1 Near-Optimal Hot-Potato Routing on Trees Costas Busch Rensselaer Polytechnic Inst. Malik Magdon Ismail Rensselaer Polytechnic Inst. Marios Mavronicolas University of Cyprus Roger wattenhofer ETH Zurich

2 Trees Trees are important in many networks (i.e. spanning trees)

3 Routing on Trees Every node generates at most one packet Packets follow shortest paths

4 Synchronous network Network Model One packet per time step Bi-directional links

5 Buffer-less nodes Time 0 Hot-Potato Routing

6 conflict Time 1

7 deflected Time 2

8 Time 3

9 Time 4

10 Hot-potato routing is interesting: Optical networks Simple hardware implementations Works well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM 1989

11 Objective: Find hot-potato algorithm which minimizes routing time The time until the last packet is delivered to its destination

12 Congestion: Maximum numbers of packets that share an edge

13 Dilation: Maximum path length

14 A lower bound on Routing Time: Congestion+Dilation We want to find an algorithm close to this lower bound

15 Our contributions: Deterministic Algorithm: node degree network size Randomized Algorithm: degree independent

16 Related Work for Trees Matching Routing [ACG94] [PRS97] [Z97] Direct Routing [AHLT98][BMMS04] Hot-Potato routing [RSW00] Most results have routing time O(n) (worst case bound for O(C+D))

17 Presentation Outiline Deterministic Algorithm Randomized Algorithm

18 Deterministic Algorithm 1. Divide time into phases according to short nodes 2. At each phase send packets to their destinations greedily

19 every subtree has at most nodes Short Node: ……

20 short node Example

21 Phase 1: …… Route packets that cross the short node

22 Phase 2: …… In each subtreeget the short nodes

23 Phase 2: …… In each subtreeget the short nodes

24 Phase 2: Route packets that cross the short nodes …………

25 There are at most phases

26 Phase k: …… Route packets that cross the short node Bound on number of packets:

27 A packet follows its path greedily Phase k:

28 conflict However, packets can conflict and get deflected deflected

29 Deflection Sequence If a packet is deflected then some other packet reaches its destination [Borodin, Rabani, Schieber 1997]

30 Since there are at most packets, there are at most deflections Worst Routing Time for a packet: deflectionsInitial distance

31 Total Routing Time Packet time In a phase Number of Phases

32 Presentation Outiline Deterministic Algorithm Randomized Algorithm

33 Randomized Algorithm Same with deterministic algorithm, with only difference: Packet conflicts are resolved according to random packet priorities

34 Packet Priorities: Low: each packet starts with a low priority High: when a packet is deflected it increases its priority with probability

35 A high priority packet can conflict with at most packets …… From those packets, are expected to be in high priority

36 A high priority packet successfully reaches its destination with probability Thus attempts to become high priority are enough.

37 Total Routing Time Packet time In a phase Number of Phases

38 Discussion We presented two near-optimal hot-potato algorithms for trees (within logarithmic factors from optimal) Open problem: Remove the logarithmic factors