## Abstract

This paper initiates the study of fault resilient network structures that mix two orthogonal protection mechanisms: (a) backup, namely, augmenting the structure with many (redundant) low-cost but fault-prone components, and (b) reinforcement, namely, acquiring high-cost but fault-resistant components. To study the trade-off between these two mechanisms in a concrete setting, we address the problem of designing a (b, r) fault-tolerant BFS (or (b, r) FT-BFS for short) structure, namely, a subgraph H of the network G consisting of two types of edges: a set E' ⊆ E of r(n) fault-resistant reinforcement edges, which are assumed to never fail, and a (larger) set E(H)\E' of b(n) fault-prone backup edges, such that subsequent to the failure of a single fault-prone backup edge e ∈ E \ E', the surviving part of H still contains a BFS spanning tree for (the surviving part of) G, satisfying dist(s, v, H \ {e}) ≤ dist(s,v, G \ {e}) for every v ∈ V and e G E\E'. We establish the following tradeoff: For every real e G (0,1], if r(n) = Θ(n^{1-∈}), then b(n) = ΘT(n^{1+∈}) is necessary and sufficient. More specifically, as shown in [14], for e = 1, FT-BFS structures (with no reinforced edges) require Θ(n^{3/2}) edges, and this is sufficient. At the other extreme, if ∈ = 0, then n - 1 reinforced edges suffice (with no need for backup). Here, we present a polynomial time algorithm that given an undirected graph G = (V, E), a source vertex s and a real ∈ ∈ (0,1], constructs a (b(n), r(n)) FT-BFS with r(n) = O(n^{1-∈}) and b(n) = O(min{1/∈ · n^{1+∈} logn,n^{3/2}}). We complement this result by providing a nearly matching lower bound, showing that there are n-vertex graphs for which any (b(n),r(n)) FT-BFS structure requires Ω(min{n^{1+∈},n^{3/2}}) backup edges when r(n) = Ω(n^{1-∈}) edges are reinforced.

Original language | English |
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Title of host publication | SPAA 2015 - Proceedings of the 27th ACM Symposium on Parallelism in Algorithms and Architectures |

Publisher | Association for Computing Machinery |

Pages | 264-273 |

Number of pages | 10 |

ISBN (Electronic) | 9781450335881 |

DOIs | |

State | Published - 13 Jun 2015 |

Externally published | Yes |

Event | 27th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2015 - Portland, United States Duration: 13 Jun 2015 → 15 Jun 2015 |

### Publication series

Name | Annual ACM Symposium on Parallelism in Algorithms and Architectures |
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Volume | 2015-June |

### Conference

Conference | 27th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2015 |
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Country/Territory | United States |

City | Portland |

Period | 13/06/15 → 15/06/15 |

### Bibliographical note

Publisher Copyright:Copyright © 2015 ACM.

## Keywords

- Fault-tolerance
- Replacement-paths
- Tree-decomposition