Worst-Case Optimal and Average-Case Efficient Geometric Ad-Hoc Routing

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

2. s Geometric Routing ??? t s MobiHoc 2003

3. Greedy Routing • Each node forwards message to “best” neighbor t s MobiHoc 2003

4. 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 MobiHoc 2003

5. 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 MobiHoc 2003

6. Related Work in Geometric Routing MobiHoc 2003

7. 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 MobiHoc 2003

8. Face Routing • Based on ideas by [Kranakis, Singh, Urrutia CCCG 1999] • Here simplified (and actually improved) MobiHoc 2003

9. Face Routing • Remark: Planar graph can easily (and locally!) be computed with the Gabriel Graph, for example Planarity is NOT an assumption MobiHoc 2003

10. Face Routing s t MobiHoc 2003

11. Face Routing s t MobiHoc 2003

12. Face Routing s t MobiHoc 2003

13. Face Routing s t MobiHoc 2003

14. Face Routing s t MobiHoc 2003

15. Face Routing s t MobiHoc 2003

16. Face Routing s t MobiHoc 2003

17. “Right Hand Rule” Face Routing Properties • 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 MobiHoc 2003

18. 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 MobiHoc 2003

19. Face Routing • Theorem: Face Routing reaches destination in O(n) steps • But: Can be very bad compared to the optimal route MobiHoc 2003

20. Bounding Searchable Area s t MobiHoc 2003

21. 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(c2) 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] MobiHoc 2003

22. 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 MobiHoc 2003

23. 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 MobiHoc 2003

24. 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 (c2) edges is traversed (c) times  (c3) steps MobiHoc 2003

25. 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?  MobiHoc 2003

26. 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 sparse critical density too dense MobiHoc 2003

27. Critical Density: Shortest Path vs. Euclidean Distance • Shortest path is significantly longer than Euclidean distance • Critical density range mandatory for the simulation of any routing algorithm (not only geometric) MobiHoc 2003

28. Randomly Generated Graphs: Critical Density Range 1.9 1 0.9 1.8 Connectivity 0.8 1.7 0.7 1.6 0.6 Greedy success 1.5 Shortest Path Span Frequency 0.5 1.4 0.4 Shortest Path Span 1.3 0.3 1.2 0.2 1.1 0.1 critical 1 0 10 15 0 5 Network Density [nodes per unit disk] MobiHoc 2003

29. Average-Case Performance: Face vs. Greedy/Face 20 1 worse 18 0.9 16 0.8 14 0.7 12 0.6 Performance 10 0.5 Frequency 8 0.4 Face Routing 6 0.3 better 4 0.2 2 0.1 critical Greedy/FaceRouting 0 0 0 5 10 15 Network Density [nodes per unit disk] MobiHoc 2003

30. Simulation on Randomly Generated Graphs 9 1 GFG/GPSR worse 0.9 8 0.8 7 0.7 6 0.6 Frequency 5 0.5 Performance 0.4 4 GOAFR+ 0.3 3 0.2 better 2 0.1 critical 1 0 0 2 4 6 8 10 12 Network Density [nodes per unit disk] MobiHoc 2003

31. CorrectRouting Worst-CaseOptimal Avg-CaseEfficient ComprehensiveSimulation Conclusion MobiHoc 2003

32. ??? Questions?Comments?Demo?