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Bounded-degree plane geometric spanners in practice [article]

Frederick Anderson, Anirban Ghosh, Matthew Graham, Lucas Mougeot, David Wisnosky
2022 arXiv   pre-print
In our experiments with bounded-degree plane geometric spanners, we find that AppxStretchFactor estimates stretch factors almost precisely.  ...  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.  ...  Bounded-degree plane geometric spanners have been an area of interest in computational geometry for a long time.  ... 
arXiv:2205.03204v1 fatcat:jfklsbxq7vflbj2kpzjuhrfizm

Geometric Spanners [chapter]

Joachim Gudmundsson, Giri Narasimhan, Michiel Smid
2008 Encyclopedia of Algorithms  
There are many open problems in this area. We only mention a few:  ...  CROSS REFERENCES Plane geometric spanners, Well-separated pair decomposition and Applications of geometric spanners.  ...  There are modifications that can be made to obtain bounded diameter or bounded degree.  ... 
doi:10.1007/978-0-387-30162-4_167 fatcat:d5zduebdmzcfbmg3vg6dncapqy

Light Euclidean Spanners with Steiner Points

Hung Le, Shay Solomon, Peter Sanders, Grzegorz Herman, Fabrizio Grandoni
2020 European Symposium on Algorithms  
In the regime of Δ ≪ 2^(1/ε), this provides an improvement over the lightness bound of [Hung Le and Shay Solomon, 2019]; this regime of parameters is of practical interest, as point sets arising in real-life  ...  spanner can be constructed in O_ε(n) time for polynomially bounded spread, where O_ε hides a factor of poly(1/(ε)).  ...  Researchers have studied random distributions of point sets, in part to explain the success of solving various geometric optimization problems in practice [9, 53, 30] , and there are many results on spanners  ... 
doi:10.4230/lipics.esa.2020.67 dblp:conf/esa/LeS20 fatcat:xhg6xti37fe5xkel5dw4ntojsu

Optimal Spanners for Unit Ball Graphs in Doubling Metrics [article]

David Eppstein, Hadi Khodabandeh
2021 arXiv   pre-print
Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple 𝒪(log^*n)  ...  -round distributed algorithm that given a unit ball graph G with n vertices and a positive constant ϵ < 1 finds a (1+ϵ)-spanner with constant bounds on its maximum degree and its lightness using only 2  ...  This, together with the well-known fact that greedy spanners have bounded-degree in the two dimensional Euclidean plane, makes greedy spanners more practical in this particular metric space.  ... 
arXiv:2106.15234v1 fatcat:qoh6b2p7djaxvd4kphycyzclz4

Light Euclidean Spanners with Steiner Points [article]

Hung Le, Shay Solomon
2020 arXiv   pre-print
In the regime of Δ≪ 2^1/ϵ, this provides an improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications  ...  Moreover, for spread polynomially bounded in 1/ϵ, this upper bound provides a quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019], We then demonstrate that such a light spanner  ...  Researchers have studied random distributions of point sets, in part to explain the success of solving various geometric optimization problems in practice [9, 56, 32] , and there are many results on spanners  ... 
arXiv:2007.11636v2 fatcat:r3iroeq3irbqjlyumscklftwrq

On Spanners and Lightweight Spanners of Geometric Graphs

Iyad A. Kanj, Ljubomir Perković, Ge Xia
2010 SIAM journal on computing (Print)  
While Delaunay graphs are good plane geometric spanners of Euclidean graphs, they may have unbounded degree.  ...  We extend this algorithm to compute a bounded-degree plane lightweight spanner of a complete Euclidean graph.  ...  The bounded-degree plane spanner G of E can be constructed in O(n log n) time by Theorem 3.11.  ... 
doi:10.1137/080737708 fatcat:7edhqnzzkjbgtl45434w4dldy4

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces [chapter]

Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, David Ilcinkas
2010 Lecture Notes in Computer Science  
In particular, Θ6-graphs are 2-spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible.  ...  Θ k -graphs are geometric graphs that appear in the context of graph navigation.  ...  Although geometric graphs are in theory specific weighted graphs, they naturally model many practical problems and in various fields of Computer Science, from Networking to Computational Geometry.  ... 
doi:10.1007/978-3-642-16926-7_25 fatcat:edkbc7ndkrcs5nxd7qqh3j3aq4

Sparse Hop Spanners for Unit Disk Graphs [article]

