on Leveled Networks Costas Busch Shailesh Kelkar Malik - - PowerPoint PPT Presentation

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Efficient Bufferless Routing on Leveled Networks Costas Busch Shailesh Kelkar Malik Magdon-Ismail Rensselaer Polytechnic Institute 1 Talk Outline Introduction Centralized Algorithm Distributed Algorithm Conclusion 2 Leveled

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1

Efficient Bufferless Routing

  • n Leveled Networks

Costas Busch Shailesh Kelkar Malik Magdon-Ismail

Rensselaer Polytechnic Institute

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2

  • Introduction
  • Centralized Algorithm
  • Distributed Algorithm
  • Conclusion

Talk Outline

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3

Leveled Networks

Level:

1 2 3 L-1 L

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4

Examples of Leveled Networks

1 2 3

Butterfly

1 2 3 4 5 6

Mesh

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5

  • Synchronous network (time steps)

Network Model

  • Bi-directional links

One packet per direction, per time step

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6

Buffer-less nodes Time 0 Packets are always moving

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7

Time 1 Buffer-less nodes Packets are always moving

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8

Time 2 Buffer-less nodes Packets are always moving

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9

Time 3 Buffer-less nodes Packets are always moving

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10

Time 4 Buffer-less nodes Packets are always moving

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11

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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12

Routing Time: the time until the last packet is absorbed Objective: Minimize Routing Time

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13

source destination

Each packet has a pre-selected path Packet path is from left to right

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14

source destination

The packet follows the pre-selected path

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15

source destination

The packet follows the pre-selected path

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16

source destination

The packet follows the pre-selected path

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17

Each packet has its own path There are packets

N

3 = N

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18

Dilation D: The maximum length

  • f any path

6 = D

Routing time:

) D ( Ω

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19

Congestion C: The maximum number of packets traversing any edge

2  C

Routing time:

) C ( Ω

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20

Lower bound on Routing Time:

) D + C ( Ω

D C 

We want algorithms with Routing Time close to: Congestion Dilation

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21

Our Contributions

  • Centralized Algorithm:

) ) log( ( D DN C O  

  • Distributed Algorithm:

)) log( ) ( log (

2

DN D DN C O   

N : number of packets

Results hold with high probability

       DN 1 1

Both algorithms are randomized

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22

Networks with buffers Leveled networks:

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

) N log + L + C ( O

Arbitrary networks:

) D + C ( O

[Leighton - Maggs - Rao, Combinatorica 94]

[BS99, LMR99, MV99, OR97, RT96]

Related Work

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23

Bufferless networks

Mesh [BRST93, BES97, BHS98, BU96, BHW00] Hypercube [BH85, BC95, FR92, H91] Trees [BMMW04, RSW00, BMMW] Leveled [BBPRRS96, B02] Vertex-symmetric [MS95] Arbitrary networks [BMM04]

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24

Most related work Leveled Networks: Arbitrary networks:

Busch, Magdon-Ismail, Mavronicolas WAOA’04

)) ( log ) ((

3

N n D C O    )) ( log ) ((

9 LN

D C O  

Busch TCS’04

Leveled networks in different routing model:

Bhatt, Bilardi, Pucci, Ranade, Rosenberg, Scwabe TC’96

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25

  • Introduction
  • Centralized Algorithm
  • Distributed Algorithm
  • Conclusion

Talk Outline

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26

Centralized Algorithm

A central node knows all the parameters of the problem and computes a packet schedule

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27

5  D

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28

Packet Grouping A

Group 1 Group 2 Group 3

D 2

Group 4 Group 5

D 2

Packet Grouping B

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29

Packets in different groups can be sent simultaneously

Group 1 Group 2 Group 3

Group 1

We focus only in one group

  • 1. Send packets of Grouping A
  • 2. Send packets of Grouping B

Packet Grouping A

Increases routing time by only a factor of 2

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30

Group 1

D 2

 set of packets

Congestion Dilation

C  D 

N   | |

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31

Partition the packets randomly and uniformly into sets

C

1

packets

C | | 

#of packets

2

3

C

C | |  C | |  C | | 

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32

C

1

packets

) log(DN

2

3

C

Congestion Benefit: congestion drops

) log(DN ) log(DN ) log(DN

New Congestion w.h.p.

