Abstract
A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper considers breadth-first search (BFS) spanning trees, and addresses the problem of designing a sparse fault-tolerant BFS tree, or FT-BFS tree for short, namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T′ of T still contains a BFS spanning tree for (the surviving part of) G. For a source node s, a target node t and an edge e ∈ G, the shortest s − t path P s,t,e that does not go through e is known as a replacement path. Thus, our FT-BFS tree contains the collection of all replacement paths P s,t,e for every t ∈ V(G) and every failed edge e ∈ E(G). Our main results are as follows. We present an algorithm that for every n-vertex graph G and source node s constructs a (single edge failure) FT-BFS tree rooted at s with \(O(n \cdot \min\{{\tt Depth}(s), \sqrt{n}\})\) edges, where Depth(s) is the depth of the BFS tree rooted at s. This result is complemented by a matching lower bound, showing that there exist n-vertex graphs with a source node s for which any edge (or vertex) FT-BFS tree rooted at s has Ω(n 3/2) edges. We then consider fault-tolerant multi-source BFS trees, or FT-MBFS trees for short, aiming to provide (following a failure) a BFS tree rooted at each source s ∈ S for some subset of sources S ⊆ V. Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every n-vertex graph and source set S ⊆ V of size σ constructs a (single failure) FT-MBFS tree T *(S) from each source s i ∈ S, with \(O(\sqrt{\sigma} \cdot n^{3/2})\) edges, and on the other hand there exist n-vertex graphs with source sets S ⊆ V of cardinality σ, on which any FT-MBFS tree from S has \(\Omega(\sqrt{\sigma}\cdot n^{3/2})\) edges. Finally, we propose an O(logn) approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no Ω(logn) approximation algorithm for these problems under standard complexity assumptions. In comparison with previous constructions our algorithm is deterministic and may improve the number of edges by a factor of up to \(\sqrt{n}\) for some instances. All our algorithms can be extended to deal with one vertex failure as well, with the same performance.
Supported in part by the Israel Science Foundation (grant 894/09), the I-CORE program of the Israel PBC and ISF (grant 4/11), the United States-Israel Binational Science Foundation (grant 2008348), the Israel Ministry of Science and Technology (infrastructures grant), and the Citi Foundation.
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References
Awerbuch, B., Bar-Noy, A., Linial, N., Peleg, D.: Compact distributed data structures for adaptive network routing. In: STOC, pp. 230–240 (1989)
Abraham, I., Chechik, S., Gavoille, C., Peleg, D.: Forbidden-Set Distance Labels for Graphs of Bounded Doubling Dimension. In: PODC, pp. 192–200 (2010)
Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in expected O(n 2) time. ACM Trans. Algorithms 2(4), 557–577 (2006)
Bernstein, A., Karger, D.: A nearly optimal oracle for avoiding failed vertices and edges. In: STOC, pp. 101–110 (2009)
Chechik, S., Langberg, M., Peleg, D., Roditty, L.: f-sensitivity distance oracles and routing schemes. Algorithmica, 861–882 (2012)
Chechik, S., Langberg, M., Peleg, D., Roditty, L.: Fault-tolerant spanners for general graphs. In: STOC, pp. 435–444 (2009)
Chechik, S.: Fault-Tolerant Compact Routing Schemes for General Graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 101–112. Springer, Heidelberg (2011)
Czumaj, A., Zhao, H.: Fault-tolerant geometric spanners. Discrete & Computational Geometry 32 (2003)
Demetrescu, C., Thorup, M., Chowdhury, R., Ramachandran, V.: Oracles for distances avoiding a failed node or link. SIAM J. Computing 37, 1299–1318 (2008)
Dinitz, M., Krauthgamer, R.: Fault-tolerant spanners: better and simpler. In: PODC, pp. 169–178 (2011)
Duan, R., Pettie, S.: Dual-failure distance and connectivity oracles. In: SODA (2009)
Feige, U.: A Threshold of ln n for Approximating Set Cover. J. ACM, 634–652 (1998)
Grandoni, F., Williams, V.V.: Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths. In: FOCS (2012)
Hershberger, J., Subhash, S.: Vickrey prices and shortest paths: What is an edge worth? In: FOCS (2001)
Hershberger, J., Subhash, S., Bhosle, A.: On the difficulty of some shortest path problems. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 343–354. Springer, Heidelberg (2003)
Levcopoulos, C., Narasimhan, G., Smid, M.: Efficient algorithms for constructing fault-tolerant geometric spanners. In: STOC, pp. 186–195 (1998)
Lukovszki, T.: New results on fault tolerant geometric spanners. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 193–204. Springer, Heidelberg (1999)
Parter, P., Peleg, D.: Sparse Fault-Tolerant BFS Trees (2013), http://arxiv.org/abs/1302.5401
Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM (2000)
Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13, 99–116 (1989)
Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Computing 18(2), 740–747 (1989)
Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. J. ACM 36, 510–530 (1989)
Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 261–272. Springer, Heidelberg (2005)
Roditty, L., Zwick, U.: Replacement paths and k simple shortest paths in unweighted directed graphs. ACM Trans. Algorithms (2012)
Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA, pp. 1–10 (2001)
Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52, 1–24 (2005)
Vazirani, V.: Approximation Algorithms. Georgia Inst. Tech. (1997)
Weimann, O., Yuster, R.: Replacement paths via fast matrix multiplication. In: FOCS (2010)
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Parter, M., Peleg, D. (2013). Sparse Fault-Tolerant BFS Trees. In: Bodlaender, H.L., Italiano, G.F. (eds) Algorithms – ESA 2013. ESA 2013. Lecture Notes in Computer Science, vol 8125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40450-4_66
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