University of Denver Department of Mathematics Department of Computer Science

1. University of Denver Department of Mathematics Department of Computer Science

2. Applications Ad hoc Wireless networks Robot Route Planning in a terrain of varied types (ex: grassland, brush land, forest, water etc.) Geometric graphs Planar graph Unit disk graph Geometric Routing

3. General graph • A graph (network) consists of nodes and edges represented as G(V, E, W) e3(5) b a e4(2) e6(2) e1(1) e e5(2) d c e2(2)

4. b a e d c Planar Graphs • A Planar graph is a graph that can be drawn in the plane such that edges do not intersect Examples: Voronoi diagram and Delaunay triangulation

5. Topics: Minimum Disk Covering Problem (MDC) Minimum Forwarding Set Problem (MFS) Two-Hop Realizability (THP) Exact Solution to Weighted Region Problem (WRP) Raster and Vector based solutions to WRP Conclusion Questions? AGENDA

6. Topics: Minimum Disk covering Problem (MDC) Minimum Forwarding Set Problem (MFS) Two-Hop realizability (THP) Exact solution to Weighted Region Problem (WRP) Raster and vector based solutions to WRP Conclusion Questions?

7. 1 1 . Minimum Disk Covering Problem (MDC) Cover Blue points with unit disks centered at Red points !! Use Minimum red disks!!

8. 1 Other Variation Cover all Blues with unit disks centered at blue points !! Using Minimum Number of disks

9. Complexity • MDC is known to be NP-complete • Reference “Unit Disk Graphs”Discrete Mathematics 86 (1990) 165–177, B.N. Clark, C.J. Colbourn and D.S. Johnson.

10. Previous work (Cont…) A 108-approximation factor algorithm for MDC is known “Selecting Forwarding Neighbors in Wireless Ad-Hoc Networks” Jrnl: Mobile Networks and Applications(2004) Gruia Calinescu ,Ion I. Mandoiu ,Peng-Jun Wan Alexander Z. Zelikovsky

11. Previous method • Tile the plane with equilateral triangles of unit side • Cover Each triangle by solving a Linear program (LP) • Round the solution to LP to obtain a factor of 6 for each triangle

12. 1 The method to cover triangle

13. Covering a triangle IF No blue points in a triangle- NOTHING TO DO!! IF∆ contains RED + BLUETHEN Unit disk centered at RED Covers the ∆ Assume BLUE + RED do not share a ∆

14. Covering a triangle cont… A T1 T3 B C T2

15. Covering a triangle cont… • Using Skyline of disks • cover each of the 3 sides with 2-approximation • combine the result to get: • 6-approximation for each ∆

16. Desired PropertyP • Skyline gives an approximation factor of 2 • No two discs intersect more than once inside a triangle • No Two discs are tangent inside the triangle

17. Unit disk intersects at most 18 triangles It can be easily verified that a Unit disk intersects at most 18 equilateral triangles in a tiling of a plane

18. Result 108-approximation • Covered each triangle with approximation factor of 6 • Optimal cover can intersect at most 18 triangles • Hence, 6 *18 = 108 - approximation

19. Improvements CAN WE • use a larger tile? • split the tile into two regions? • get better than 6-approximation by different tiling? • cover the plane instead of tiling?

20. Can we use a larger tile? • If tile is larger than a unit diameter !! • Unit disc inside Tile cannot cover the tile • Hence we cannot use previous method

21. Split the tile into two regions v0 n = 2m +1 n = 5; m = 2 v1 v4 v3 v2

22. Different shape Tile? • Each side with 2-approx. factor • Hence 8 for a square • Unit disk can intersect 14 such squares • 14 * 8 =112 • No Gain by such method

23. Different shape Tile? • Each side with 2-approx. factor • Hence 12 for a hexagon • Unit disk can intersect 12 such hexagons • 12 * 12 =144 • No Gain by such method

24. Our Approach • How about using a unit diameter hexagon as a tile • Split the tile into 3 regions around the hexagon • Does this give a better bound?

25. Hexagon- split it into 3 regions • Partition Hexagon into 3 regions (Similar to triangle) • Obtain 2-approximation for each side 6-approximation for hexagon • Unit disk intersects 12 hexagons • Hence, 6 * 12 = 72-approximation T1 T3 T2

26. Covering • Instead of tiling the plane, how about covering the plane?

27. Conclusion of MDC • Conjecture: A unit disk will intersect at least 12 tiles of any covering of R2 by unit diameter tiles • Each tile has an approximation of 6 by the known method • Cannot do better than 72 by the method used

28. Topics: Minimum Disk covering Problem (MDC) Minimum Forwarding Set Problem (MFS) Two-Hop realizability (THP) Exact solution to Weighted Region Problem (WRP) Raster and vector based solutions to WRP Conclusion Questions?

29. 2. Minimum Forwarding Set Problem (MFS) s ONE-HOP REGION A Cover blue points with unit disks centered at red points, now all red points are inside a unit disk

30. Previous work (MFS) • Despite its simplicity, complexity is unknown • 3- and 6-approximation algorithms known • Algorithm is based on property P

31. Desired Property P Again • No two discs intersect more than once along their border inside a region Q • No Two discs are tangent inside a region Q • A disk intersect exactly twice along their border with Q P1 Q P3

32. PropertyP • Property P applies if the region is outside of disk radius Unit disk A s Q

33. Redundant points • Remove redundant points Redundant point x y s

34. Bell and Cover of node x • Remove points inside the Bell- Bell Elimination Algorithm (BEA)

35. Analysis • Assume points to be uniformly distributed • BEA eliminates all the points inside the disk of radius • Need about 75 points • Therefore exact solution

36. Empirical result

37. Distance of one-hop neighbors • Extra region

38. Approximation factor

39. Topics: Minimum Disk covering Problem (MDC) Minimum Forwarding Set Problem (MFS) Two-Hop realizability (THP) Exact solution to Weighted Region Problem (WRP) Raster and vector based solutions to WRP Conclusion Questions?

40. b a c a b c s Degree of at most 2 • Two-hop to bipartite graph 2 1 3 1 2 3 4 4

41. a b d c 2 5 3 . Two-hop realizability • Result:A bipartite graph having a degree of at most 2 is two-hop realizable two-hop neighbors one-hop neighbors 1 3 4

42. Topics: Minimum Disk covering Problem (MDC) Minimum Forwarding Set Problem (MFS) Two-Hop realizability (THP) Exact solution to Weighted Region Problem (WRP) Raster and vector based solutions to WRP Conclusion Questions?

43. Objective - Find an optimal path from START to GOAL Complexity of WRP is unknown 4. Weighted region problem (WRP)

44. Planar Graphs • Planar sub-division considered as planar graph

45. Shortest path G(V, E, W) • Dijkstra algorithm finds a shortest path from a source vertex to all other vertices • Running time O(|V| log |V|+ |E|) • Linear time for planar graphs

46. WRP - General case Notations f = weight of face f e = weight of edge e, where e = f  f’ ≤ min {f, f’} A weight of  implies A path cannot cross that face or edge Note that all optimal paths must be piecewise linear!!

47. Snell’s Law Cost function Optimal point of incidence

48. Weight  w R w v v R 0/1/ Special case WRP • Construct a critical graph G • Run Dijkstra on G Weight 0

49. Convex Polygon C • Exact path when s in C and t is arbitrary • Construct “Exact Weighted” Graph • Add edges that contribute to exact path • Run Dijkstra shortest path Algorithm

50. Critical edges C Critical points s t