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Routing in Unit Disk Graphs

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 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.

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Notes

  1. 1.

    Alternatively, a unit disk graph is the intersection graph of a set of disks of radii 1/2.

  2. 2.

    By Lemma 4.2 if t is not a neighbor of s then \(\sigma (v)\) cannot be a neighbor of s, and therefore m must exist.

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Author information

Correspondence to Wolfgang Mulzer.

Additional information

This work is supported in part by GIF project 1161 & DFG projects MU/3501/1. A preliminary version appeared as Haim Kaplan, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Routing in Unit Disk Graphs. Proc. 12th LATIN, 2016.

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Kaplan, H., Mulzer, W., Roditty, L. et al. Routing in Unit Disk Graphs. Algorithmica 80, 830–848 (2018). https://doi.org/10.1007/s00453-017-0308-2

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Keywords

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