LATIN 2016: LATIN 2016: Theoretical Informatics pp 536-548

# Routing in Unit Disk Graphs

• Haim Kaplan
• Wolfgang Mulzer
• Liam Roditty
• Paul Seiferth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

## 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 site r (initially, $$r = s$$), a header h (initially empty), and the target label $$\ell (t)$$, the scheme R may consult the current routing table $$\rho (r)$$ to compute a new site $$r'$$ and a new header $$h'$$, where $$r'$$ is a neighbor of r. The packet is then routed to $$r'$$, and the process 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 of R and the shortest path in $${{\mathrm{UD}}}(S)$$, over all pairs of 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.

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## Authors and Affiliations

• Haim Kaplan
• 1
• Wolfgang Mulzer
• 2
• Liam Roditty
• 3
• Paul Seiferth
• 2
Email author
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