Abstract

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose length is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per vertex coordinate via Schnyder's embedding scheme [W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st Annual ACM--SIAM Symposium on Discrete Algorithms (SODA 1990), ACM, New York, SIAM, Philadelphia, 1990, pp. 138--148], our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of $15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the half-$\theta_6$-graph.

Keywords

  1. online routing
  2. local routing
  3. competitive routing
  4. geometric spanner
  5. theta-graph

MSC codes

  1. 52C99
  2. 68M10
  3. 68R10
  4. 68W27
  5. 90B18

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Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 1626 - 1649
ISSN (online): 1095-7111

History

Submitted: 23 September 2014
Accepted: 14 August 2015
Published online: 3 November 2015

Keywords

  1. online routing
  2. local routing
  3. competitive routing
  4. geometric spanner
  5. theta-graph

MSC codes

  1. 52C99
  2. 68M10
  3. 68R10
  4. 68W27
  5. 90B18

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