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33

Before partitioning Congestion C Edge

   

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34

After partitioning

1

2

3

C

Expected one packet from each packet set

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35

1

Expected Congestion 1 (Congestion w.h.p.)

) log(DN

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36 1

F

2

F

3

F

k

F

We partition the levels into frames

)) (log(DN 

D 2

# number of frames:

         ) log(DN D

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37 1

F

2

F

3

F

k

F

We send packets from frame to frame

1

Wave

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38 1

F

2

F

3

F

k

F

We send packets from frame to frame

1

Wave Wave Duration:

)) (log(DN 

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39 1

F

2

F

3

F

k

F

We send packets from frame to frame

1

Wave

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40 1

F

2

F

3

F

k

F

We send packets from frame to frame

1

Wave

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41 1

F

2

F

3

F

k

F

We send packets from frame to frame

1

Wave

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42

A packet follows its path from source to Destination along the wave

Injection wave

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43

A packet follows its path from source to Destination along the wave

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A packet follows its path from source to Destination along the wave

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45

A packet follows its path from source to Destination along the wave

Absorption wave

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46

A packet follows its path from source to Destination along the wave

Absorption wave

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47

Sending packets of different packet sets simultaneously

Wave 1

1

2

3

C

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48

Sending packets of different packet sets simultaneously

Wave 1

1

2

3

C

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49

Sending packets of different packet sets simultaneously

Wave 1

1

2

3

C

Wave 2

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50

Sending packets of different packet sets simultaneously

Wave 1

1

2

3

C

Wave 2

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51

Sending packets of different packet sets simultaneously

Wave 1

1

2

3

C

Wave 2 Wave 3

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52

Sending packets of different packet sets simultaneously

1

2

3

C

Wave 1 Wave 2 Wave 3

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53

Sending packets of different packet sets simultaneously

1

2

3

C

Wave 2 Wave 3 Wave C

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54

Sending packets of different packet sets simultaneously

1

2

3

C

Wave 2 Wave 3 Wave C

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55

Sending packets of different packet sets simultaneously

1

2

3

C

Wave 3 Wave C

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56

Sending packets of different packet sets simultaneously

1

2

3

C

Wave 3 Wave C

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57

Sending packets of different packet sets simultaneously

1

2

3

C

Wave C

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58

Sending packets of different packet sets simultaneously

1

2

3

C

Wave C

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59

Sending packets of different packet sets simultaneously

1

2

3

C

Wave C

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60

1

2

3

C

All packets have been absorbed!

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61

Routing Time = Time until last wave C leaves the network

) ) log( ( ) log( ) log( ) log( 2 D DN C DN DN D DN C    

Time when wave C enters the network Time that wave C needs to traverse the network

#frames

Wave duration Wave duration

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62

Frame

i

F

Oscilation Time t Simulates buffering

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63

Frame

i

F

Oscilation Time

1  t

Simulates buffering

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64

Frame

i

F

Oscilation Time

2  t

Simulates buffering

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65

Frame

i

F

Oscilation Time

 , 4 , 3   t t

Simulates buffering

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66

Wave Packet propagation during a wave Frame

i

F

Frame

1  i

F

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67

Frame

i

F

Frame

1  i

F

Wave Packet propagation during a wave

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68

Conflict Graph

Each node is a packet Two packets are adjacent if their paths use a common edge in the frames and

i

F

1  i

F

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69

Frame

i

F

Frame

1  i

F

Share edge

Conflict graph

a

b

c a

b

c

Example

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70

The degree of any node is bounded by

) log(DN

Thus the conflict graph can be colored with

1 ) log(   DN 

colors

(A consequence of packet partitioning) 1 1 1 1 2 2 2 2 3 3

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71

Wave Frame

i

F

Frame

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

We send packets of each color seperately

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72

Wave Frame

i

F

Frame

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

First send packets of color 1

1 1 1 1 1

Packet paths don’t conflict Time needed:

