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1 Costas Busch Õ(Congestion + Dilation) Hot-Potato Routing on Leveled Networks Costas Busch Rensselaer Polytechnic Institute
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2 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline
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3 Leveled Networks Level: 0123L-1L
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4 Examples of Leveled Networks 0123 Butterfly 0 1 2 3456 Mesh
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5 Synchronous network One packet per link direction Network Model One time step per link traversal Bi-directional links
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6 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline
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7 Congestion + Dilation source destination Each packet has a pre-selected path
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8 source destination The packet follows the pre-selected path
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9 source destination The packet follows the pre-selected path
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10 source destination The packet follows the pre-selected path
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11 source destination The packet follows the pre-selected path
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12 source destination The packet follows the pre-selected path
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13 source destination The packet follows the pre-selected path
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14 absorbed The packet follows the pre-selected path
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15 Each packet has its own path There are packets
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16 Dilation D: The maximum length of any path Routing time:
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17 Congestion C: The maximum number of packets traversing any edge Routing time:
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18 Lower bound on routing time: There could be paths with length Lower bound:
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19 Networks with buffers
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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
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21 What about buffer-less networks?
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22 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline
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23 Hot-Potato Routing Nodes are buffer-less:
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24 Nodes are buffer-less:
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25 Nodes are buffer-less: Time 0
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26 Nodes are buffer-less: Time 1 conflict
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27 deflected Nodes are buffer-less: Time 2
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28 Nodes are buffer-less: Time 3 conflict
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29 Nodes are buffer-less: Time 4 deflected
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30 Nodes are buffer-less: Time 5
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31 Nodes are buffer-less: Time 6
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32 original pre-selected path Path lengths may grow arbitrarily large
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33 actual path followed Path lengths may grow arbitrarily large original pre-selected path
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34 Hot-potato routing is interesting: Optical networks Simple hardware implementations Work well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM 1989
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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) :
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36 Our contribution: A new hot-potato routing algorithm with time With high probability:
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37 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline
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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
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39 packets
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40 Partition the packets randomly and uniformly into sets packets
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41 Partition the packets randomly and uniformly into sets congestion
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42 We route each set in a separate frame
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43 We route each set in a separate frame
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44 We route each set in a separate frame
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45 pre-selected path Routing a packet
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46 Phase 1 Routing a packet
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47 Routing a packet Phase 2
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48 Routing a packet Phase 3
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49 Routing a packet Phase 4
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50 Routing a packet Phase 5
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51 Routing a packet Phase 6
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52 Routing a packet Phase 7
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53 Routing a packet Phase 8
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54 Routing a packet Phase 9
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55 Routing a packet Phase 10
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56 Routing a packet Phase 11
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57 Routing a packet Phase 12
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58 Routing a packet set
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59 Phase 1 Routing a packet set
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60 Routing a packet set Phase 2
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61 Routing a packet set Phase 3
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62 Routing a packet set Phase 4
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63 Routing a packet set Phase 5
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64 Routing a packet set Phase 6
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65 Routing a packet set Phase 7
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66 Routing a packet set Phase 8
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67 Routing a packet set Phase 9
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68 Routing a packet set Phase 10
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69 Routing a packet set Phase 11
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70 Routing a packet set Phase 12
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71 Routing many packet sets
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72 Phase 1 Routing many packet sets
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73 Routing many packet sets Phase 2
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74 Routing many packet sets Phase 3
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75 Routing many packet sets Phase 4
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76 Routing many packet sets Phase 5
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77 Routing many packet sets Phase 6
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78 Routing many packet sets Phase 7
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79 Routing many packet sets Phase 8
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80 Routing many packet sets Phase 9
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81 Routing many packet sets Phase 10
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82 Routing many packet sets Phase 11
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83 Routing many packet sets Phase 12
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84 Routing many packet sets Phase 13
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85 Routing many packet sets Phase 14
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86 Routing many packet sets Phase 15
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87 Routing many packet sets Phase 16
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88 Routing many packet sets Phase 17
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89 Routing many packet sets Phase 18
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90 #frames X frame length + Total time: #phases phase duration X X
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91 Phase k Start Packets may appear anywhere
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92 Phase k End Packets are accumulated at the right end
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93 Phase k+1 Start
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94 Phase k+1 Start Free from packets
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95 Phase k+1 Start Packets from set are injected here
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96 Phase k+1 End
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97 Phase k+2 Start
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98 Phase k
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99 Phase k -- Round 1 Target Column
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100 Phase k -- Round 1 Target Column
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101 Phase k -- Round 2 Target Column
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102 Phase k -- Round 2 Target Column
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103 Phase k -- Round 3 Target Column
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104 Phase k+1
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105 Phase k
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106 Phase k -- Round 1 Target Column
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107 Phase k -- Round 1 Target Column
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108 Phase k -- Round 2 Target Column
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109 Phase k -- Round 3 Target Column
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110 Phase k+1
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111 Duration of Round: Number of packets that Reach target column at each round: #Rounds needed:
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112 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline
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113 Future Work Decrease logarithmic factors Generalize to arbitrary networks
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