1 Oblivious Routing in Wireless networks Costas Busch Rensselaer Polytechnic Institute Joint work with: Malik Magdon-Ismail and Jing Xi

2 Outline of Presentation Introduction Network Model Oblivious Algorithm Discussion Analysis

3 Routing: choose paths from sources to destinations

4 Edge congestion maximum number of paths that use any edge Node congestion maximum number of paths that use any node

5 Length of chosen path Length of shortest path Stretch= shortest path chosen path

6 Oblivious Routing Each packet path choice is independent of other packet path choices

7 Path choices: Probability of choosing a path:

8 Benefits of oblivious routing: Appropriate for dynamic packet arrivals Distributed Needs no global coordination

9 Related Work Valiant [SICOMP’82]: First oblivious routing algorithms for permutations on butterfly and hypercube butterflybutterfly (reversed)

10 d-dimensional Grid: Lower bound for oblivious routing: Maggs, Meyer auf der Heide, Voecking, Westermann [FOCS’97]:

11 Azar et al. [STOC03] Harrelson et al. [SPAA03] Bienkowski et al. [SPAA03] Arbitrary Graphs: constructive Racke [FOCS’02]: existential result

12 Hierarchical clustering Approach:

13

14 At the lowest level every node is a cluster

15 source destination

16 Pick random node

17 Pick random node

18 Pick random node

19 Pick random node

20 Pick random node

21 Pick random node

22 Pick random node

23

24 Adjacent nodes may follow long paths Big stretch Problem:

25 An Impossibility Result Stretch and congestion cannot be minimized simultaneously in arbitrary graphs

26 Each path has length paths Length 1 Source of packets Destination of all packets Example graph: nodes

27 packets in one path Stretch = Edge congestion =

28 1 packet per path Stretch = Edge congestion =

29 Contribution Oblivious algorithm for special graphs embedded in the 2-dimensional plane Constant stretch Small congestion degree Busch, Magdon-Ismail, Xi [SPAA 2005]:

30 Embeddings in wide, closed-curved areas

31 Our algorithm is appropriate for various wireless network topologies Transmission radius

32 Basic Idea source destination

33 Pick a random intermediate node

34 Construct path through intermediate node

35 Stretch = Previous results for Grids: Busch, Magdon-Ismail, Xi [IPDPS’05] For d=2, a similar result given by C. Scheideler

36 Outline of Presentation Introduction Network Model Oblivious Algorithm Discussion Analysis

37 Network Surrounding area

38 space point space point Perpendicular bisector geodesic

39 space point space point

40 Area wideness:

41 space point graph node Coverage Radius : maximum distance from a space point to the closest node

42 there exist For all pair of nodes Shortest path length: Euclidian distance:

43 Consequences of (max transmission radius in wireless networks) edge Max Euclidian distance between adjacent nodes

44 Consequences of nodes Min Euclidian Distance between any pair of nodes:

45 Small and large Good Network embeddings: Suppose they are constants

46 Outline of Presentation Introduction Network Model Oblivious Algorithm Discussion Analysis

47 Every pair of nodes is assigned a default path default path Examples: Shortest paths Geographic routing paths (GPSR)

48 The algorithm source destination

49 geodesic Perpendicular bisector

50 Pick random space point

51 Find closest node to point

52 default path default path Connect intermediate node to source and destination

53 Outline of Presentation Introduction Network Model Oblivious Algorithm Discussion Analysis

54 Consider an arbitrary set of packets: Suppose the oblivious algorithm gives paths:

55 We will show: optimal congestion

56 Theorem: Proof: Consider an arbitrary path and show that:

57 default path default path shortest path

58 we show this is constant when default paths are shortest paths

59 Default path (shortest) Similarly:

60 shortest path

61 For constants: End of Proof

62 Theorem: Proof: Consider some arbitrary node and estimate congestion on Expected case: denotes

63 Deviation of default paths: maximum distance from geodesic geodesic

64 Consider some path from to

65 the use of depends on the choice of space point one choice

66 another choice

67 cone affecting If you choose node in the cone the respective path may use

68 cone affecting If you choose node outside the cone the respective path does not use

69 cone affecting Segment of space points affecting

70 Probability of using node : cone affecting

71 It can be shown that: constant

72 for simplicity assume:

73 : constants

74 Divide area into concentric circles

75 Max Euclidian distance between any two nodes = Longest path has at most nodes

76 Maximum ring radius

77 = number of packets that can affect = number of paths that use Ring We will bound

78

79 Expected congestion:

80 We have proven we prove next

81 we showed earlier

82 Similarly, each packet that affects traverses distance at least

83 Area

84 Total number of nodes used Area

85 Average node utilization Area

86 #nodes in area = Area

87 Average node utilization average node utilization

88 We have proven:

89 Considering all the rings: End of Proof

90 Recap Constant stretch Small congestion We presented a simple oblivious algorithm which has: when the parameters of the Euclidian embedding are constants

91 Outline of Presentation Introduction Network Model Oblivious Algorithm Discussion Analysis

92 Holes

93 Arbitrary closed shapes there is no