)) (log(DN 

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73

Wave Frame

i

F

Frame

1  i

F

2 2 2 2 3 3 3

Similarly, send packets of color 2

1 1 1 1 1

Packet paths don’t conflict

2 2 2 2

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74

Wave Frame

i

F

Frame

1  i

F

3 3 3

Similarly, send packets of color 3

1 1 1 1 1

Packet paths don’t conflict

2 2 2 2 3 3 3

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75

Wave Frame

i

F

Frame

1  i

F

All packets have been delivered

1 1 1 1 1 2 2 2 2 3 3 3

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76

Wave time: Colors X 2 Frame size

)) ( (log ) log( 2 ) log(

2 DN

DN DN   

We can speed up the process by pipelining different colors:

)) (log(DN 

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77

i

F

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Frame Frame

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78

i

F

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Frame Frame

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79

i

F

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Frame Frame

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80

i

F

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

deflected

Frame Frame

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81

i

F

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Boat 2

Back In position

Frame Frame

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82

i

F

1  i

F

1 1 1 1 1 2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Boat 2

Frame Frame

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83

i

F

1  i

F

2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Boat 2

1 1 1 1 1

Frame Frame

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84

i

F

1  i

F

2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Boat 2

1 1 1 1 1

Frame Frame

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85

i

F

1  i

F

2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Boat 2

1 1 1 1 1

Boat 3

Frame Frame

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86

i

F

1  i

F

2 2 2 2 3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

Boat 2

1 1 1 1 1

Boat 3

Frame Frame

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87

i

F

1  i

F

3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

2 2 2 2

Boat 2

1 1 1 1 1

Boat 3

Frame Frame

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88

i

F

1  i

F

3 3 3

Pipelining using Boats

Boat 1

Packets of color follow boat

i i

2 2 2 2

Boat 2

1 1 1 1 1

Boat 3

Frame Frame

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89

i

F

1  i

F

3 3 3

Pipelining using Boats Packets of color follow boat

i i

2 2 2 2

Boat 2

1 1 1 1 1

Boat 3

Frame Frame

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90

i

F

1  i

F

3 3 3

Pipelining using Boats Packets of color follow boat

i i

2 2 2 2

Boat 2

1 1 1 1 1

Boat 3

Frame Frame

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91

i

F

1  i

F

Pipelining using Boats Packets of color follow boat

i i

2 2 2 2 1 1 1 1 1 3 3 3

Boat 3

Frame Frame

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92

i

F

1  i

F

Pipelining using Boats Packets of color follow boat

i i

2 2 2 2 1 1 1 1 1 3 3 3

Boat 3

Frame Frame

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93

i

F

1  i

F

Pipelining using Boats Packets of color follow boat

i i

2 2 2 2 1 1 1 1 1 3 3 3

Frame Frame

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94

Wave time: Time until last boat reaches target level

)) (log( ) log( ) log( 4 ) log( 2 4 DN DN DN DN      

Number of colors Frame size

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95

  • Introduction
  • Centralized Algorithm
  • Distributed Algorithm
  • Conclusion

Talk Outline

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96

In the distributed version, we assume that every node knows parameters

N D C , ,

Nodes do not know the packet paths, except for the packets in them.

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97

The distributed algorithm is the same with the centralized, except for one thing: The conflict graph is colored in a distributed manner

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98

Distributed coloring – Basic Idea

  • 1. each packet chooses a random color

Between 0 and 2log(DN)

During a wave:

  • 2. each packet assumes the color is

correct and follows the respective boat

  • 3. For packets that conflict, the

process repeats

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99

This process repeats a logarithmic number of times, thus it gives an extra logarithmic factor in the performance

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100

  • Introduction
  • Centralized Algorithm
  • Distributed Algorithm
  • Conclusion

Talk Outline

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101

Conclusion

  • Centralized algorithm
  • Distributed Algorithm

Uses conflict graph coloring

) ) log( ( D DN C O  

Worse by a logarithmic factor Implements distributed coloring