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

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

3 Leveled Networks Level: 0123L-1L

4 Examples of Leveled Networks 0123 Butterfly Mesh

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

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

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

8 source destination The packet follows the pre-selected path

9 source destination The packet follows the pre-selected path

10 source destination The packet follows the pre-selected path

11 source destination The packet follows the pre-selected path

12 source destination The packet follows the pre-selected path

13 source destination The packet follows the pre-selected path

14 absorbed The packet follows the pre-selected path

15 Each packet has its own path There are packets

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

17 Congestion C: 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 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline

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: Time 1 conflict

27 deflected Nodes are buffer-less: Time 2

28 Nodes are buffer-less: Time 3 conflict

29 Nodes are buffer-less: Time 4 deflected

30 Nodes are buffer-less: Time 5

31 Nodes are buffer-less: Time 6

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

33 actual path followed Path lengths may grow arbitrarily large 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) Bhatt et al.: ESA 1993 Only work known (but for different problem) :

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

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

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

39 packets

40 Partition the packets randomly and uniformly into sets packets

41 Partition the packets randomly and uniformly into sets 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 pre-selected path Routing a packet

46 Phase 1 Routing a packet

47 Routing a packet Phase 2

48 Routing a packet Phase 3

49 Routing a packet Phase 4

50 Routing a packet Phase 5

51 Routing a packet Phase 6

52 Routing a packet Phase 7

53 Routing a packet Phase 8

54 Routing a packet Phase 9

55 Routing a packet Phase 10

56 Routing a packet Phase 11

57 Routing a packet Phase 12

58 Routing a packet set

59 Phase 1 Routing a packet set

60 Routing a packet set Phase 2

61 Routing a packet set Phase 3

62 Routing a packet set Phase 4

63 Routing a packet set Phase 5

64 Routing a packet set Phase 6

65 Routing a packet set Phase 7

66 Routing a packet set Phase 8

67 Routing a packet set Phase 9

68 Routing a packet set Phase 10

69 Routing a packet set Phase 11

70 Routing a packet set Phase 12

71 Routing many packet sets

72 Phase 1 Routing many packet sets

73 Routing many packet sets Phase 2

74 Routing many packet sets Phase 3

75 Routing many packet sets Phase 4

76 Routing many packet sets Phase 5

77 Routing many packet sets Phase 6

78 Routing many packet sets Phase 7

79 Routing many packet sets Phase 8

80 Routing many packet sets Phase 9

81 Routing many packet sets Phase 10

82 Routing many packet sets Phase 11

83 Routing many packet sets Phase 12

84 Routing many packet sets Phase 13

85 Routing many packet sets Phase 14

86 Routing many packet sets Phase 15

87 Routing many packet sets Phase 16

88 Routing many packet sets Phase 17

89 Routing many packet sets Phase 18

90 #frames X frame length + Total time: #phases phase duration X X

91 Phase k Start Packets may appear anywhere

92 Phase k End Packets are accumulated at the right end

93 Phase k+1 Start

94 Phase k+1 Start Free from packets

95 Phase k+1 Start Packets from set are injected here

96 Phase k+1 End

97 Phase k+2 Start

98 Phase k

99 Phase k -- Round 1 Target Column

100 Phase k -- Round 1 Target Column

101 Phase k -- Round 2 Target Column

102 Phase k -- Round 2 Target Column

103 Phase k -- Round 3 Target Column

104 Phase k+1

105 Phase k

106 Phase k -- Round 1 Target Column

107 Phase k -- Round 1 Target Column

108 Phase k -- Round 2 Target Column

109 Phase k -- Round 3 Target Column

110 Phase k+1

111 Duration of Round: Number of packets that Reach target column at each round: #Rounds needed:

112 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline

113 Future Work Decrease logarithmic factors Generalize to arbitrary networks