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 $r$, for all
stretch bounds.
For stretch $k \geq 3$ we design a simple transformation that converts every
$k$-spanner construction with at most $f(n)$ edges into an $r$-fault-tolerant
$k$-spanner construction with at most $O(r^3 \log n) \cdot f(2n/r)$ edges.
Applying this to standard greedy spanner constructions gives $r$-fault tolerant
$k$-spanners with $\tilde O(r^{2} n^{1+\frac{2}{k+1}})$ edges. The previous
construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends
similarly on $n$ but exponentially on $r$ (approximately like $k^r$).
For the case $k=2$ and unit-length edges, an $O(r \log n)$-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 $O(\log n)$,
which is, notably, independent of the number of faults $r$. 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