## Abstract

Let S⊂ R^{2} be a set of n sites. The unit disk graph UD (S) on S has vertex set S and an edge between two distinct sites s, t∈ S if and only if s and t have Euclidean distance | st| ≤ 1. A routing scheme R for UD (S) assigns to each site s∈ S a labelℓ(s) and a routing tableρ(s). For any two sites s, t∈ S, the scheme R must be able to route a packet from s to t in the following way: given a current siter (initially, r= s), a headerh (initially empty), and the labelℓ(t) of the target, the scheme R consults the routing table ρ(r) to compute a neighbor r^{′} of r, a new header h^{′}, and the label ℓ(t^{′}) of an intermediate target t^{′}. (The label of the original target may be stored at the header h^{′}.) The packet is then routed to r^{′}, and the procedure is repeated until the packet reaches t. The resulting sequence of sites is called the routing path. The stretch of R is the maximum ratio of the (Euclidean) length of the routing path produced by R and the shortest path in UD (S) , over all pairs of distinct sites in S. For any given ε> 0 , we show how to construct a routing scheme for UD (S) with stretch 1 + ε using labels of O(log n) bits and routing tables of O(ε^{- 5}log ^{2}nlog ^{2}D) bits, where D is the (Euclidean) diameter of UD (S). The header size is O(log nlog D) bits.

Original language | English |
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Pages (from-to) | 830-848 |

Number of pages | 19 |

Journal | Algorithmica |

Volume | 80 |

Issue number | 3 |

DOIs | |

State | Published - 1 Mar 2018 |

### Bibliographical note

Publisher Copyright:© 2017, Springer Science+Business Media New York.

## Keywords

- Routing scheme
- Unit disk graph
- Well-separated pair decomposition