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

2. 2 Geometric Routing 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

3. 3 General graph A graph (network) consists of nodes and edges represented as G(V, E, W) a b c d e e 1 (1) e 3 (5) e 4 (2) e 2 (2) e 6 (2) e 5 (2)

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

5. 5 AGENDA Topics: 1. Minimum Disk Covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop Realizability (THP) 4. Exact Solution to Weighted Region Problem (WRP) 5. Raster and Vector based solutions to WRP Conclusion Questions?

6. 6 Topics: 1. Minimum Disk covering Problem (MDC) Minimum Disk covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop realizability (THP) 4. Exact solution to Weighted Region Problem (WRP) 5. Raster and vector based solutions to WRP Conclusion Questions?

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

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

9. 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. 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 CalinescuGruia Calinescu,Ion I. Mandoiu,Peng-Jun WanIon I. MandoiuPeng-Jun Wan Alexander Z. Zelikovsky

11. 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. 12 The method to cover triangle 1

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

14. 14 Covering a triangle cont… A CB T1T1 T2T2 T3T3

15. 15 Covering a triangle cont… 1.Using Skyline of disks 2.cover each of the 3 sides with 2-approximation 3.combine the result to get: 6-approximation for each ∆

16. 16 Desired Property Desired Property P 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. 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. 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. 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. 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. 21 Split the tile into two regions v0v0 v1v1 v2v2 v3v3 v4v4 n = 2m +1 n = 5; m = 2

22. 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. 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. 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. 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 T1T1 T2T2 T3T3

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

27. 27 Conclusion of MDC Conjecture: A unit disk will intersect at least 12 tiles of any covering of R 2 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. 28 Topics: 1. Minimum Disk covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop realizability (THP) 4. Exact solution to Weighted Region Problem (WRP) 5. Raster and vector based solutions to WRP Conclusion Questions?

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

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

31. 31 Desired Property P Again 1.No two discs intersect more than once along their border inside a region Q 2.No Two discs are tangent inside a region Q 3.A disk intersect exactly twice along their border with Q P1 P3 Q

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

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

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

35. 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. 36 Empirical result

37. 37 Distance of one-hop neighbors Extra region

38. 38 Approximation factor

39. 39 Topics: 1. Minimum Disk covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop realizability (THP) 4. Exact solution to Weighted Region Problem (WRP) 5. Raster and vector based solutions to WRP Conclusion Questions?

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

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

42. 42 Topics: 1. Minimum Disk covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop realizability (THP) 4. Exact solution to Weighted Region Problem (WRP) 5. Raster and vector based solutions to WRP Conclusion Questions?

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

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

45. 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. 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. 47 Snell’s Law Cost function Optimal point of incidence

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

49. 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. 50 Critical points CC Critical edges s t

51. 51 Snell points 11 x s t

52. 52 Border points t s

53. 53 Topics: 1. Minimum Disk covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop realizability (THP) 4. Exact solution to Weighted Region Problem (WRP) 5. Raster and vector based solutions to WRP Conclusion Questions?

54. 54 5. WRP - General case  -optimal path  -optimal path from s to t is specified by users path within a factor of (1+  ) from the optimal

55. 55 Transform weighted planar graph to uniform rectangular grid Make a graph with nodes and edges - nodes : raster cells - edges : the possible paths between the nodes Find the optimal path by running Dijkstra ’ s algorithm Raster-based algorithms 8 connected 16 connected 32 connected

56. 56 Raster-based algorithms… Advantages - Simple to implement - Well suited for grid input data - Easy to add other cost criteria Drawbacks - Errors in distance estimate, since we measure grid distance instead of Euclidean distance - Error factor : 4-connectivity:√2 8-connectivity:(√2+1)/5

57. 57 Distortions - bends in raster paths

58. 58 Approximate by a straight line Reduce deviation errors

59. 59 Compare vector vs. Raster Raster1178.6850 secs Straight1130.5665 secs Vector(  =.1) 1128.27< 1 sec

60. 60 Topics: 1. Minimum Disk covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop realizability (THP) 4. Exact solution to Weighted Region Problem (WRP) 5. Raster and vector based solutions to WRP Conclusion Questions?

61. 61 Conclusion Improved approximation to MDC Bell elimination algorithm Two-hop realizability Exact solutions to special cases of WRP Straight optimal raster paths

62. 62 Topics: 1. Minimum Disk covering Problem (MDC) 2. Minimum Forwarding Set Problem (MFS) 3. Two-Hop realizability (THP) 4. Exact solution to Weighted Region Problem (WRP) 5. Raster and vector based solutions to WRP Conclusion Questions?