840 research outputs found
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph
Data-sensitive metrics adapt distances locally based the density of data
points with the goal of aligning distances and some notion of similarity. In
this paper, we give the first exact algorithm for computing a data-sensitive
metric called the nearest neighbor metric. In fact, we prove the surprising
result that a previously published -approximation is an exact algorithm.
The nearest neighbor metric can be viewed as a special case of a
density-based distance used in machine learning, or it can be seen as an
example of a manifold metric. Previous computational research on such metrics
despaired of computing exact distances on account of the apparent difficulty of
minimizing over all continuous paths between a pair of points. We leverage the
exact computation of the nearest neighbor metric to compute sparse spanners and
persistent homology. We also explore the behavior of the metric built from
point sets drawn from an underlying distribution and consider the more general
case of inputs that are finite collections of path-connected compact sets.
The main results connect several classical theories such as the conformal
change of Riemannian metrics, the theory of positive definite functions of
Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop
novel proof techniques based on the combination of screw functions and
Lipschitz extensions that may be of independent interest.Comment: 15 page
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
Fault-Tolerant Spanners: Better and Simpler
A natural requirement of many distributed structures is fault-tolerance:
after some failures, whatever remains from the structure should still be
effective for whatever remains from the network. In this paper we examine
spanners of general graphs that are tolerant to vertex failures, and
significantly improve their dependence on the number of faults , for all
stretch bounds.
For stretch we design a simple transformation that converts every
-spanner construction with at most edges into an -fault-tolerant
-spanner construction with at most edges.
Applying this to standard greedy spanner constructions gives -fault tolerant
-spanners with edges. The previous
construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends
similarly on but exponentially on (approximately like ).
For the case and unit-length edges, an -approximation
algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010],
where several spanner results are obtained using a common approach of rounding
a natural flow-based linear programming relaxation. Here we use a different
(stronger) LP relaxation and improve the approximation ratio to ,
which is, notably, independent of the number of faults . We further
strengthen this bound in terms of the maximum degree by using the \Lovasz Local
Lemma.
Finally, we show that most of our constructions are inherently local by
designing equivalent distributed algorithms in the LOCAL model of distributed
computation.Comment: 17 page
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
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