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Õ(Congestion + Dilation) Hot-Potato Routing on Leveled Networks

Costas BuschCostas BuschRensselaer Polytechnic Institute

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•Leveled Networks•Congestion + Dilation•Hot-Potato Routing•Our Algorithm•Future Work

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Leveled Networks

Level: 0 1 2 3 L-1 L

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Examples of Leveled Networks

0 1 2 3

Butterfly

0

1

2

3 4 5 6

Mesh

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•Synchronous network

•One packet per link direction

Network Model

•One time step per link traversal

•Bi-directional links

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•Leveled Networks•Congestion + Dilation•Hot-Potato Routing•Our Algorithm•Future Work

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Congestion + Dilation

sourcedestination

Each packet has a pre-selected path

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sourcedestination

The packet follows the pre-selected path

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sourcedestination

The packet follows the pre-selected path

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sourcedestination

The packet follows the pre-selected path

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sourcedestination

The packet follows the pre-selected path

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sourcedestination

The packet follows the pre-selected path

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sourcedestination

The packet follows the pre-selected path

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absorbed

The packet follows the pre-selected path

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Each packet has its own path

There are packets N

3=N

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Dilation D: The maximum length of any path

6=D

Routing time: )D(Ω

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Congestion C: The maximum number of packets traversing any edge

2CRouting time: )C(Ω

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Lower bound on routing time:

)D+C(Ω

)L+C(Ω

LThere could bepaths with length

Lower bound:

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Networks with buffers

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Networks with buffers

Leveled networks:

•Leighton, Maggs, Ranade, Rao: J. Algorithms 1992

)Nlog+L+C(O

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

)D+C(O

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What about buffer-less networks?

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•Leveled Networks•Congestion + Dilation•Hot-Potato Routing•Our Algorithm•Future Work

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Hot-Potato Routing

Nodes are buffer-less:

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Nodes are buffer-less:

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Nodes are buffer-less:

Time 0

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Nodes are buffer-less:

Time 1

conflict

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deflected

deflected

Nodes are buffer-less:

Time 2

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Nodes are buffer-less:

Time 3

conflict

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Nodes are buffer-less:

Time 4

deflected

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Nodes are buffer-less:

Time 5

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Nodes are buffer-less:

Time 6

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original pre-selected path

Path lengths may grow arbitrarily large

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actual path followed

Path lengths may grow arbitrarily large

original pre-selected path

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Hot-potato routing is interesting:

•Optical networks

•Simple hardware implementations

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

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Goal: Find a hot-potato algorithm with time close to optimal )L+C(O

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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Our contribution:

A new hot-potato routing algorithmwith time )L+C(O~

))LN(log•)L+C((O 9

With high probability: LN1

1-

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•Leveled Networks•Congestion + Dilation•Hot-Potato Routing•Our Algorithm•Future Work

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Our Algorithm

•Online algorithm:

•Local algorithm:

routing decisions are takenat real time

routing decisions at a node depend only on the node’s state

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N packets

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Partition the packets randomly and uniformly into setsC

1S CS

N packets

2S 3S

CN packets

CN

CN

CN

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Partition the packets randomly and uniformly into setsC

1S CS2S 3S

)LNlog( )LNlog( )LNlog( )LNlog(

C congestion

congestion

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1S CS2S 3S

We route each set in a separate frame

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1F 2F 3F CFWe route each set in a separate frame

)LN(log2

1S CS2S 3S

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1F 2F 3F CFWe route each set in a separate frame

1S CS2S 3S

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pre-selected path

kS

Routing a packet

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kF

Phase 1kS

Routing a packet

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kF

kS

Routing a packet

Phase 2

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kF

kS

Routing a packet

Phase 3

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kF

kS

Routing a packet

Phase 4

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kF

kS

Routing a packet

Phase 5

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kF

kS

Routing a packet

Phase 6

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kF

kS

Routing a packet

Phase 7

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kF

kS

Routing a packet

Phase 8

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kF

kS

Routing a packet

Phase 9

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kF

kS

Routing a packet

Phase 10

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kF

kS

Routing a packet

Phase 11

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kS

Routing a packet

Phase 12

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kS

Routing a packet set

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kS

kF

Phase 1

Routing a packet set

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kS

kFRouting a packet set

Phase 2

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kS

kFRouting a packet set

Phase 3

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kS

kFRouting a packet set

Phase 4

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kS

kFRouting a packet set

Phase 5

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kS

kFRouting a packet set

Phase 6

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kS

kFRouting a packet set

Phase 7

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kS

kFRouting a packet set

Phase 8

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kS

kFRouting a packet set

Phase 9

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kS

kFRouting a packet set

Phase 10

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kS

kFRouting a packet set

Phase 11

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kS

Routing a packet set

Phase 12

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3S

Routing many packet sets

2S1S

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3F

Phase 1

3S2S1S

Routing many packet sets

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3S2S1S

3FRouting many packet sets

Phase 2

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3S2S1S

3FRouting many packet sets

Phase 3

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3S2S1S

3F2FRouting many packet sets

Phase 4

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3S2S1S

3F2FRouting many packet sets

Phase 5

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3S2S1S

3F2FRouting many packet sets

Phase 6

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3S2S1S

3F2F1FRouting many packet sets

Phase 7

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3S2S1S

3F2F1FRouting many packet sets

Phase 8

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3S2S1S

3F2F1FRouting many packet sets

Phase 9

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3S2S1S

3F2F1FRouting many packet sets

Phase 10

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3S2S1S

3F2F1FRouting many packet sets

Phase 11

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3S2S1S

2F1FRouting many packet sets

Phase 12

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3S2S1S

2F1FRouting many packet sets

Phase 13

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3S2S1S

2F1FRouting many packet sets

Phase 14

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3S2S1S

1FRouting many packet sets

Phase 15

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3S2S1S

1FRouting many packet sets

Phase 16

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3S2S1S

1FRouting many packet sets

Phase 17

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3S2S1S

Routing many packet sets

Phase 18

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)LNlog(C

)LN(log2

#frames Xframe length + L )LN(log9

Total time: #phases phase duration

)LN(log•L+)LN(log•C 89

)L+C(O~

X

X

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iF

Phase k

Start

Packets may appear anywhere

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iF

Phase k

End

Packets are accumulated at the right end

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iF

Phase k+1

Start

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iF

Phase k+1

Start

Free from packets

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iF

Phase k+1

Start

Packets from set are injected hereiS

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iF

Phase k+1

End

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iF

Phase k+2

Start

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iFPhase k

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iFPhase k -- Round 1

Target Column

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Phase k -- Round 1

Target Column

iF

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iFPhase k -- Round 2

Target Column

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iFPhase k -- Round 2

Target Column

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iFPhase k -- Round 3

Target Column

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iFPhase k+1

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iFPhase k

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iFPhase k -- Round 1

Target Column

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iFPhase k -- Round 1

Target Column

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iFPhase k -- Round 2

Target Column

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iFPhase k -- Round 3

Target Column

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iFPhase k+1

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Duration of Round: )LN(log6

Number of packets that Reach target column at each round:

)LNlog(Packets Remaining

#Rounds needed: )LN(log2

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•Leveled Networks•Congestion + Dilation•Hot-Potato Routing•Our Algorithm•Future Work

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Future Work

•Decrease logarithmic factors

•Generalize to arbitrary networks