71 research outputs found
A Unified and Fine-Grained Approach for Light Spanners
Seminal works on light spanners from recent years provide near-optimal
tradeoffs between the stretch and lightness of spanners in general graphs,
minor-free graphs, and doubling metrics. In FOCS'19 the authors provided a
"truly optimal" tradeoff for Euclidean low-dimensional spaces. Some of these
papers employ inherently different techniques than others. Moreover, the
runtime of these constructions is rather high.
In this work, we present a unified and fine-grained approach for light
spanners. Besides the obvious theoretical importance of unification, we
demonstrate the power of our approach in obtaining (1) stronger lightness
bounds, and (2) faster construction times. Our results include:
_ -minor-free graphs: A truly optimal spanner construction and a fast
construction.
_ General graphs: A truly optimal spanner -- almost and a linear-time
construction with near-optimal lightness.
_ Low dimensional Euclidean spaces: We demonstrate that Steiner points help
in reducing the lightness of Euclidean -spanners almost
quadratically for .Comment: We split this paper into two papers: arXiv:2106.15596 and
arXiv:2111.1374
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
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
A Spanner for the Day After
We show how to construct -spanner over a set of
points in that is resilient to a catastrophic failure of nodes.
Specifically, for prescribed parameters , the
computed spanner has edges, where . Furthermore, for any , and
any deleted set of points, the residual graph is -spanner for all the points of except for
of them. No previous constructions, beyond the trivial clique
with edges, were known such that only a tiny additional fraction
(i.e., ) lose their distance preserving connectivity.
Our construction works by first solving the exact problem in one dimension,
and then showing a surprisingly simple and elegant construction in higher
dimensions, that uses the one-dimensional construction in a black box fashion
Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings
Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive
orderings (LSO) for Euclidean space. A -LSO is a collection
of orderings such that for every there is an
ordering , where all the points between and w.r.t.
are in the -neighborhood of either or . In essence, LSO
allow one to reduce problems to the -dimensional line. Later, Filtser and Le
[STOC 2022] developed LSO's for doubling metrics, general metric spaces, and
minor free graphs.
For Euclidean and doubling spaces, the number of orderings in the LSO is
exponential in the dimension, which made them mainly useful for the low
dimensional regime. In this paper, we develop new LSO's for Euclidean,
, and doubling spaces that allow us to trade larger stretch for a much
smaller number of orderings. We then use our new LSO's (as well as the previous
ones) to construct path reporting low hop spanners, fault tolerant spanners,
reliable spanners, and light spanners for different metric spaces.
While many nearest neighbor search (NNS) data structures were constructed for
metric spaces with implicit distance representations (where the distance
between two metric points can be computed using their names, e.g. Euclidean
space), for other spaces almost nothing is known. In this paper we initiate the
study of the labeled NNS problem, where one is allowed to artificially assign
labels (short names) to metric points. We use LSO's to construct efficient
labeled NNS data structures in this model
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