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

Transcription

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

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

3 Leveled Networks Level: L-1 L 3

4 Examples of Leveled Networks Butterfly Mesh 4

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

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

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

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

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

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

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

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

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

14 The packet follows the pre-selected path absorbed 14

15 There are N packets Each packet has its own path N = 3 15

16 Dilation D: The maximum length of any path Routing time: Ω(D) D = 6 16

17 Congestion C: The maximum number of packets traversing any edge Routing time: Ω(C) C 2 17

18 Lower bound on routing time: Ω(C + D) There could be paths with length L Lower bound: Ω(C + L) 18

19 Networks with buffers 19

20 Networks with buffers Leveled networks: O(C + L +logn) Leighton, Maggs, Ranade, Rao: J. Algorithms 1992 Arbitrary networks: O(C + D) 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

21 What about buffer-less networks? 21

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

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

24 Nodes are buffer-less: 24

25 Nodes are buffer-less: Time 0 25

26 Nodes are buffer-less: conflict Time 1 26

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

28 Nodes are buffer-less: conflict Time 3 28

29 Nodes are buffer-less: deflected Time 4 29

30 Nodes are buffer-less: Time 5 30

31 Nodes are buffer-less: Time 6 31

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

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

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

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

36 Our contribution: A new hot-potato routing algorithm with time O ~ (C + L) O((C + L) log 9 (LN)) With high probability: 1-1 LN 36

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

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 38

39 N packets 39

40 Partition the packets randomly and uniformly into C sets N packets S1 S2 S3 SC N C N C N C packets N C 40

41 Partition the packets randomly and uniformly into C sets C congestion S1 S2 S3 SC congestion log( LN) log( LN) log( LN) log(ln) 41

42 We route each set in a separate frame S1 S2 S3 SC 42

43 We route each set in a separate frame F1 F 2 F 3 F C log 2 (LN) S1 S2 S3 SC 43

44 We route each set in a separate frame F1 F 2 F 3 F C S1 S2 S3 SC 44

45 Routing a packet pre-selected path S k 45

46 Routing a packet F k S k Phase 1 46

47 Routing a packet F k S k Phase 2 47

48 Routing a packet F k S k Phase 3 48

49 Routing a packet F k S k Phase 4 49

50 Routing a packet F k S k Phase 5 50

51 Routing a packet F k S k Phase 6 51

52 Routing a packet F k S k Phase 7 52

53 Routing a packet F k S k Phase 8 53

54 Routing a packet F k S k Phase 9 54

55 Routing a packet F k S k Phase 10 55

56 Routing a packet F k S k Phase 11 56

57 Routing a packet S k Phase 12 57

58 Routing a packet set S k 58

59 Routing a packet set F k S k Phase 1 59

60 Routing a packet set F k S k Phase 2 60

61 Routing a packet set F k S k Phase 3 61

62 Routing a packet set F k S k Phase 4 62

63 Routing a packet set F k S k Phase 5 63

64 Routing a packet set F k S k Phase 6 64

65 Routing a packet set F k S k Phase 7 65

66 Routing a packet set F k S k Phase 8 66

67 Routing a packet set F k S k Phase 9 67

68 Routing a packet set F k S k Phase 10 68

69 Routing a packet set F k S k Phase 11 69

70 Routing a packet set S k Phase 12 70

71 Routing many packet sets S 1 S 2 S 3 71

72 Routing many packet sets F 3 S 1 S 2 S 3 Phase 1 72

73 Routing many packet sets F 3 S 1 S 2 S 3 Phase 2 73

74 Routing many packet sets F 3 S 1 S 2 S 3 Phase 3 74

75 Routing many packet sets F 2 F 3 S 1 S 2 S 3 Phase 4 75

76 Routing many packet sets F 2 F 3 S 1 S 2 S 3 Phase 5 76

77 Routing many packet sets F 2 F 3 S 1 S 2 S 3 Phase 6 77

78 Routing many packet sets F 1 F 2 F 3 S 1 S 2 S 3 Phase 7 78

79 Routing many packet sets F 1 F 2 F 3 S 1 S 2 S 3 Phase 8 79

80 Routing many packet sets F 1 F 2 F 3 S 1 S 2 S 3 Phase 9 80

81 Routing many packet sets F 1 F 2 F 3 S 1 S 2 S 3 Phase 10 81

82 Routing many packet sets F 1 F 2 F 3 S 1 S 2 S 3 Phase 11 82

83 Routing many packet sets F 1 F 2 S 1 S 2 S 3 Phase 12 83

84 Routing many packet sets F 1 F 2 S 1 S 2 S 3 Phase 13 84

85 Routing many packet sets F 1 F 2 S 1 S 2 S 3 Phase 14 85

86 Routing many packet sets F 1 S 1 S 2 S 3 Phase 15 86

87 Routing many packet sets F 1 S 1 S 2 S 3 Phase 16 87

88 Routing many packet sets F 1 S 1 S 2 S 3 Phase 17 88

89 Routing many packet sets S 1 S 2 S 3 Phase 18 89

90 Total time: #phases X phase duration #frames X frame C log(ln) + length L X log 9 (LN) log 2 (LN) C log 9 (LN) + L log 8 (LN) O ~ (C + L) 90

91 Phase k Start F i Packets may appear anywhere 91

92 Phase k End F i Packets are accumulated at the right end 92

93 Phase k+1 Start F i 93

94 Phase k+1 Start F i Free from packets 94

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

96 Phase k+1 End F i 96

97 Phase k+2 Start F i 97

98 Phase k F i 98

99 Phase k -- Round 1 F i Target Column 99

100 Phase k -- Round 1 F i Target Column 100

101 Phase k -- Round 2 F i Target Column 101

102 Phase k -- Round 2 F i Target Column 102

103 Phase k -- Round 3 F i Target Column 103

104 Phase k+1 F i 104

105 Phase k F i 105

106 Phase k -- Round 1 F i Target Column 106

107 Phase k -- Round 1 F i Target Column 107

108 Phase k -- Round 2 F i Target Column 108

109 Phase k -- Round 3 F i Target Column 109

110 Phase k+1 F i 110

111 Duration of Round: log 6 (LN) Number of packets that Reach target column at each round: Remaining Packets log(ln) #Rounds needed: log 2 (LN) 111

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

113 Future Work Decrease logarithmic factors Generalize to arbitrary networks 113