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Geometric spanners with small chromatic number
2009
Computational geometry
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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]
2007
arXiv
pre-print
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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]
Approximation and Online Algorithms
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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]
2013
arXiv
pre-print
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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
2013
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
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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
2013
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
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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
2013
SIAM journal on computing (Print)
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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]
2007
arXiv
pre-print
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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
2009
SIAM journal on computing (Print)
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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]
2003
Lecture Notes in Computer Science
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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]
2020
arXiv
pre-print
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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]
2000
Handbook of Computational Geometry
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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
2009
Theoretical Computer Science
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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]
2008
Lecture Notes in Computer Science
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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
2009
Discrete Mathematics, Algorithms and Applications (DMAA)
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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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