1. Geometric Ad-Hoc Routing: Of Theory and Practice Fabian Kuhn Roger Wattenhofer Yan Zhang Aaron Zollinger

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

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

4. PODC 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. PODC 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. PODC 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 Kuhn, Wattenhofer, Zhang, Zollinger PODC 2003 GOAFR+Improved GOAFR for average case, analysis of cost metrics

7. PODC 20037 Overview Introduction –What is Geometric Routing? –Greedy Routing Correct Geometric Routing: Face Routing Efficient Geometric Routing –Worst-Case Optimality: Adaptively Bound Searchable Area –Average-Case Efficiency: GOAFR+ Analysis of Cost Metrics –Linearly Bounded vs. Super-Linear Cost Metrics Conclusions

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

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

10. PODC 200310 Face Routing st

11. PODC 200311 Face Routing st

12. PODC 200312 Face Routing st

13. PODC 200313 Face Routing st

14. PODC 200314 Face Routing st

15. PODC 200315 Face Routing st

16. PODC 200316 Face Routing st

17. PODC 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. PODC 200318 Overview Introduction –What is Geometric Routing? –Greedy Routing Correct Geometric Routing: Face Routing Efficient Geometric Routing –Worst-Case Optimality: Adaptively Bound Searchable Area –Average-Case Efficiency: GOAFR+ Analysis of Cost Metrics –Linearly Bounded vs. Super-Linear Cost Metrics Conclusions

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

20. PODC 200320 Bounding Searchable Area ts

21. PODC 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 an optimal path from source to destination. Theorem: AFR algorithm is asymptotically worst-case optimal. [Kuhn, Wattenhofer, Zollinger DIALM 2002]

22. PODC 200322 Overview Introduction –What is Geometric Routing? –Greedy Routing Correct Geometric Routing: Face Routing Efficient Geometric Routing –Worst-Case Optimality: Adaptively Bound Searchable Area –Average-Case Efficiency: GOAFR+ Analysis of Cost Metrics –Linearly Bounded vs. Super-Linear Cost Metrics Conclusions

23. PODC 200323 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 GOAFR+ – Greedy Other Adaptive Face Routing Counters p: closer to t than u Counters q: farther from t than u Fall back to greedy routing if p >  q

24. PODC 200324 GOAFR+ –Early fallback technique with counters –Bounding searchable area with circle centered at t 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? GOAFR+ Is Worst-Case Optimal 

25. PODC 200325 Simulation on Randomly Generated Graphs GFG/GPSR 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 GOAFR GOAFR+

26. PODC 200326 Overview Introduction –What is Geometric Routing? –Greedy Routing Correct Geometric Routing: Face Routing Efficient Geometric Routing –Worst-Case Optimality: Adaptively Bound Searchable Area –Average-Case Efficiency: GOAFR+ Analysis of Cost Metrics –Linearly Bounded vs. Super-Linear Cost Metrics Conclusions

27. PODC 200327 Super-Linear Cost Metrics Energy metric c(d) = d 2 Linearly Bounded Cost Metrics Link/hop metric c(d) ´ 1 Euclidean metric c(d) = d Dropping  (1)-model / civilized graphs Cost metric: nondecreasing function c: ]0,1]  R + Analysis of Cost Metrics c(d) d d c(d) ¸ m ¢ d

28. PODC 200328 Super-linear cost metrics No geometric routing algorithm can perform competitively Linearly bounded cost metrics Backbone graph constructible for general Unit Disk Graphs All linearly bounded cost metrics asymptotically equivalent Asymptotically optimal geometric routing Linearly Bounded vs. Super-Linear Cost Metrics

29. PODC 200329 Conclusion “Geometric Ad-Hoc Routing: Of Theory and Practice“ Average-case efficiency Drop assumption on distance between nodes Asymptotic worst-case optimality Analysis of cost metrics GOAFR+  (1)-model

30. Questions? Comments? Demo?