Õ(Congestion + Dilation) Hot-Potato Routing on Leveled Networks Costas Busch Rensselaer Polytechnic Institute

1. Õ(Congestion + Dilation) Hot-Potato Routing on Leveled NetworksCostas BuschRensselaer Polytechnic Institute

2. Talk Outline • Leveled Networks • Congestion + Dilation • Hot-Potato Routing • Our Algorithm • Future Work

3. Leveled Networks Level: 0 1 2 3 L-1 L

4. Examples of Leveled Networks 0 1 2 3 3 4 5 6 2 1 0 Butterfly Mesh

5. Network Model • Synchronous network • One packet per link direction • One time step per link traversal • Bi-directional links

6. Talk Outline • Leveled Networks • Congestion + Dilation • Hot-Potato Routing • Our Algorithm • Future Work

7. Congestion + Dilation Each packet has a pre-selected path source destination

8. The packet follows the pre-selected path source destination

9. The packet follows the pre-selected path source destination

10. The packet follows the pre-selected path source destination

11. The packet follows the pre-selected path source destination

12. The packet follows the pre-selected path source destination

13. The packet follows the pre-selected path source destination

14. The packet follows the pre-selected path absorbed

15. There are packets Each packet has its own path

16. Dilation D: The maximum length of any path Routing time:

17. CongestionC: The maximum number of packets traversing any edge Routing time:

18. Lower bound on routing time: There could be paths with length Lower bound:

19. Networks with buffers

20. Networks with buffers Leveled networks: • Leighton, Maggs, Ranade, Rao: J. Algorithms 1992 Arbitrary networks: • Leighton, Maggs, Rao: Combinatorica 1994 • Leighton, Maggs, Richa: Combinatorica 1999 • Meyer auf der Heide, Sheideler: J. Algorithms 1999 • Ostrovsky, Rabani: STOC 1997 • Rabani, Tardos: STOC 1996

21. What about buffer-less networks?

22. Talk Outline • Leveled Networks • Congestion + Dilation • Hot-Potato Routing • Our Algorithm • Future Work

23. Hot-Potato Routing Nodes are buffer-less:

24. Nodes are buffer-less:

25. Nodes are buffer-less: Time 0

26. Nodes are buffer-less: conflict Time 1

27. Nodes are buffer-less: deflected deflected Time 2

28. Nodes are buffer-less: conflict Time 3

29. Nodes are buffer-less: deflected Time 4

30. Nodes are buffer-less: Time 5

31. Nodes are buffer-less: Time 6

32. Path lengths may grow arbitrarily large original pre-selected path

33. Path lengths may grow arbitrarily large actual path followed original pre-selected path

34. Hot-potato routing is interesting: • Optical networks • Simple hardware implementations • Work well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM 1989

35. Goal: Find a hot-potato algorithm with time close to optimal No previous work for leveled networks (and arbitrary networks) Only work known (but for different problem): Bhatt et al.: ESA 1993

36. Our contribution: A new hot-potato routing algorithm with time With high probability:

37. Talk Outline • Leveled Networks • Congestion + Dilation • Hot-Potato Routing • Our Algorithm • Future Work

38. Our Algorithm • Online algorithm: routing decisions are taken at real time • Local algorithm: routing decisions at a node depend only on the node’s state

39. packets

40. Partition the packets randomly and uniformly into sets packets packets

41. Partition the packets randomly and uniformly into sets congestion congestion

42. We route each set in a separate frame

43. We route each set in a separate frame

44. We route each set in a separate frame

45. Routing a packet pre-selected path

46. Routing a packet Phase 1

47. Routing a packet Phase 2

48. Routing a packet Phase 3

49. Routing a packet Phase 4

50. Routing a packet Phase 5