509 research outputs found
Robust Geometric Spanners
Highly connected and yet sparse graphs (such as expanders or graphs of high
treewidth) are fundamental, widely applicable and extensively studied
combinatorial objects. We initiate the study of such highly connected graphs
that are, in addition, geometric spanners. We define a property of spanners
called robustness. Informally, when one removes a few vertices from a robust
spanner, this harms only a small number of other vertices. We show that robust
spanners must have a superlinear number of edges, even in one dimension. On the
positive side, we give constructions, for any dimension, of robust spanners
with a near-linear number of edges.Comment: 18 pages, 8 figure
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
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
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
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)
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