Adrian Dumitrescu, Anirban Ghosh, Csaba D. Tóth
2021 arXiv   pre-print
A unit disk graph G on a given set P of points in the plane is a geometric graph where an edge exists between two points p,q ∈ P if and only if |pq| ≤ 1.  ...  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 for any positive integer k.  ...  For packet routing and other applications, a bounded-degree plane geometric spanner of the wireless network is often desired but not always feasible [7] .  ... 
arXiv:2002.07840v3 fatcat:jbenhv2z2fcfjn4ztvv3i52cqa

Localized Topology Control Algorithm with No Geometric Information for Ad Hoc Sensor Networks

Kamrul Islam, Selim G. Akl
2008 2008 Second International Conference on Sensor Technologies and Applications (sensorcomm 2008)  
We provide simulation results and show that on random graphs the resulting topology is spanner and the average degree of the subgraph is low.  ...  Assuming sensor nodes are deployed in the plane, it is shown that the resulting subgraph obtained by our algorithm is connected, symmetric and contains few edge crossings.  ...  In fact, a node in the network topology can have a logarithmic bound on the degree in the original graph.  ... 
doi:10.1109/sensorcomm.2008.72 fatcat:rv6edo3vszh6lp3q5eyp3dfx4a

On the Edge Crossings of the Greedy Spanner [article]

David Eppstein, Hadi Khodabandeh
2021 arXiv   pre-print
In this paper, we study the edge crossings of the greedy spanner for points in the Euclidean plane.  ...  Our results imply that greedy spanners for points in the plane have separators of size 𝒪(√(n)), that their planarizations have linear size, and that a separator hierarchy for these graphs can be constructed  ...  Despite the simplicity of Algorithm 1, Farshi and Gudmundsson [29] observed that in practice, greedy spanners are surprisingly good in terms of the number of edges, weight, maximum vertex degree, and  ... 
arXiv:2002.05854v2 fatcat:jtvuww2vbvavrooiw5gwcvyenu

δ-Greedy t-spanner

A. Karim Abu-Affash, Gali Bar-On, Paz Carmi
2021 Computational geometry  
The δ-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice.  ...  We show how to construct such a spanner for a set of n points in the plane in O (n 2 log n) time.  ...  Then, in Section 2.2, we prove that the resulting graph is indeed a t-spanner with bounded degree.  ... 
doi:10.1016/j.comgeo.2021.101807 fatcat:4rklm5hluvdwni2ewkyvm4luva

On the Power of the Semi-Separated Pair Decomposition [chapter]

Mohammad Ali Abam, Paz Carmi, Mohammad Farshi, Michiel Smid
2009 Lecture Notes in Computer Science  
In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set S ⊂ R d of n imprecise  ...  Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in O(n 2 log 2 n) time using O(n log n) space and answers a query in  ...  They proposed an algorithm and showed that it works well in practice, but no theoretical bound was provided.  ... 
doi:10.1007/978-3-642-03367-4_1 fatcat:pmyzrjil3vdefgwqludjbpisci

δ-Greedy t-spanner [article]

Gali Bar-On, Paz Carmi
2017 arXiv   pre-print
The δ-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice.  ...  We show how to construct such a spanner for a set of n points in the plane in O(n^2 n) time.  ...  Then, in Section 2.2 we prove that the resulting graph is indeed a t-spanner with bounded degree.  ... 
arXiv:1702.05900v1 fatcat:zvz3ypeflbea3i56yfql2iqsoe

On the power of the semi-separated pair decomposition

Mohammad Ali Abam, Paz Carmi, Mohammad Farshi, Michiel Smid
2013 Computational geometry  
In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set S ⊂ R d of n imprecise  ...  Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in O(n 2 log 2 n) time using O(n log n) space and answers a query in  ...  They proposed an algorithm and showed that it works well in practice, but no theoretical bound was provided.  ... 
doi:10.1016/j.comgeo.2013.02.003 fatcat:ffbecq4ka5bllek2lkhtdhwria

Experimental Study of Geometric t-Spanners [chapter]

Mohammad Farshi, Joachim Gudmundsson
2005 Lecture Notes in Computer Science  
Recently low-weight spanners found interesting practical  ...  We say that a geometric network G(V, E) is a (geometric) t-spanner for V , if for each pair of points u, v ∈ V , there exists a path in G between u and v of weight at most t · |uv|.  ...  We start with a directed √ t-spanner with bounded out-degree, denoted − → G . We will use the Θgraph which easily can be seen to have out-degree k, but linear in-degree.  ... 
doi:10.1007/11561071_50 fatcat:dkbk5xgtyza4jcnc6m4n4ewwiy
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