## 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 addresses the problem of designing a fault-tolerant (α,β) approximate BFS structure (or FT-ABFS structure for short), namely, a subgraph H of the network G such that subsequent to the failure of some subset F of edges or vertices, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, satisfying dist(s,v,H\F) ≤ α-dist(s,v,G\F)+β for every v ∈ V. We first consider multiplicative (α, 0) FT-ABFS structures resilient to a failure of a single edge or vertex, and present an algorithm that given an n-vertex unweighted undirected graph G and a source s constructs a (3,0) FT-ABFS structure rooted at s with at most 3n edges (improving by an O(logn) factor on the near-tight result of [3]). Assuming at most f edge failures, for constant integer f > 1, we prove that there exists a (poly-time constructible) (3(f + l), (/ + 1) logn) FT-ABFS structure with O(fn) edges. We then consider additive (1, β) FT-ABFS structures. In contrast to the linear size of (α, 0) FT-ABFS structures, we show that for every β ∈ [1,O(logn)] there exists an n-vertex graph G with a source s for which any (1,β) FT-ABFS structure rooted at s has Ω(n^{1+ε(β)}) edges, for some function ε(β) ∈ (0,1). In particular, (1,3) FT-ABFS structures admit a lower bound of Ω(n^{5/4}) edges. These lower bounds demonstrate an interesting dichotomy between multiplicative and additive spanners; whereas (α, 0) FT-ABFS structures of size O(n) exist (for α ≥ 3), their additive counterparts, (1, β) FT-ABFS structures, are of super-linear size. Our lower bounds are complemented by an upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source s constructs a (1,4) FT-ABFS structure rooted at s with at most O(n^{4/3}) edges.

Original language | English |
---|---|

Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery |

Pages | 1073-1092 |

Number of pages | 20 |

ISBN (Print) | 9781611973389 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: 5 Jan 2014 → 7 Jan 2014 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
---|

### Conference

Conference | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
---|---|

Country/Territory | United States |

City | Portland, OR |

Period | 5/01/14 → 7/01/14 |