1. Stefan Rührup 1 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Competitive Time and Traffic Analysis of Position-based Routing using a Cell-Structure Stefan Rührup and Christian Schindelhauer Heinz Nixdorf Institute University of Paderborn Germany IEEE WMAN‘05

2. Stefan Rührup 2 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Outline Part I: Topology control for position-based routing –Position-based routing: greedy forwarding and recovery –Topology issues in position-based routing –Abstracting from graph theory: the cell structure approach Part II: Performance measures and algorithms –Competitive performance measures –Single-path versus multi-path routing strategies

3. Stefan Rührup 3 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Part I Topology Control for Position-based Routing Part I Topology Control for Position-based Routing

4. Stefan Rührup 4 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Position-based routing in a nutshell Given: Source, location of the destination Task: Deliver a message to the destination Assumptions: A node can determine its own position Each node knows the positions of the neighbors The position of the target is known transmission range source target (x,y)

5. Stefan Rührup 5 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Greedy forwarding and recovery (1) With position information one can forward a message in the "right" direction (greedy forwarding) Example: s t no routing tables, no flooding! transmission range progress boundary (circle around the destination)

6. Stefan Rührup 6 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity barrier ? Greedy forwarding and recovery (2) Greedy forwarding is stopped by barriers (local minima) Recovery strategy: Traverse the border of a barrier... until a forwarding progress is possible (right-hand rule) transmission range s t greedy recovery greedy routing time depends on the size of barriers! right-hand rule needs planar topology!

7. Stefan Rührup 7 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity The Cell Structure transmission radius (Unit Disk Graph) v Define a grid consisting of l  l squares

8. Stefan Rührup 8 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity The Cell Structure transmission radius (Unit Disk Graph) v nodes exchange beacon messages  node v knows positions of ist neighbors nodes exchange beacon messages  node v knows positions of ist neighbors

9. Stefan Rührup 9 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity The Cell Structure v node celllink cellbarrier cell each node classifies the cells in ist transmission range

10. Stefan Rührup 10 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity The Cell Structure v node celllink cellbarrier cell each node includes the classification in its beacon messages (only constant overhead) each node includes the classification in its beacon messages (only constant overhead)

11. Stefan Rührup 11 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Routing based on the Cell Structure Routing based on the cell structure uses cell paths cell path = sequence of orthogonally neighboring cells Paths in the original network (here: unit disk graph) and cell paths are equivalent up to a constant factor no planarization strategy needed (required for recovery using the right-hand rule)

12. Stefan Rührup 12 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity node celllink cellbarrier cell Routing based on the Cell Structure v virtual forwarding using cells w physical forwarding from v to w, if visibility range is exceeded

13. Stefan Rührup 13 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Part II Performance Measures and Algorithms Part II Performance Measures and Algorithms

14. Stefan Rührup 14 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Performance Measures barriers make routing difficult what is the worst case scenario? it depends... how difficult is a scenario? what would the best algorithm do?  comparative ratios

15. Stefan Rührup 15 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity How difficult is a scenario? barrier

16. Stefan Rührup 16 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity How difficult is a scenario? perimeter barrier perimeter ( p ) = number of border cells

17. Stefan Rührup 17 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity What would the best algorithm do? length of shortest barrier-free cell path ( h )

18. Stefan Rührup 18 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity competitive ratio: competitive time ratio of a routing algorithm – h = length of shortest barrier-free path –algorithm needs T rounds to deliver a message Competitive Ratio solution of the algorithm optimal offline solution cf. [Borodin, El-Yanif, 1998] h T single-path „ “

19. Stefan Rührup 19 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity optimal (offline) solution for traffic: h messages (length of shortest path) this is unfair, because... –offline algorithm knows the barriers –but every online algorithm has to pay exploration costs exploration costs: sum of perimeters of all barriers ( p ) comparative traffic ratio cf. [Koutsoupias, Papadimitriou 2000] Comparative Ratios M = # messages used h = length of shortest path p = sum of perimeters h+p

20. Stefan Rührup 20 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Comparative Ratios measure for time efficiency: competitive time ratio measure for traffic efficiency: comparative traffic ratio Combined comparative ratio time efficiency and traffic efficiency

21. Stefan Rührup 21 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Algorithms under Comparative Measures Sinlge-path strategies: no parallelism, traffic-efficient (time = traffic) example: GuideLine/Recovery –follow a guide line connecting source and target –traverse all barriers intersecting the guide line Time and Traffic: Multi-path strategies: speed-up by parallel exploration, increasing traffic example: Expanding Ring Search –start flooding with restricted search depth –if target is not in reach then repeat with double search depth Time:Traffic:

22. Stefan Rührup 22 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Algorithms under Comparative Measures GuideLine/Recovery (single-path) Expanding Ring Search (multi-path) traffictime scenario maze open space GuideLine/Recovery (single-path) Expanding Ring Search (multi-path) time ratio traffic ratio combined ratio Is that good? It depends... on the

23. Stefan Rührup 23 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity The Alternating Algorithm... uses a combination of both strategies: 1.i = 1 2.d = 2 i 3.start GuideLine/Recovery with time-to-live = d 3/2 4.if the target is not reached then start Flooding with time-to-live = d 5.if the target is not reached then i = 2 · i goto line 2 Combined comparative ratio:

24. Stefan Rührup 24 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Conclusion cell structure abstracts from graph theoretical issues neighborhood information (= cell classification) causes only constant overhead in beacon messages implicit planarization, well-suited for position-based routing comparative performance measures in relation to the difficulty of the scenario (optimal distance & perimeter of barriers) time and traffic efficiency

25. Stefan Rührup 25 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Thank you for your attention! Questions... Thank you for your attention! Questions... Stefan Rührup sr@upb.de Tel.: +49 5251 60-6722 Fax: +49 5251 60-6482 Algorithms and Complexity Heinz Nixdof Institute University of Paderborn Fürstenallee 11 33102 Paderborn, Germany

26. Stefan Rührup 26 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Learning the neighborhood Does a node have to know his neighborhood? example: –worst case for reactive protocols (at least n messages needed) in the discrete time model beacon-less routing requires continuous time space [Heissenbüttel, Braun, 2003] –proactivity helps! we consider k-hop-proactive protocols (exchange of information over k hops at regular intervals)

27. Stefan Rührup 27 HEINZ NIXDORF INSTITUTE University of Paderborn, Germany Algorithms and Complexity Algorithms under Comparative Mesures GuideLine/Recovery (single-path) Time and Traffic: bad behaviour in a maze, i.e. if Expanding Ring Search (multi-path) Time:Traffic: bad behaviour in open space, i.e. if