## Abstract

Let G=(V,E) be an n-vertices m-edges directed graph with edge weights in the range [1,W] for some parameter W, and sμ V be a designated source. In this article, we address several variants of the problem of maintaining the (1+ϵ)-approximate shortest path from s to each vμ V{s} in the presence of a failure of an edge or a vertex. From the graph theory perspective, we show that G has a subgraph H with Õ(ϵ -1} nlog W) edges such that for any x,vμ V, the graph H \ x contains a path whose length is a (1+ϵ)-approximation of the length of the shortest path from s to v in G \ x. We show that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of ω (ϵ -1 n log W) edges. Demetrescu, Thorup, Chowdhury, and Ramachandran (SICOMP 2008) showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is ω (m). Parter and Peleg (ESA 2013) showed that even in the restricted case of unweighted undirected graphs, the size of any subgraph for the exact shortest path is at least ω (n1.5). Therefore, a (1+ϵ)-approximation is the best one can hope for. We consider also the data structure problem and show that there exists an φ(ϵ -1 n log W) size oracle that for any vμ V reports a (1+ϵ)-approximate distance of v from s on a failure of any xμ V in O(log log 1+ϵ (nW)) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of ω (ϵ -1 nlog W log -1 n). Finally, we present two distributed algorithms. We present a single-source routing scheme that can route on a (1+ϵ)-approximation of the shortest path from a fixed source s to any destination t in the presence of a fault. Each vertex has a label and a routing table of φ(ϵ -1 log W) bits. We present also a labeling scheme that assigns each vertex a label of φ(ϵ -1log W) bits. For any two vertices x,vμ V, the labeling scheme outputs a (1+ϵ)-approximation of the distance from s to v in G \ x using only the labels of x and v.

Original language | English |
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Article number | 3397532 |

Journal | ACM Transactions on Algorithms |

Volume | 16 |

Issue number | 4 |

DOIs | |

State | Published - Sep 2020 |

### Bibliographical note

Publisher Copyright:© 2020 ACM.

### Funding

A preliminary version of this article appeared in the Proceedings of SODA 2018. This work was partially supported by Israel Science Foundation (ISF) and University Grants Commission (UGC) of India. Authors’ addresses: S. Baswana, Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India; email: [email protected]; K. Choudhary, Department of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel; email: [email protected]; M. Hussain, Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India; email: [email protected]; L. Roditty, Department of Computer Science, Bar Ilan University, Ramat-Gan 52900, Israel; email: [email protected]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. © 2020 Association for Computing Machinery. 1549-6325/2020/07-ART44 $15.00 https://doi.org/10.1145/3397532

Funders | Funder number |
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University Grants Commission | |

Israel Science Foundation |

## Keywords

- Fault-tolerant
- approximate-distances
- oracle
- routing
- subgraph