Fault Tolerant Spanners for General Graphs

• Authors: S. Chechik;M. Langberg;D. Peleg;L. Roditty
• Venue: SIAM Journal on Computing
• Year: 2010

This paper concerns graph spanners that are resistant to vertex or edge failures. In the failure-free setting, it is known how to efficiently construct a $(2k-1)$-spanner of size $O(n^{1+1/k})$, and this size-stretch trade-off is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [C. Levcopoulos, G. Narasimhan, and M. Smid, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 186-195]. A subgraph $H$ is an $f$-vertex fault tolerant $k$-spanner of the graph $G$ if for any set $F\subseteq V$ of size at most $f$ and any pair of vertices $u,v\in V\setminus F$, the distances in $H$ satisfy $\delta_{H\setminus F}(u,v)\leq k\cdot\delta_{G\setminus F}(u,v)$. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [A. Czumaj and H. Zhao, Discrete Comput. Geom., 32 (2004), pp. 207-230]. This paper also raised as an open problem the question of whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an $f$-vertex fault tolerant $(2k-1)$-spanner of size $O(f^{2}k^{f+1}\cdot n^{1+1/k}\log^{1-1/k}n)$. Interestingly, the stretch of the spanner remains unchanged, while the size of the spanner increases only by a factor that depends on the stretch $k$, on the number of potential faults $f$, and on logarithmic terms in $n$. In addition, we consider the simpler setting of $f$-edge fault tolerant spanners (defined analogously). We present an $f$-edge fault tolerant $(2k-1)$-spanner with edge set of size $O(f\cdot n^{1+1/k})$ (only $f$ times larger than standard spanners). For both edge and vertex faults, our results are shown to hold when the given graph $G$ is weighted.