1 costas busch Õ(congestion + dilation) hot-potato routing on leveled networks costas busch...

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1 Costas Busch (Congestion + Dilation) Hot-Potato Routing on Leveled Networks Costas Busch Rensselaer Polytechnic Institute Slide 2 2 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline Slide 3 3 Leveled Networks Level: 0123L-1L Slide 4 4 Examples of Leveled Networks 0123 Butterfly 0 1 2 3456 Mesh Slide 5 5 Synchronous network One packet per link direction Network Model One time step per link traversal Bi-directional links Slide 6 6 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline Slide 7 7 Congestion + Dilation source destination Each packet has a pre-selected path Slide 8 8 source destination The packet follows the pre-selected path Slide 9 9 source destination The packet follows the pre-selected path Slide 10 10 source destination The packet follows the pre-selected path Slide 11 11 source destination The packet follows the pre-selected path Slide 12 12 source destination The packet follows the pre-selected path Slide 13 13 source destination The packet follows the pre-selected path Slide 14 14 absorbed The packet follows the pre-selected path Slide 15 15 Each packet has its own path There are packets Slide 16 16 Dilation D: The maximum length of any path Routing time: Slide 17 17 Congestion C: The maximum number of packets traversing any edge Routing time: Slide 18 18 Lower bound on routing time: There could be paths with length Lower bound: Slide 19 19 Networks with buffers Slide 20 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 Slide 21 21 What about buffer-less networks? Slide 22 22 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline Slide 23 23 Hot-Potato Routing Nodes are buffer-less: Slide 24 24 Nodes are buffer-less: Slide 25 25 Nodes are buffer-less: Time 0 Slide 26 26 Nodes are buffer-less: Time 1 conflict Slide 27 27 deflected Nodes are buffer-less: Time 2 Slide 28 28 Nodes are buffer-less: Time 3 conflict Slide 29 29 Nodes are buffer-less: Time 4 deflected Slide 30 30 Nodes are buffer-less: Time 5 Slide 31 31 Nodes are buffer-less: Time 6 Slide 32 32 original pre-selected path Path lengths may grow arbitrarily large Slide 33 33 actual path followed Path lengths may grow arbitrarily large original pre-selected path Slide 34 34 Hot-potato routing is interesting: Optical networks Simple hardware implementations Work well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM 1989 Slide 35 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) : Slide 36 36 Our contribution: A new hot-potato routing algorithm with time With high probability: Slide 37 37 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline Slide 38 38 Our Algorithm Online algorithm: Local algorithm: routing decisions are taken at real time routing decisions at a node depend only on the nodes state Slide 39 39 packets Slide 40 40 Partition the packets randomly and uniformly into sets packets Slide 41 41 Partition the packets randomly and uniformly into sets congestion Slide 42 42 We route each set in a separate frame Slide 43 43 We route each set in a separate frame Slide 44 44 We route each set in a separate frame Slide 45 45 pre-selected path Routing a packet Slide 46 46 Phase 1 Routing a packet Slide 47 47 Routing a packet Phase 2 Slide 48 48 Routing a packet Phase 3 Slide 49 49 Routing a packet Phase 4 Slide 50 50 Routing a packet Phase 5 Slide 51 51 Routing a packet Phase 6 Slide 52 52 Routing a packet Phase 7 Slide 53 53 Routing a packet Phase 8 Slide 54 54 Routing a packet Phase 9 Slide 55 55 Routing a packet Phase 10 Slide 56 56 Routing a packet Phase 11 Slide 57 57 Routing a packet Phase 12 Slide 58 58 Routing a packet set Slide 59 59 Phase 1 Routing a packet set Slide 60 60 Routing a packet set Phase 2 Slide 61 61 Routing a packet set Phase 3 Slide 62 62 Routing a packet set Phase 4 Slide 63 63 Routing a packet set Phase 5 Slide 64 64 Routing a packet set Phase 6 Slide 65 65 Routing a packet set Phase 7 Slide 66 66 Routing a packet set Phase 8 Slide 67 67 Routing a packet set Phase 9 Slide 68 68 Routing a packet set Phase 10 Slide 69 69 Routing a packet set Phase 11 Slide 70 70 Routing a packet set Phase 12 Slide 71 71 Routing many packet sets Slide 72 72 Phase 1 Routing many packet sets Slide 73 73 Routing many packet sets Phase 2 Slide 74 74 Routing many packet sets Phase 3 Slide 75 75 Routing many packet sets Phase 4 Slide 76 76 Routing many packet sets Phase 5 Slide 77 77 Routing many packet sets Phase 6 Slide 78 78 Routing many packet sets Phase 7 Slide 79 79 Routing many packet sets Phase 8 Slide 80 80 Routing many packet sets Phase 9 Slide 81 81 Routing many packet sets Phase 10 Slide 82 82 Routing many packet sets Phase 11 Slide 83 83 Routing many packet sets Phase 12 Slide 84 84 Routing many packet sets Phase 13 Slide 85 85 Routing many packet sets Phase 14 Slide 86 86 Routing many packet sets Phase 15 Slide 87 87 Routing many packet sets Phase 16 Slide 88 88 Routing many packet sets Phase 17 Slide 89 89 Routing many packet sets Phase 18 Slide 90 90 #frames X frame length + Total time: #phases phase duration X X Slide 91 91 Phase k Start Packets may appear anywhere Slide 92 92 Phase k End Packets are accumulated at the right end Slide 93 93 Phase k+1 Start Slide 94 94 Phase k+1 Start Free from packets Slide 95 95 Phase k+1 Start Packets from set are injected here Slide 96 96 Phase k+1 End Slide 97 97 Phase k+2 Start Slide 98 98 Phase k Slide 99 99 Phase k -- Round 1 Target Column Slide 100 100 Phase k -- Round 1 Target Column Slide 101 101 Phase k -- Round 2 Target Column Slide 102 102 Phase k -- Round 2 Target Column Slide 103 103 Phase k -- Round 3 Target Column Slide 104 104 Phase k+1 Slide 105 105 Phase k Slide 106 106 Phase k -- Round 1 Target Column Slide 107 107 Phase k -- Round 1 Target Column Slide 108 108 Phase k -- Round 2 Target Column Slide 109 109 Phase k -- Round 3 Target Column Slide 110 110 Phase k+1 Slide 111 111 Duration of Round: Number of packets that Reach target column at each round: #Rounds needed: Slide 112 112 Leveled Networks Congestion + Dilation Hot-Potato Routing Our Algorithm Future Work Talk Outline Slide 113 113 Future Work Decrease logarithmic factors Generalize to arbitrary networks

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