Universal Bufferless Routing

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Universal Bufferless Routing
Marios Mavronicolas University of Cyprus Costas
Busch Rensselaer Polytechnic Inst.
Malik Magdon Ismail Rensselaer Polytechnic
Inst.

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Routing Problem

• Arbitrary network

• Arbitrary set of packets

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Network Model

• Synchronous network (time steps)

• One packet per time step

• Bi-directional links

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Network Model
Buffer-less nodes
Time 0
Packets are always moving
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Buffer-less nodes
Time 1
Packets are always moving
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Buffer-less nodes
Time 2
Packets are always moving
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Buffer-less nodes
Time 3
Packets are always moving
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Buffer-less nodes
Time 4
Packets are always moving
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Bufferless routing is interesting

• Optical networks
• Simple hardware implementations
• Works well in practice

Bartzis et al. EUROPAR 2000 Maxemchuck INFOCOM
1989
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the time until the last Packet is absorbed
Routing Time
Objective
Minimize Routing Time
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Congestion
Maximum number of packets that cross an edge
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Dilation
Maximum path length
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Lower bound on Routing Time
Congestion
Dilation
We want to find algorithms with routing time
close to lower bound
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Our contribution
A buffer-less algorithm with routing time
nodes
packets
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Related Work
For routing with buffersthere exist optimal
algorithms
Leighton - Maggs - Rao, Combinatorica
94 BS99, LMR99, MV99, OR97, RT96
What about buffer-less routing?
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Related Work
Buffer-less Routing has been studied for
specific topologies
Mesh BRST93, BES97, BHS98,
BU96, BHW00 Hypercube BH85, BC95,
FR92, H91 Trees BMMW04,
RSW00 Leveled BBPRRS96,
B02 Vertex-symmetric MS95
No previous work for time on
arbitrary networks
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Presentation Outiline

• Algorithm
• Analysis
• Conclusion

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Partition the Network into Regions

• Each edge belongs to only one region

• Regions share nodes

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Construct Region Graph
Regions are connected if they share a node in
the original graph
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Translate Packet Paths
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Translate Packet Paths
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Execute a Buffered Algorithm on Region Graph
Packet Buffer
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Simulate Buffered Algorithm
Buffered Algorithm
Buffers
No Buffers
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Region Graph
Original Graph
Buffered Algorithm Action for a Packet
Bufferless Algorithm Simulation Action
1 time step
1 phase

• Circulare in region

• Buffer in node

• Move from
• region to

• Move from
• node to

• Inject/Absorb
• at source/destination
• in region

• Inject/Absorb
• at node

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Simulation- Circulation
Packet Buffer
Follow repeatetly an Euler tour
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Simulation- Circulation
Packet Buffer
Euler tour (abstract)
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Simulation- Circulation
Packet Buffer
Multiple packets
Pipeline packets on Euler Tour
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Simulation of move
Packet Move
Packet moves to new region when it reaches
common node
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Simulation of move
Packet Move
Packet moves to new region when it reaches
common node
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Simulation of move
Packet Move
Packet moves to new region when it reaches
common node
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Simulation of move
Packet Move
Insert packet when there is an empty spot in
multiple packets in
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Oscillate here until empty slot is found
Packet Move
Slow Circulation
Fast Circulation
Sending Mode
Receiving Mode
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Color the Region Graph
0
1
0
2
1
Let be the maximum color
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Change Colors
12
9
12
6
9
Replace each color with bits
original color
complement
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Each phase consists of rounds
At round a node is in sending mode if its
color bit is equal to 1
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Round 0
Round 1
sending
12
9
12
9
12
12
6
9
6
9
Round 2
Round 3
12
9
12
9
12
12
6
9
6
9
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There is a round that
is receiving
is sending
And vice-versa
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Presentation Outiline

• Algorithm
• Analysis
• Conclusion

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We give a buffered algorithm with
Buffer size
Routing time
Time Steps
nodes, node congestion, dilation in region
graph
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We partition original graph so that each region
has edges
edges
Buffer size
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It holds
Edges in region
Edge congestion
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Each simulation phase has duration
duration of round
number of rounds
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Total Routing Time
Number of phases
Phase Duration
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Presentation Outiline

• Algorithm
• Analysis
• Conclusion

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We presented the first bufferless algorithm for
arbitrary networks within logarithmic factors
from optimal

• Future research
• Improve logarithmic factors
• Maintain original paths