Constructing Plane Spanners of Bounded Degree and Low Weight

Bose, Prosenjit; Gudmundsson, Joachim; Smid, Michiel

doi:10.1007/3-540-45749-6_24

Cited by 57 publications

(82 citation statements)

References 13 publications

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“…This guarantee turns out to be particularly important for routing [37,1], computing almost shortest paths [22,23], and in other applications. Another parameter that plays an important role in many applications is the maximum (vertex) degree of the constructed tree [6,14,8,37].…”

Section: Background and Main Resultsmentioning

confidence: 99%

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Dinitz

Elkin

Solomon

2009

Discrete Comput Geom

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We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight, and a spanning tree T ′ with weight w(T ′ ) = O(k) · w(M ST (M )) and unweighted diameter O(k · n 1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently.We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic 1-dimensional Euclidean space.Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n) · w(M ST (M )). In SODA'05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors.Finally, our upper bounds imply improved approximation algorithms for the minimum h-hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.

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Section: Background and Main Resultsmentioning

confidence: 99%

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Dinitz

Elkin

Solomon

2009

Discrete Comput Geom

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“…We denote this number byn(v) (son(v) = N 71 (v) ). We show that the processing time is bounded by O n(v) 3 .…”

Section: Processing Timementioning

confidence: 93%

Local Construction and Coloring of Spanners of Location Aware Unit Disk Graphs

Wiese

Kranakis

2009

Discrete Math. Algorithm. Appl.

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Abstract. We look at the problem of coloring locally specially constructed spanners of unit disk graphs. First we present a local approximation algorithm for the vertex coloring problem in Unit Disk Graphs (UDGs) which uses at most four times as many colors as an optimal solution requires. 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. The output consists of the spanner and the 4-coloring. The computed spanner also has the properties that it is planar, the degree of a vertex in the spanner is at most 5 and the angles between two edges are at least π/3. By enlarging the locality distance (i.e. the size of the neighborhood which a vertex has to explore in order to compute its color) we can ensure the total weight of the spanner to be arbitrarily close to the weight of a minimum spanning tree.We prove that a local algorithm cannot compute a bipartite spanner of a unit disk graph and therefore our algorithm needs at most one color more than any local algorithm for the task requires. Moreover, we prove that there is no local algorithm for 3-coloring UDGs or spanners of UDGs, even if the 3-colorability of the graph (or the spanner respectively) is guaranteed in advance.

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“…Bose et al [9] were the first to show that there always exists a plane t-spanner of degree at most 27 on any set of points in the Euclidean plane where t ≈ 10.02. The result was subsequently improved in [5,6,7,12,27,33] in terms of degree.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds on the Dilation of Plane Spanners

Dumitrescu

Ghosh

2016

Int. J. Comput. Geom. Appl.

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Abstract(I) We exhibit a set of 23 points in the plane that has dilation at least 1.4308, improving the previous best lower bound of 1.4161 for the worst-case dilation of plane spanners.(II) For every n ≥ 13, there exists an n-element point set S such that the degree 3 dilation of S equals 1 + √ 3 = 2.7321 . . . in the domain of plane geometric spanners. In the same domain, we show that for every n ≥ 6, there exists a an n-element point set S such that the degree 4 dilation of S equals 1 + (5 − √ 5)/2 = 2.1755 . . . The previous best lower bound of 1.4161 holds for any degree.(III) For every n ≥ 6, there exists an n-element point set S such that the stretch factor of the greedy triangulation of S is at least 2.0268.

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Constructing Plane Spanners of Bounded Degree and Low Weight

Cited by 57 publications

References 13 publications

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Local Construction and Coloring of Spanners of Location Aware Unit Disk Graphs

Lower Bounds on the Dilation of Plane Spanners

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