1. Worst-Case Optimal and Average-Case Efficient Geometric Ad-Hoc Routing Fabian Kuhn Roger Wattenhofer Aaron Zollinger

2. MobiHoc 20032 Geometric Routing t s s ??????

3. MobiHoc 20033 Greedy Routing Each node forwards message to “best” neighbor t s

4. MobiHoc 20034 Greedy Routing Each node forwards message to “best” neighbor But greedy routing may fail: message may get stuck in a “dead end” Needed: Correct geometric routing algorithm t ? s

5. MobiHoc 20035 What is Geometric Routing? A.k.a. location-based, position-based, geographic, etc. Each node knows its own position and position of neighbors Source knows the position of the destination No routing tables stored in nodes! Geometric routing is important: –GPS/Galileo, local positioning algorithm, overlay P2P network, Geocasting –Most importantly: Learn about general ad-hoc routing

6. MobiHoc 20036 Related Work in Geometric Routing Kleinrock et al.Various 1975ff MFR et al. Geometric Routing proposed Kranakis, Singh, Urrutia CCCG 1999 Face Routing First correct algorithm Bose, Morin, Stojmenovic, Urrutia DialM 1999 GFGFirst average-case efficient algorithm (simulation but no proof) Karp, KungMobiCom 2000 GPSRA new name for GFG Kuhn, Wattenhofer, Zollinger DialM 2002 AFRFirst worst-case analysis. Tight  (c 2 ) bound. Kuhn, Wattenhofer, Zollinger MobiHoc 2003 GOAFRWorst-case optimal and average- case efficient, percolation theory

7. MobiHoc 20037 Overview Introduction –What is Geometric Routing? –Greedy Routing Correct Geometric Routing: Face Routing Efficient Geometric Routing –Adaptively Bound Searchable Area –Lower Bound, Worst-Case Optimality –Average-Case Efficiency –Critical Density –GOAFR Conclusions

8. MobiHoc 20038 Face Routing Based on ideas by [Kranakis, Singh, Urrutia CCCG 1999] Here simplified (and actually improved)

9. MobiHoc 20039 Face Routing Remark: Planar graph can easily (and locally!) be computed with the Gabriel Graph, for example Planarity is NOT an assumption

10. MobiHoc 200310 Face Routing st

11. MobiHoc 200311 Face Routing st

12. MobiHoc 200312 Face Routing st

13. MobiHoc 200313 Face Routing st

14. MobiHoc 200314 Face Routing st

15. MobiHoc 200315 Face Routing st

16. MobiHoc 200316 Face Routing st

17. MobiHoc 200317 All necessary information is stored in the message –Source and destination positions –Point of transition to next face Completely local: –Knowledge about direct neighbors‘ positions sufficient –Faces are implicit Planarity of graph is computed locally (not an assumption) –Computation for instance with Gabriel Graph Face Routing Properties “Right Hand Rule”

18. MobiHoc 200318 Overview Introduction –What is Geometric Routing? –Greedy Routing Correct Geometric Routing: Face Routing Efficient Geometric Routing –Adaptively Bound Searchable Area –Lower Bound, Worst-Case Optimality –Average-Case Efficiency –Critical Density –GOAFR Conclusions

19. MobiHoc 200319 Face Routing Theorem: Face Routing reaches destination in O(n) steps But: Can be very bad compared to the optimal route

20. MobiHoc 200320 Bounding Searchable Area ts

21. MobiHoc 200321 Adaptively Bound Searchable Area What is the correct size of the bounding area? –Start with a small searchable area –Grow area each time you cannot reach the destination –In other words, adapt area size whenever it is too small → Adaptive Face Routing AFR Theorem: AFR Algorithm finds destination after O(c 2 ) steps, where c is the cost of the optimal path from source to destination. Theorem: AFR Algorithm is asymptotically worst-case optimal. [Kuhn, Wattenhofer, Zollinger DIALM 2002]

22. MobiHoc 200322 Overview Introduction –What is Geometric Routing? –Greedy Routing Correct Geometric Routing: Face Routing Efficient Geometric Routing –Adaptively Bound Searchable Area –Lower Bound, Worst-Case Optimality –Average-Case Efficiency –Critical Density –GOAFR Conclusions

23. MobiHoc 200323 GOAFR – Greedy Other Adaptive Face Routing AFR Algorithm is not very efficient (especially in dense graphs) Combine Greedy and (Other Adaptive) Face Routing –Route greedily as long as possible –Overcome “dead ends” by use of face routing –Then route greedily again Similar as GFG/GPSR, but adaptive

24. MobiHoc 200324 Early Fallback to Greedy Routing? We could fall back to greedy routing as soon as we are closer to t than the local minimum But: “Maze” with  (c 2 ) edges is traversed  (c) times   (c 3 ) steps

25. MobiHoc 200325 GOAFR Is Worst-Case Optimal GOAFR traverses complete face boundary: Theorem: GOAFR is asymptotically worst-case optimal. Remark: GFG/GPSR is not –Searchable area not bounded –Immediate fallback to greedy routing GOAFR’s average-case efficiency? 

26. MobiHoc 200326 Average Case Not interesting when graph not dense enough Not interesting when graph is too dense Critical density range (“percolation”) –Shortest path is significantly longer than Euclidean distance too sparsetoo densecritical density

27. MobiHoc 200327 Shortest path is significantly longer than Euclidean distance Critical density range mandatory for the simulation of any routing algorithm (not only geometric) Critical Density: Shortest Path vs. Euclidean Distance

28. MobiHoc 200328 Randomly Generated Graphs: Critical Density Range 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 05 1015 Network Density [nodes per unit disk] Shortest Path Span 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency Greedy success Connectivity Shortest Path Span critical

29. MobiHoc 200329 Average-Case Performance: Face vs. Greedy/Face 0 2 4 6 8 10 12 14 16 18 20 051015 Network Density [nodes per unit disk] Performance 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency better worse Face Routing Greedy/Face Routing critical

30. MobiHoc 200330 Simulation on Randomly Generated Graphs GFG/GPSR GOAFR+ better worse 1 2 3 4 5 6 7 8 9 024681012 Network Density [nodes per unit disk] Performance 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency critical

31. MobiHoc 200331 Conclusion Greedy Routing (MFR) ( ) Face Routing GFG/GPSR AFR GOAFR Correct Routing Avg-Case Efficient Worst-Case Optimal Comprehensive Simulation

32. Questions? Comments? Demo?