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Near-Optimal Hot-Potato Routing on Trees Costas Busch Malik Magdon Ismail Marios Mavronicolas Roger wattenhofer Rensselaer Polytechnic Inst. Rensselaer Polytechnic Inst. University of Cyprus ETH Zurich 1 Trees Trees are important in many networks (i.e. spanning trees) 2 Routing on Trees •Every node generates at most one packet •Packets follow shortest paths 3 Network Model •Synchronous network •Bi-directional links •One packet per time step 4 Hot-Potato Routing Time 0 Buffer-less nodes 5 conflict Time 1 6 Time 2 deflected 7 Time 3 8 Time 4 9 Hot-potato routing is interesting: •Optical networks •Simple hardware implementations •Works well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM 1989 10 Objective: Find hot-potato algorithm which minimizes routing time The time until the last packet is delivered to its destination 11 Congestion: Maximum numbers of packets that share an edge C 3 12 Dilation: Maximum path length D 6 13 A lower bound on Routing Time: Congestion+Dilation (C D) We want to find an algorithm close to this lower bound 14 Our contributions: •Deterministic Algorithm: O ((C D )logn ) node degree network size 2 O (( C D ) log n) •Randomized Algorithm: degree independent 15 Related Work for Trees •Matching Routing •Direct Routing •Hot-Potato routing [ACG94] [PRS97] [Z97] [AHLT98][BMMS04] [RSW00] Most results have routing time O(n) (worst case bound for O(C+D)) 16 Presentation Outiline •Deterministic Algorithm •Randomized Algorithm 17 Deterministic Algorithm 1. Divide time into phases according to short nodes 2. At each phase send packets to their destinations greedily 18 Short Node: n 2 n 2 …… every subtree has at most n 2 n 2 nodes 19 Example short node 6 6 n 14 7 2 2 20 Phase 1: Route packets that cross the short node …… 21 Phase 2: In each subtree get the short nodes …… 22 Phase 2: In each subtree get the short nodes …… 23 Phase 2: Route packets that cross the short nodes ………… 24 There are at most logn n phases n 2 n 4 1 25 Phase k: Route packets that cross the short node C …… Bound on number of packets: C 26 Phase k: A packet follows its path greedily 27 However, packets can conflict and get deflected conflict deflected 28 Deflection Sequence [Borodin, Rabani, Schieber 1997] p p1 p2 p j 2 pj 1 p j If a packet p is deflected then some other packet p j reaches its destination 29 Since there are at most C packets, there are at most C deflections Worst Routing Time for a packet: 2C D deflections Initial distance 30 Total Routing Time (2C D ) logn Packet time In a phase Number of Phases 31 Presentation Outiline •Deterministic Algorithm •Randomized Algorithm 32 Randomized Algorithm Same with deterministic algorithm, with only difference: Packet conflicts are resolved according to random packet priorities 33 Packet Priorities: Low: each packet starts with a low priority High: when a packet is deflected it increases its priority with probability 1 4(C D ) 34 A high priority packet can conflict with at most 2C packets C C …… From those packets, O (1) are expected to be in high priority 35 A high priority packet successfully reaches its destination 1 with probability 2 Thus O (logn ) attempts to become high priority are enough. 36 Total Routing Time (C D ) logn logn Packet time In a phase Number of Phases 37 Discussion We presented two near-optimal hot-potato algorithms for trees (within logarithmic factors from optimal) Open problem: Remove the logarithmic factors 38