88 research outputs found
Balancing Degree, Diameter and Weight in Euclidean Spanners
In this paper we devise a novel \emph{unified} construction of Euclidean
spanners that trades between the maximum degree, diameter and weight
gracefully. For a positive integer k, our construction provides a
(1+eps)-spanner with maximum degree O(k), diameter O(log_k n + alpha(k)),
weight O(k \cdot log_k n \cdot log n) \cdot w(MST(S)), and O(n) edges. Note
that for k= n^{1/alpha(n)} this gives rise to diameter O(alpha(n)), weight
O(n^{1/alpha(n)} \cdot log n \cdot alpha(n)) \cdot w(MST(S)) and maximum degree
O(n^{1/alpha(n)}), which improves upon a classical result of Arya et al.
\cite{ADMSS95}; in the corresponding result from \cite{ADMSS95} the spanner has
the same number of edges and diameter, but its weight and degree may be
arbitrarily large. Also, for k = O(1) this gives rise to maximum degree O(1),
diameter O(log n) and weight O(log^2 n) \cdot w(MST(S)), which reproves another
classical result of Arya et al. \cite{ADMSS95}. Our bound of O(log_k n +
alpha(k)) on the diameter is optimal under the constraints that the maximum
degree is O(k) and the number of edges is O(n). Our bound on the weight is
optimal up to a factor of log n. Our construction also provides a similar
tradeoff in the complementary range of parameters, i.e., when the weight should
be smaller than log^2 n, but the diameter is allowed to grow beyond log n.
For random point sets in the d-dimensional unit cube, we "shave" a factor of
log n from the weight bound. Specifically, in this case our construction
achieves maximum degree O(k), diameter O(log_k n + alpha(k)) and weight that is
with high probability O(k \cdot log_k n) \cdot w(MST(S)).
Finally, en route to these results we devise optimal constructions of
1-spanners for general tree metrics, which are of independent interest.Comment: 27 pages, 7 figures; a preliminary version of this paper appeared in
ESA'1
Incubators vs Zombies: Fault-Tolerant, Short, Thin and Lanky Spanners for Doubling Metrics
Recently Elkin and Solomon gave a construction of spanners for doubling
metrics that has constant maximum degree, hop-diameter O(log n) and lightness
O(log n) (i.e., weight O(log n)w(MST). This resolves a long standing conjecture
proposed by Arya et al. in a seminal STOC 1995 paper.
However, Elkin and Solomon's spanner construction is extremely complicated;
we offer a simple alternative construction that is very intuitive and is based
on the standard technique of net tree with cross edges. Indeed, our approach
can be readily applied to our previous construction of k-fault tolerant
spanners (ICALP 2012) to achieve k-fault tolerance, maximum degree O(k^2),
hop-diameter O(log n) and lightness O(k^3 log n)
Optimal Euclidean spanners: really short, thin and lanky
In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and
lanky", Arya et al. devised a construction of Euclidean (1+\eps)-spanners
that achieves constant degree, diameter , and weight , and has running time . This construction
applies to -point constant-dimensional Euclidean spaces. Moreover, Arya et
al. conjectured that the weight bound can be improved by a logarithmic factor,
without increasing the degree and the diameter of the spanner, and within the
same running time.
This conjecture of Arya et al. became a central open problem in the area of
Euclidean spanners.
In this paper we resolve the long-standing conjecture of Arya et al. in the
affirmative. Specifically, we present a construction of spanners with the same
stretch, degree, diameter, and running time, as in Arya et al.'s result, but
with optimal weight .
Moreover, our result is more general in three ways. First, we demonstrate
that the conjecture holds true not only in constant-dimensional Euclidean
spaces, but also in doubling metrics. Second, we provide a general tradeoff
between the three involved parameters, which is tight in the entire range.
Third, we devise a transformation that decreases the lightness of spanners in
general metrics, while keeping all their other parameters in check. Our main
result is obtained as a corollary of this transformation.Comment: A technical report of this paper was available online from April 4,
201
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
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
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