## Abstract

We consider the Backup Placement problem in networks in the CONGEST distributed setting. Given a network graph G= (V, E) , the goal of each vertex v∈ V is selecting a neighbor, such that the maximum number of vertices in V that select the same vertex is minimized. The backup placement problem was introduced by Halldorsson, Kohler, Patt-Shamir, and Rawitz, who obtained in 2015 an O(log n/ log log n) approximation with randomized polylogarithmic time. Their algorithm remained state-of-the-art for general graphs, as well as for specific graph topologies. In the current paper, we obtain significantly improved algorithms for various graph topologies. Specifically, we show that O(1)-approximation to optimal backup placement can be computed deterministically in O(1) rounds (and even just one round) in wireless networks, certain social networks, claw-free graphs, and, more precisely, in any graph with neighborhood independence bounded by a constant. We also consider graphs such as trees, forests, planar graphs and, more precisely, graphs of constant arboricity. For such graphs, we obtain constant approximation to optimal backup placement in O(log n) deterministic rounds. Clearly, our constant-time algorithms for graphs with constant neighborhood independence are asymptotically optimal. Moreover, we show that our algorithms for graphs with constant arboricity are not far from optimal as well by proving several lower bounds. Specifically, in unoriented trees, optimal backup placement requires Ω (log n) time and polylogarithmic-approximate backup placement requires Ω(logn/loglogn) time. These lower bounds are applicable in particular to graphs of constant arboricity.

Original language | English |
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Article number | 5 |

Pages (from-to) | 455-473 |

Number of pages | 19 |

Journal | Distributed Computing |

Volume | 35 |

Issue number | 5 |

Early online date | 24 Mar 2022 |

DOIs | |

State | Published - Oct 2022 |

### Bibliographical note

Funding Information:This work was supported by the Open University of Israel’s Research Fund and ISF grant 724/15. The authors are grateful to Michael Elkin for very helpful suggestions. The authors would also like to thank the reviewers for all of their careful, constructive, and insightful comments.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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## Keywords

- Backup placement
- Graph theory
- Wireless networks