57 research outputs found
On Euclidean Steiner (1+?)-Spanners
Lightness and sparsity are two natural parameters for Euclidean (1+?)-spanners. Classical results show that, when the dimension d ? ? and ? > 0 are constant, every set S of n points in d-space admits an (1+?)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ? > 0 for constant d ? ? have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+?)-spanner. They gave upper bounds of O?(?^{-(d+1)/2}) for the minimum lightness in dimensions d ? 3, and O?(?^{-(d-1))/2}) for the minimum sparsity in d-space for all d ? 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+?)-spanners of lightness O(?^{-1}log?) in the plane, where ? ? ?(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points.
In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+?)-spanners. Using a new geometric analysis, we establish lower bounds of ?(?^{-d/2}) for the lightness and ?(?^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ? 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+?)-spanners of lightness O(?^{-1}log n) for n points in Euclidean plane
Light Euclidean Steiner Spanners in the Plane
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio
of the spanner weight to the weight of the minimum spanning tree of a finite
set of points in . In a recent breakthrough, Le and Solomon
(2019) established the precise dependencies on and of the minimum lightness of -spanners, and
observed that additional Steiner points can substantially improve the
lightness. Le and Solomon (2020) constructed Steiner -spanners
of lightness in the plane, where is the \emph{spread} of the point set, defined as the ratio
between the maximum and minimum distance between a pair of points. They also
constructed spanners of lightness in
dimensions . Recently, Bhore and T\'{o}th (2020) established a lower
bound of for the lightness of Steiner
-spanners in , for . The central open
problem in this area is to close the gap between the lower and upper bounds in
all dimensions .
In this work, we show that for every finite set of points in the plane and
every , there exists a Euclidean Steiner
-spanner of lightness ; this matches the
lower bound for . We generalize the notion of shallow light trees, which
may be of independent interest, and use directional spanners and a modified
window partitioning scheme to achieve a tight weight analysis.Comment: 29 pages, 14 figures. A 17-page extended abstract will appear in the
Proceedings of the 37th International Symposium on Computational Geometr
Light Euclidean Spanners with Steiner Points
The FOCS'19 paper of Le and Solomon, culminating a long line of research on
Euclidean spanners, proves that the lightness (normalized weight) of the greedy
-spanner in is for any
and any (where
hides polylogarithmic factors of ), and also shows the
existence of point sets in for which any -spanner
must have lightness . Given this tight bound on the
lightness, a natural arising question is whether a better lightness bound can
be achieved using Steiner points.
Our first result is a construction of Steiner spanners in with
lightness , where is the spread of the
point set. In the regime of , 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 (e.g., for various random distributions) have polynomially bounded
spread, while in spanner applications often controls the precision,
and it sometimes needs to be much smaller than . Moreover, for
spread polynomially bounded in , 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 can be constructed in
time for polynomially bounded spread, where
hides a factor of . Finally, we extend the
construction to higher dimensions, proving a lightness upper bound of
for any and any .Comment: 23 pages, 2 figures, to appear in ESA 2
Light Spanners
A -spanner of a weighted undirected graph , is a subgraph
such that for all . The sparseness of
the spanner can be measured by its size (the number of edges) and weight (the
sum of all edge weights), both being important measures of the spanner's
quality -- in this work we focus on the latter.
Specifically, it is shown that for any parameters and ,
any weighted graph on vertices admits a
-stretch spanner of weight at most , where is the weight of a minimum
spanning tree of . Our result is obtained via a novel analysis of the
classic greedy algorithm, and improves previous work by a factor of .Comment: 10 pages, 1 figure, to appear in ICALP 201
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