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Competitive Routing on a Bounded-Degree Plane Spanner Prosenjit Bose, Rolf Fagerberg, Andr e van Renssen and Sander Verdonschot Carleton University, University of Southern Denmark August 4, 2012 Sander Verdonschot (Carleton University)

  1. Competitive Routing on a Bounded-Degree Plane Spanner Prosenjit Bose, Rolf Fagerberg, Andr´ e van Renssen and Sander Verdonschot Carleton University, University of Southern Denmark August 4, 2012 Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 1 / 10

  2. Geometric Spanners Given: Goal: Set of points in the plane Approximate the complete Euclidean graph Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 2 / 10

  3. Geometric Spanners Given: Goal: Set of points in the plane Approximate the complete Euclidean graph Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 2 / 10

  4. Geometric Spanners Given: Goal: Set of points in the plane Approximate the complete Euclidean graph shortest path ≤ k · Euclidean distance Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 2 / 10

  5. Competitive Routing Given: Goal: Geometric spanner Find a short path between any two vertices Using only local information path length ≤ r · Euclidean distance Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 3 / 10

  6. Previous Work Half- θ 6 -graph (Bonichon et al. 2010) Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 4 / 10

  7. Previous Work Bounded-degree variants (Bonichon et al. 2010) Half- θ 6 -graph (Bonichon et al. 2010) Competitive routing (Bose et al. 2012) Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 4 / 10

  8. Previous Work Bounded-degree variants (Bonichon et al. 2010) Competitive routing on Half- θ 6 -graph bounded-degree variants (Bonichon et al. 2010) (This result) Competitive routing (Bose et al. 2012) Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 4 / 10

  9. Half- θ 6 -graph 6 Cones around each vertex: 3 positive, 3 negative Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

  10. Half- θ 6 -graph Connect to ‘closest’ vertex in each positive cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

  11. Half- θ 6 -graph Connect to ‘closest’ vertex in each positive cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

  12. Half- θ 6 -graph Connect to ‘closest’ vertex in each positive cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

  13. Bounded Degree Negative cones can have unbounded in-degree. Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  14. Bounded Degree Negative cones can have unbounded in-degree. Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  15. Bounded Degree Consecutive vertices are connected by a canonical path . Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  16. Bounded Degree Keep the edge to the closest vertex... Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  17. Bounded Degree Keep the edge to the closest vertex and the extreme edges. Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  18. Bounded Degree Keep the edge to the closest vertex and the extreme edges. Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  19. Bounded Degree Edges on the canonical path are always extreme. Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  20. Bounded Degree There is an approximation path for every removed edge. Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  21. Bounded Degree Result: A 3-spanner of the half- θ 6 -graph. Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

  22. Routing Algorithm If t lies in a positive cone: In the half- θ 6 -graph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  23. Routing Algorithm If t lies in a positive cone: In the bounded-degree subgraph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  24. Routing Algorithm If t lies in a positive cone: In the bounded-degree subgraph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  25. Routing Algorithm If t lies in a positive cone: In the bounded-degree subgraph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  26. Routing Algorithm If t lies in a positive cone: In the bounded-degree subgraph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  27. Routing Algorithm If t lies in a positive cone: In the bounded-degree subgraph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  28. Routing Algorithm If t lies in a positive cone: In the bounded-degree subgraph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  29. Routing Algorithm If t lies in a positive cone: In the bounded-degree subgraph. Follow the edge in that cone Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

  30. Routing Algorithm If t lies in a negative cone and In the half- θ 6 -graph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  31. Routing Algorithm If t lies in a negative cone and In the half- θ 6 -graph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  32. Routing Algorithm If t lies in a negative cone and In the half- θ 6 -graph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  33. Routing Algorithm If t lies in a negative cone and In the half- θ 6 -graph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  34. Routing Algorithm If t lies in a negative cone and In the half- θ 6 -graph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  35. Routing Algorithm If t lies in a negative cone and In the bounded-degree subgraph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  36. Routing Algorithm If t lies in a negative cone and In the bounded-degree subgraph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  37. Routing Algorithm If t lies in a negative cone and In the bounded-degree subgraph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  38. Routing Algorithm If t lies in a negative cone and In the bounded-degree subgraph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

  39. Routing Algorithm If t lies in a negative cone and In the bounded-degree subgraph. we did not mark a side yet: Follow an edge in that cone Follow an edge to the shorter side Follow an edge to the longer side and mark the shorter side Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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