TY - JOUR

T1 - Fault Tolerant Approximate BFS Structures with Additive Stretch

AU - Parter, Merav

AU - Peleg, David

N1 - Supported in part by the Israel Science Foundation (Grant 894/09), the United States-Israel Binational Science Foundation (Grant 2008348), the I-CORE program of the Israel PBC and ISF (Grant 4/11), the Israel Ministry of Science and Technology (infrastructures grant), and the Citi Foundation.

PY - 2020/12/1

Y1 - 2020/12/1

N2 - This paper addresses the problem of designing a ββ-additive fault-tolerant approximate BFS (or FT-ABFS for short) structure, namely, a subgraph H of the network G such that subsequent to the failure of a single edge e, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, whose distances satisfy dist(s,v,H∖{e})≤dist(s,v,G∖{e})+βdist(s,v,H∖{e})≤dist(s,v,G∖{e})+β for every v∈Vv∈V. It was shown in Parter and Peleg (SODA, 2014), that for every β∈[1,O(logn)]β∈[1,O(logn)] there exists an n-vertex graph G with a source s for which any ββ-additive FT-ABFS structure rooted at s has Ω(n1+ϵ(β))Ω(n1+ϵ(β)) edges, for some function ϵ(β)∈(0,1)ϵ(β)∈(0,1). In particular, 3-additive FT-ABFS structures admit a lower bound of Ω(n5/4)Ω(n5/4) edges. In this paper we present the first upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source s constructs a 4-additive FT-ABFS structure rooted at s with at most O(n4/3)O(n4/3) edges. The main technical contribution of our algorithm is in adapting the path-buying strategy used in Baswana et al. (ACM Trans Algorithms 7:A5, 2010) and Cygan et al. (Proceedings of the 30th symposium on theoretical aspects of computer science, pp 209–220, 2013) to failure-prone settings.

AB - This paper addresses the problem of designing a ββ-additive fault-tolerant approximate BFS (or FT-ABFS for short) structure, namely, a subgraph H of the network G such that subsequent to the failure of a single edge e, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, whose distances satisfy dist(s,v,H∖{e})≤dist(s,v,G∖{e})+βdist(s,v,H∖{e})≤dist(s,v,G∖{e})+β for every v∈Vv∈V. It was shown in Parter and Peleg (SODA, 2014), that for every β∈[1,O(logn)]β∈[1,O(logn)] there exists an n-vertex graph G with a source s for which any ββ-additive FT-ABFS structure rooted at s has Ω(n1+ϵ(β))Ω(n1+ϵ(β)) edges, for some function ϵ(β)∈(0,1)ϵ(β)∈(0,1). In particular, 3-additive FT-ABFS structures admit a lower bound of Ω(n5/4)Ω(n5/4) edges. In this paper we present the first upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source s constructs a 4-additive FT-ABFS structure rooted at s with at most O(n4/3)O(n4/3) edges. The main technical contribution of our algorithm is in adapting the path-buying strategy used in Baswana et al. (ACM Trans Algorithms 7:A5, 2010) and Cygan et al. (Proceedings of the 30th symposium on theoretical aspects of computer science, pp 209–220, 2013) to failure-prone settings.

KW - Additive spanners

KW - Fault tolerance

KW - Replacement paths

UR - http://www.scopus.com/inward/record.url?scp=85088707358&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s00453-020-00734-2

DO - https://doi.org/10.1007/s00453-020-00734-2

M3 - مقالة

SN - 0178-4617

VL - 82

SP - 3458

EP - 3491

JO - Algorithmica

JF - Algorithmica

IS - 12

ER -