3,987 research outputs found
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)
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
On a family of strong geometric spanners that admit local routing strategies
We introduce a family of directed geometric graphs, denoted \paz, that
depend on two parameters and . For and , the \paz graph is a strong
-spanner, with . The out-degree of a node
in the \paz graph is at most . Moreover, we show that routing can be
achieved locally on \paz. Next, we show that all strong -spanners are also
-spanners of the unit disk graph. Simulations for various values of the
parameters and indicate that for random point sets, the
spanning ratio of \paz is better than the proven theoretical bounds
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