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Geometric spanners with small chromatic number

Prosenjit Bose, Paz Carmi, Mathieu Couture, Anil Maheshwari, Michiel Smid, Norbert Zeh
2009 Computational geometry  
We also show that for any > 0, there exists a (1 + )t(k)-spanner for P that has O(|P |) edges and chromatic number at most k.  ...  Given an integer k ≥ 2, we consider the problem of computing the smallest real number t(k) such that for each set P of points in the plane, there exists a t(k)-spanner for P that has chromatic number at  ...  The number of edges in these graphs can be reduced from quadratic to linear with a slight increase in the spanning ratio by applying the general technique of Gudmundsson et al. [4] .  ... 
doi:10.1016/j.comgeo.2008.04.003 fatcat:r6msrwefljbtvhnybnitwk6qpa

Geometric Spanners With Small Chromatic Number [article]

Prosenjit Bose, Paz Carmi, Mathieu Couture, Anil Maheshwari, Michiel Smid, Norbert Zeh
2007 arXiv   pre-print
We also show that for any ϵ >0, there exists a (1+ϵ)t(k)-spanner for P that has O(|P|) edges and chromatic number at most k.  ...  Given an integer k ≥ 2, we consider the problem of computing the smallest real number t(k) such that for each set P of points in the plane, there exists a t(k)-spanner for P that has chromatic number at  ...  For these reasons, it is desirable to have geometric graphs that have both small chromatic number and small stretch factor.  ... 
arXiv:0711.0114v1 fatcat:ms3nqtgsuzbydexvgannrrucqu

Geometric Spanners with Small Chromatic Number [chapter]

Prosenjit Bose, Paz Carmi, Mathieu Couture, Anil Maheshwari, Michiel Smid, Norbert Zeh
Approximation and Online Algorithms  
We also show that for any > 0, there exists a (1 + )t(k)-spanner for P that has O(|P |) edges and chromatic number at most k.  ...  Given an integer k ≥ 2, we consider the problem of computing the smallest real number t(k) such that for each set P of points in the plane, there exists a t(k)-spanner for P that has chromatic number at  ...  The number of edges in these graphs can be reduced from quadratic to linear with a slight increase in the spanning ratio by applying the general technique of Gudmundsson et al. [4] .  ... 
doi:10.1007/978-3-540-77918-6_7 dblp:conf/waoa/BoseCCMSZ07 fatcat:r5fllo2cerdj5avdg46uj5zede

Robust Geometric Spanners [article]

Prosenjit Bose, Vida Dujmovic, Pat Morin, Michiel Smid
2013 arXiv   pre-print
Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices.  ...  On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.  ...  diameter [8, 9] , planar spanners [5, 21, 23, 28] , spanners of low chromatic number [13] , fault-tolerant spanners [2, 22, 30, 31] , low-power spanners [4, 33, 36] , kinetic spanners [1, 3] , angle-constrained  ... 
arXiv:1204.4679v2 fatcat:fl2flb6qy5at3jw3rkt5nkyihu

Robust geometric spanners

Prosenjit Bose, Vida Dujmović, Pat Morin, Michiel Smid
2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13  
Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices.  ...  On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.  ...  diameter [8, 9] , planar spanners [5, 21, 23, 29] , spanners of low chromatic number [13] , fault-tolerant spanners [2, 22, 31, 32] , lowpower spanners [4, 34, 37] , kinetic spanners [1, 3] , angle-constrained  ... 
doi:10.1145/2493132.2462381 fatcat:2jlv5ebocvgkfbcu5bc43vbvci

Robust geometric spanners

Prosenjit Bose, Vida Dujmović, Pat Morin, Michiel Smid
2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13  
Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices.  ...  On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.  ...  diameter [8, 9] , planar spanners [5, 21, 23, 29] , spanners of low chromatic number [13] , fault-tolerant spanners [2, 22, 31, 32] , lowpower spanners [4, 34, 37] , kinetic spanners [1, 3] , angle-constrained  ... 
doi:10.1145/2462356.2462381 dblp:conf/compgeom/BoseDMS13 fatcat:g6jmxnr3q5arrnp7ii4c3b6x54

Robust Geometric Spanners

Prosenjit Bose, Vida Dujmović, Pat Morin, Michiel Smid
2013 SIAM journal on computing (Print)  
Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices.  ...  On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges. Fig. 1.1.  ...  diameter [8, 9] , planar spanners [5, 21, 23, 29] , spanners of low chromatic number [13] , fault-tolerant spanners [2, 22, 31, 32] , lowpower spanners [4, 34, 37] , kinetic spanners [1, 3] , angle-constrained  ... 
doi:10.1137/120874473 fatcat:wg7nxatgenemdkvpuiwjlohtki

Spanners of Complete k-Partite Geometric Graphs [article]

