TY - JOUR

T1 - Fault tolerant spanners for general graphs

AU - Chechik, S.

AU - Langberg, Michael

AU - Peleg, D.

AU - Roditty, L.

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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(n1+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 ⊇ V of size at most f and any pair of vertices u, v ∈ V \ F, the distances in H satisfy δH\F(u, v) ≤ k · δG/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(f2kf+1 · n1+1/k log1-1/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·n1+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.

AB - 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(n1+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 ⊇ V of size at most f and any pair of vertices u, v ∈ V \ F, the distances in H satisfy δH\F(u, v) ≤ k · δG/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(f2kf+1 · n1+1/k log1-1/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·n1+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.

KW - Fault tolerance

KW - Graphs

KW - Spanners

UR - http://www.scopus.com/inward/record.url?scp=77956014931&partnerID=8YFLogxK

U2 - 10.1137/090758039

DO - 10.1137/090758039

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AN - SCOPUS:77956014931

SN - 0097-5397

VL - 39

SP - 3403

EP - 3423

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 7

ER -