Fault-tolerant spanners: better and simpler

A natural requirement for many distributed structures is <i>fault-tolerance</i>: after some failures in the underlying network, 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 <i>r</i> for all stretch bounds. For stretch <i>k</i> e 3 we design a simple transformation that converts <i>every k</i>-spanner construction with at most <i>f</i>(<i>n</i>) edges into an <i>r</i>-fault-tolerant <i>k</i>-spanner construction with at most <i>O</i>(<i>r</i>³ log <i>n</i>) Å <i>f</i>(2<i>n/r</i>) edges. Applying this to standard greedy spanner constructions gives <i>r</i>-fault tolerant <i>k</i>-spanners with Õ(<i>r</i>² <i>n</i>^{1+2/<i>k</i>+1}) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on <i>n</i> but <i>exponentially</i> on <i>r</i> (approximately like <i>k^r</i>). For the case of <i>k</i>=2 and unit edge-lengths, an <i>O</i>(<i>r</i> log <i>n</i>)-approximation is known from recent work of Dinitz and Krauthgamer [STOC 2011], in which 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 <i>O</i>(log <i>n</i>), which is, notably, <i>independent</i> of the number of faults <i>r</i>. 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 <i>LOCAL</i> model of distributed computation.