Prosenjit Bose, Paz Carmi, Mathieu Couture, Anil Maheshwari, Pat Morin, Michiel Smid
2007 arXiv   pre-print
We address the following problem: Given a complete k-partite geometric graph K whose vertex set is a set of n points in R^d, compute a spanner of K that has a "small" stretch factor and "few" edges.  ...  The latter result is optimal: We show that for any 2 ≤ k ≤ n - Θ(√(n n)), spanners with O(n n) edges and stretch factor less than 3 do not exist for all complete k-partite geometric graphs.  ...  Compute a t-spanner of the complete k-partite graph K C 1 ...C k that has a "small" number of edges and whose stretch factor t is "small".  ... 
arXiv:0712.0554v1 fatcat:safhlubecjaarcfymjylbendj4

Spanners of Complete k-Partite Geometric Graphs

Prosenjit Bose, Paz Carmi, Mathieu Couture, Anil Maheshwari, Pat Morin, Michiel Smid
2009 SIAM journal on computing (Print)  
We address the following problem: Given a complete k-partite geometric graph K whose vertex set is a set of n points in R d , compute a spanner of K that has a "small" stretch factor and "few" edges.  ...  The latter result is optimal: We show that there exist complete k-partite geometric graphs K such that every subgraph of K with a subquadratic number of edges has stretch factor at least 3.  ...  Compute a t-spanner of the k-partite complete graph K S 1 ...S k that has a "small" number of edges and whose stretch factor t is "small".  ... 
doi:10.1137/070707130 fatcat:d72mofioujhrvnfiqnbu54kj3e

Constructing Sparse t-Spanners with Small Separators [chapter]

Joachim Gudmundsson
2003 Lecture Notes in Computer Science  
The spanner also has some additional properties; low weight and a linear number of edges.  ...  Given a set of n points S in the plane and a real value t > 1 we show how to construct in time O(n log n) a t-spanner G of S such that there exists a set of vertices S of size O( √ n log n) whose removal  ...  Thus this graph has a constant chromatic number, and consequently a constant number of independent sets. Hence, E i,j is subdivided into a constant number of groups, denoted E i,j,k .  ... 
doi:10.1007/978-3-540-45077-1_9 fatcat:enyqrhg7ebbmpdqt7tynklergi

Vertex Fault-Tolerant Geometric Spanners for Weighted Points [article]

Sukanya Bhattacharjee, R. Inkulu
2020 arXiv   pre-print
in the relative interior of the free space of a polygonal domain P, we detail an algorithm to compute a k-vertex fault-tolerant (4+ϵ)-spanner network with O(kn√(h+1)/ϵ^2 n) edges.  ...  Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer k ≥1, and a real number ϵ> 0, we present algorithms for computing a spanner network  ...  Apart from the small number of edges, spanners with additional properties such as small weight, bounded degree, small diameter, planar network, etc., were also considered.  ... 
arXiv:2011.03354v1 fatcat:bjmwwrc42vcbvaon5rowr7vjy4

Spanning Trees and Spanners [chapter]

David Eppstein
2000 Handbook of Computational Geometry  
We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.  ...  Intuitively, each node of T is then involved only in a small number of pairs, corresponding to nodes with similar sized bounding boxes within a similar distance from the node. Lemma 7 (Callahan) .  ...  For geometric graphs in any dimension, they construct spanners with constant dilation and weight O(log n) times that of the minimum spanning tree.  ... 
doi:10.1016/b978-044482537-7/50010-3 fatcat:gitonpgfozgfribivszd6gf5cy

Local edge colouring of Yao-like subgraphs of Unit Disk Graphs

J. Czyzowicz, S. Dobrev, E. Kranakis, J. Opatrny, J. Urrutia
2009 Theoretical Computer Science  
The number of colours used is 2kl + 1 and this is optimal for local algorithms (since the maximal degree is 2kl and a colouring with 2kl colours can only be constructed by a global algorithm), thus showing  ...  of hops away from it, and hence the algorithm terminates in a constant number of steps.  ...  The minimal such number of colors required is called edge chromatic number of the graph.  ... 
doi:10.1016/j.tcs.2008.11.008 fatcat:v2u4d6fq7fhkdpodooib34mh2e

Local Construction and Coloring of Spanners of Location Aware Unit Disk Graphs [chapter]

Andreas Wiese, Evangelos Kranakis
2008 Lecture Notes in Computer Science  
The output consists of the spanner and the 4-coloring.  ...  Next we look at the colorability of spanners of UDGs. In particular we present a local algorithm for constructing a 4-colorable spanner of a unit disk graph.  ...  There are several properties which are desirable for a spanner, e.g. connectivity, planarity, small node degree, small stretch factor and small total weight.  ... 
doi:10.1007/978-3-540-92248-3_33 fatcat:fotpcq5wqjfdvc4naiqwdprb6e

LOCAL CONSTRUCTION AND COLORING OF SPANNERS OF LOCATION AWARE UNIT DISK GRAPHS

ANDREAS WIESE, EVANGELOS KRANAKIS
2009 Discrete Mathematics, Algorithms and Applications (DMAA)  
The output consists of the spanner and the 4-coloring.  ...  Next we look at the colorability of spanners of UDGs. In particular we present a local algorithm for constructing a 4-colorable spanner of a unit disk graph.  ...  There are several properties which are desirable for a spanner, e.g. connectivity, planarity, small node degree, small stretch factor and small total weight.  ... 
doi:10.1142/s1793830909000415 fatcat:olveez45a5csjoxz4g67igdw6q
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