Unit Disk Graphsmentioning

confidence: 99%

“…(3) It was shown in [29,Lemma 4] that every D ∈ D that contains a point in A also contains some point in A ∩ ∂hull(A). As the center of D is below the x-axis, if D contains a point p = (p x , p y ) above the x-axis, then D also contains all points (p x , p y ) where 0 ≤ p y ≤ p y .…”

Section: Let D D ∈ D Suppose That ∂D Intersects ∂Hull(a) At Points Wi...mentioning

confidence: 99%

Hop-Spanners for Geometric Intersection Graphs

Conroy¹,

Tóth²

2021

Preprint

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A t-spanner of a graph G = (V, E) is a subgraph H = (V, E ) that contains a uv-path of length at most t for every uv ∈ E. It is known that every n-vertex graph admits a (2k − 1)-spanner with O(n 1+1/k ) edges for k ≥ 1. This bound is the best possible for 1 ≤ k ≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture.We study t-spanners for t ∈ {2, 3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results:(1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(n log n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(n log n) edges, and this bound is the best possible. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(n log n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(n log 2 n) edges.

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Bounded-Degree Plane Geometric Spanners in Practice

Anderson

Ghosh

Graham

et al. 2023

ACM J. Exp. Algorithmics

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The construction of bounded-degree plane geometric spanners has been a focus of interest since 2002 when Bose, Gudmundsson, and Smid proposed the first algorithm to construct such spanners. To date, eleven algorithms have been designed with various trade-offs in degree and stretch-factor. We have implemented these sophisticated spanner algorithms in C ++ using the CGAL library and experimented with them using large synthetic and real-world pointsets. Our experiments have revealed their practical behavior and real-world efficacy. We share the implementations via GitHub for broader uses and future research. We design and engineer EstimateStretchFactor , a simple practical algorithm, that can estimate stretch-factors (obtains lower bounds on the exact stretch-factors) of geometric spanners – a challenging problem for which no practical algorithm is known yet. In our experiments with bounded-degree plane geometric spanners, we found that EstimateStretchFactor estimated stretch-factors almost precisely. Further, it gave linear runtime performance in practice for the pointset distributions considered in this work, making it much faster than the naive Dijkstra-based algorithm for calculating stretch-factors.

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Sparse hop spanners for unit disk graphs

Cited by 3 publications

References 37 publications

Bounded-degree plane geometric spanners in practice

Bounded-degree plane geometric spanners in practice

Hop-Spanners for Geometric Intersection Graphs

Bounded-Degree Plane Geometric Spanners in Practice

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