TY - JOUR

T1 - Packing plane spanning graphs with short edges in complete geometric graphs

AU - Aichholzer, Oswin

AU - Hackl, Thomas

AU - Korman, Matias

AU - Pilz, Alexander

AU - van Renssen, André

AU - Roeloffzen, Marcel

AU - Rote, Günter

AU - Vogtenhuber, Birgit

N1 - Funding Information:
This research was initiated during the 10th European Research Week on Geometric Graphs (GGWeek 2013), Illgau, Switzerland. We would like to thank all participants for fruitful discussions. O.A. A.P. and B.V. were partially supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18. T.H. was supported by the Austrian Science Fund (FWF): P23629-N18 ?Combinatorial Problems on Geometric Graphs?. M.K. was supported in part by the ELC project (MEXT KAKENHI No. 17K12635) and National Science Foundation (US) award CCF-1423615. A.P. is supported by an Erwin Schr?dinger fellowship, Austrian Science Fund (FWF): J-3847-N35. A.v.R. and M.R. were supported by JST ERATO Grant Number JPMJER1201, Japan.
Funding Information:
This research was initiated during the 10th European Research Week on Geometric Graphs (GGWeek 2013), Illgau, Switzerland. We would like to thank all participants for fruitful discussions. O.A., A.P., and B.V. were partially supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18 . T.H. was supported by the Austrian Science Fund (FWF): P23629-N18 ‘Combinatorial Problems on Geometric Graphs’. M.K. was supported in part by the ELC project ( MEXT KAKENHI No. 17K12635 ) and National Science Foundation (US) award CCF-1423615 . A.P. is supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35 . A.v.R. and M.R. were supported by JST ERATO Grant Number JPMJER1201 , Japan.
Funding Information:
This research was initiated during the 10th European Research Week on Geometric Graphs (GGWeek 2013), Illgau, Switzerland. We would like to thank all participants for fruitful discussions. O.A. A.P. and B.V. were partially supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18. T.H. was supported by the Austrian Science Fund (FWF): P23629-N18 ‘Combinatorial Problems on Geometric Graphs’. M.K. was supported in part by the ELC project (MEXT KAKENHI No. 17K12635)and National Science Foundation (US)award CCF-1423615. A.P. is supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35. A.v.R. and M.R. were supported by JST ERATO Grant Number JPMJER1201, Japan.
Publisher Copyright:
© 2019

PY - 2019/9

Y1 - 2019/9

N2 - Given a set of points in the plane, we want to establish a connected spanning graph between these points, called connection network, that consists of several disjoint layers. Motivated by sensor networks, our goal is that each layer is connected, spanning, and plane. No edge in this connection network is too long in comparison to the length needed to obtain a spanning tree. We consider two different approaches. First we show an almost optimal centralized approach to extract two layers. Then we consider a distributed model in which each point can compute its adjacencies using only information about vertices at most a predefined distance away. We show a constant factor approximation with respect to the length of the longest edge in the graphs. In both cases the obtained layers are plane.

AB - Given a set of points in the plane, we want to establish a connected spanning graph between these points, called connection network, that consists of several disjoint layers. Motivated by sensor networks, our goal is that each layer is connected, spanning, and plane. No edge in this connection network is too long in comparison to the length needed to obtain a spanning tree. We consider two different approaches. First we show an almost optimal centralized approach to extract two layers. Then we consider a distributed model in which each point can compute its adjacencies using only information about vertices at most a predefined distance away. We show a constant factor approximation with respect to the length of the longest edge in the graphs. In both cases the obtained layers are plane.

KW - Bottleneck edge

KW - Geometric graphs

KW - Graph packing

KW - Minimum spanning tree

KW - Plane graphs

UR - http://www.scopus.com/inward/record.url?scp=85065057729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065057729&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2019.04.001

DO - 10.1016/j.comgeo.2019.04.001

M3 - Article

AN - SCOPUS:85065057729

VL - 82

SP - 1

EP - 15

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -