Competitive Routing on a Bounded-Degree Plane Spanner Prosenjit - - PowerPoint PPT Presentation

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Competitive Routing on a Bounded-Degree Plane Spanner Prosenjit - - PowerPoint PPT Presentation

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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)

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Competitive Routing

  • n 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

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SLIDE 2

Geometric Spanners

Given: Set of points in the plane Goal: Approximate the complete Euclidean graph

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 2 / 10

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Geometric Spanners

Given: Set of points in the plane Goal: Approximate the complete Euclidean graph

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 2 / 10

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SLIDE 4

Geometric Spanners

Given: Set of points in the plane Goal: Approximate the complete Euclidean graph shortest path ≤ k · Euclidean distance

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 2 / 10

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Competitive Routing

Given: Geometric spanner Using only local information Goal: Find a short path between any two vertices path length ≤ r · Euclidean distance

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 3 / 10

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SLIDE 6

Previous Work

Half-θ6-graph (Bonichon et al. 2010)

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 4 / 10

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SLIDE 7

Previous Work

Half-θ6-graph (Bonichon et al. 2010) Bounded-degree variants (Bonichon et al. 2010) Competitive routing (Bose et al. 2012)

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 4 / 10

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SLIDE 8

Previous Work

Half-θ6-graph (Bonichon et al. 2010) Bounded-degree variants (Bonichon et al. 2010) Competitive routing (Bose et al. 2012) Competitive routing on bounded-degree variants (This result)

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 4 / 10

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SLIDE 9

Half-θ6-graph

6 Cones around each vertex: 3 positive, 3 negative

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

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Half-θ6-graph

Connect to ‘closest’ vertex in each positive cone

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

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SLIDE 11

Half-θ6-graph

Connect to ‘closest’ vertex in each positive cone

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

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Half-θ6-graph

Connect to ‘closest’ vertex in each positive cone

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 5 / 10

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Bounded Degree

Negative cones can have unbounded in-degree.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

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Bounded Degree

Negative cones can have unbounded in-degree.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

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SLIDE 15

Bounded Degree

Consecutive vertices are connected by a canonical path.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

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SLIDE 16

Bounded Degree

Keep the edge to the closest vertex...

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

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SLIDE 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

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SLIDE 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

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SLIDE 19

Bounded Degree

Edges on the canonical path are always extreme.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

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Bounded Degree

There is an approximation path for every removed edge.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

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Bounded Degree

Result: A 3-spanner of the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 6 / 10

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Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 23

Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 24

Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 25

Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 26

Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 27

Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 28

Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 29

Routing Algorithm

If t lies in a positive cone: Follow the edge in that cone In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 7 / 10

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SLIDE 30

Routing Algorithm

If t lies in a negative cone and 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 In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 31

Routing Algorithm

If t lies in a negative cone and 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 In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 32

Routing Algorithm

If t lies in a negative cone and 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 In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 33

Routing Algorithm

If t lies in a negative cone and 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 In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 34

Routing Algorithm

If t lies in a negative cone and 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 In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 35

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 36

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 37

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 38

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 39

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 41

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 42

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 43

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 44

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 45

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 46

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 47

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 48

Routing Algorithm

If t lies in a negative cone and 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 In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 8 / 10

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SLIDE 49

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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SLIDE 50

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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SLIDE 51

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the half-θ6-graph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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SLIDE 52

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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SLIDE 53

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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SLIDE 54

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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SLIDE 55

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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SLIDE 56

Routing Algorithm

If t lies in a negative cone and we marked a side: Follow the edge closest to the marked side In the bounded-degree subgraph.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 9 / 10

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Conclusion

Bounded-degree spanners allow competitive routing.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 10 / 10

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Conclusion

Bounded-degree spanners allow competitive routing. Routing ratio can be improved by storing information at vertices.

Sander Verdonschot (Carleton University) Competitive Bounded-Degree Routing August 4, 2012 10 / 10