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

2 Length of chosen path Length of shortest path Stretch= shortest path chosen path

3 source destination

4 Pick random node

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10 Pick random node

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12 Adjacent nodes may follow long paths Big stretch Problem:

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

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

15 packets in one path Stretch = Edge congestion =

16 1 packet per path Stretch = Edge congestion =

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

18 Basic Idea source destination

19 Pick a random intermediate node

20 Construct path through intermediate node

21 Outline of Presentation Introduction Network Model Oblivious Algorithm Analysis

22 Network Surrounding area

23 space point space point Perpendicular bisector geodesic

24 space point space point

25 Area wideness:

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

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

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

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

30 Outline of Presentation Introduction Network Model Oblivious Algorithm Analysis

31 Every pair of nodes is assigned a default path default path Examples: Shortest paths

32 The algorithm source destination

33 geodesic Perpendicular bisector

34 Pick random space point

35 Find closest node to point

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

37 Outline of Presentation Introduction Network Model Oblivious Algorithm Analysis

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

39 We will show: optimal congestion

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

41 default path default path shortest path

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

43 Default path (shortest) Similarly:

44 shortest path

45 For constants: End of Proof

46 Theorem: denotes