Algorithmica

pp 1–19

# Routing in Unit Disk Graphs

• Haim Kaplan
• Wolfgang Mulzer
• Liam Roditty
• Paul Seiferth
Article
• 25 Downloads
Part of the following topical collections:
1. Special Issue on Theoretical Informatics

## Abstract

Let $$S \subset \mathbb {R}^2$$ be a set of n sites. The unit disk graph $${{\mathrm{UD}}}(S)$$ on S has vertex set S and an edge between two distinct sites $$s,t \in S$$ if and only if s and t have Euclidean distance $$|st| \le 1$$. A routing scheme R for $${{\mathrm{UD}}}(S)$$ assigns to each site $$s \in S$$ a label$$\ell (s)$$ and a routing table$$\rho (s)$$. For any two sites $$s, t \in S$$, 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 label$$\ell (t)$$ of the target, the scheme R consults the routing table $$\rho (r)$$ to compute a neighbor $$r'$$ of r, a new header $$h'$$, and the label $$\ell (t')$$ 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 $${{\mathrm{UD}}}(S)$$, over all pairs of distinct sites in S. For any given $$\varepsilon > 0$$, we show how to construct a routing scheme for $${{\mathrm{UD}}}(S)$$ with stretch $$1+\varepsilon$$ using labels of $$O(\log n)$$ bits and routing tables of $$O(\varepsilon ^{-5}\log ^2 n \log ^2 D)$$ bits, where D is the (Euclidean) diameter of $${{\mathrm{UD}}}(S)$$. The header size is $$O(\log n \log D)$$ bits.

### Keywords

Routing scheme Unit disk graph Well-separated pair decomposition

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## Copyright information

© Springer Science+Business Media New York 2017

## Authors and Affiliations

• Haim Kaplan
• 1
• Wolfgang Mulzer
• 2
• Liam Roditty
• 3
• Paul Seiferth
• 2
1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
3. 3.Department of Computer ScienceBar Ilan UniversityRamat GanIsrael