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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

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Trees

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

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Routing on Trees

•Every node generates at most one packet•Packets follow shortest paths

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•Synchronous network

Network Model

•One packet per time step

•Bi-directional links

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Buffer-less nodes

Time 0

Hot-Potato Routing

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conflict

Time 1

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deflected

Time 2

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Time 3

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Time 4

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Hot-potato routing is interesting:

•Optical networks

•Simple hardware implementations

•Works well in practice:Bartzis et al.: EUROPAR 2000Maxemchuck: INFOCOM 1989

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Objective: Find hot-potato algorithm

which minimizes routing time

The time until the last packetis delivered to its destination

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Congestion:Maximum numbers ofpackets that share an edge

3C

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Dilation: Maximum path length

6D

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A lower bound on Routing Time:

Congestion+Dilation

)( DC

We want to find an algorithm close to this lower bound

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Our contributions:

•Deterministic Algorithm: )log)(( nDCO

node degree

network size

•Randomized Algorithm: )log)(( 2nDCO

degree independent

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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))

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Presentation Outiline

•Deterministic Algorithm•Randomized Algorithm

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Deterministic Algorithm

1. Divide time into phases according to short nodes

2. At each phase send packets to their destinations greedily

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every subtree has at most nodes 2

n

Short Node:

……2n

2n

2n

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7214

2

n

short node

6

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Example

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Phase 1:

……

Route packets that cross the short node

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Phase 2:

……

In each subtreeget the short nodes

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Phase 2:

……

In each subtreeget the short nodes

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Phase 2:Route packets that cross the short nodes

…………

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n

2n

4n

1

There are at most

nlog phases

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Phase k:

……

Route packets that cross the short node

C

Bound on number of packets:

C

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A packet follows its path greedily

Phase k:

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conflict

However, packets can conflict and get deflected

deflected

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p

1p 2p2jp 1jp jp

Deflection Sequence

If a packet is deflectedthen some other packet reaches its destination

p

jp

[Borodin, Rabani, Schieber 1997]

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Since there are at most packets,there are at most deflections

CC

Worst Routing Time for a packet:

DC 2

deflections Initial distance

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Total Routing Time

nDC log)2(

Packet timeIn a phase

Number of Phases

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Presentation Outiline

•Deterministic Algorithm•Randomized Algorithm

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Randomized Algorithm

Same with deterministic algorithm,with only difference:

Packet conflicts are resolvedaccording to random packet priorities

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Packet Priorities:

Low: each packet starts with

a low priority

High: when a packet is deflected

it increases its priority

with probability )(4

1DC

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A high priority packet can conflict with at most packets C2

……

C

From those packets, are expected to be in high priority

)1(O

C

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A high priority packet successfully reaches its destination with probability

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Thus attempts to becomehigh priority are enough.

)(lognO

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Total Routing Time

nnDC loglog)(

Packet timeIn a phase

Number of Phases

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Discussion

We presented two near-optimalhot-potato algorithms for trees(within logarithmic factors from optimal)

Open problem: Remove the logarithmic factors