1 Optimal Oblivious Routing in Hole-Free Networks Costas Busch Louisiana State University Malik Magdon-Ismail Rensselaer Polytechnic Institute

2 Routing: choose paths from sources to destinations

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

4 Length of chosen path Length of shortest path Stretch= shortest path chosen path

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

6 Path choices: Probability of choosing a path:

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

8 Hole-free network

9 Our contribution in this work: Oblivious routing in hole-free networks Constant stretch Small congestion

10 Holes

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

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

13 Azar et al. [STOC03] Harrelson et al. [SPAA03] Bienkowski et al. [SPAA03] Arbitrary Graphs (existential result): Constructive Results: Racke [FOCS’02]: Racke [STOC’08]:

14 Hierarchical clustering General Approach:

15 Hierarchical clustering General Approach:

16 At the lowest level every node is a cluster

17 source destination

18 Pick random node

19 Pick random node

20 Pick random node

21 Pick random node

22 Pick random node

23 Pick random node

24 Pick random node

25

26 Adjacent nodes may follow long paths Big stretch Problem:

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

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

29 packets in one path Stretch = Edge congestion =

30 1 packet per path Stretch = Edge congestion =

31 Result for Grids: Busch, Magdon-Ismail, Xi [TC’08] For d=2, a similar result given by C. Scheideler

32 Special graphs embedded in the 2-dimensional plane: Constant stretch Small congestion degree Busch, Magdon-Ismail, Xi [SPAA 2005]:

33 Embeddings in wide, closed-curved areas

34 Graph models appropriate for various wireless network topologies Transmission radius

35 Basic Idea source destination

36 Pick a random intermediate node

37 Construct path through intermediate node

38 However, algorithm does not extend to arbitrary closed shapes

39 Our contribution in this work: Oblivious routing in hole-free networks

40 Approach: route within square areas

41 grid

42 simple area in grid (hole-free area)

43 Hole-free network

44 Canonical square decomposition

45 Canonical square decomposition

46 Canonical square decomposition

47 Canonical square decomposition

48

49

50 Shortest path

51 Canonical square sequence

52 A random path in canonical squares

53 Path has constant stretch

54 Random 2-bend paths or 1-bend paths in square sequence