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ICON 2004 - IEEE International Conference On Networks November 16-19, 2004, Singapore, Hilton E FFICIENT L IQUID S CHEDULE S EARCH S TRATEGIES FOR C OLLECTIVE C OMMUNICATIONS Emin Gabrielyan, Roger D. Hersch Swiss Federal Institute of Technology

slide-1
SLIDE 1
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

ICON 2004 - IEEE International Conference On Networks November 16-19, 2004, Singapore, Hilton

EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES

FOR COLLECTIVE COMMUNICATIONS Emin Gabrielyan, Roger D. Hersch Swiss Federal Institute of Technology - Lausanne

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SLIDE 2
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

l11 l12 l1 l10 l2 l3 l4 l5 l6 l7 l8 l9

R1 R3 R2 R4 R5 T1 T2 T3 T4 T5 R1 R3 R2 R4 R5 T1 T2 T3 T4 T5

Example: 25 transmissions to be carried out

slide-3
SLIDE 3
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

Round-robin schedule

R1 R3 R2 R4 R5 T1 T2 T3 T4 T5 R1 R3 R2 R4 R5 T1 T2 T3 T4 T5 R1 R3 R2 R4 R5 T1 T2 T3 T4 T5 R1 R3 R2 R4 R5 T1 T2 T3 T4 T5 R1 R3 R2 R4 R5 T1 T2 T3 T4 T5

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SLIDE 4
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

phase 1 phase 2 phase 3.1 phase 3.2 phase 4.1 phase 4.2

Round-robin Throughput

phase 5

Troundrobin 25 7 ⁄ 1Gbps ⋅ 3.57Gbps = =

c

  • n

g e s t i

  • n

c

  • n

g e s t i

  • n
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SLIDE 5
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

time frame 1 time frame 2 time frame 3 time frame 4 time frame 5 time frame 6

Liquid schedule

Tliquid 25 6 ⁄ 1Gbps ⋅ 4.16Gbps = =

slide-6
SLIDE 6
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

R1 R3 R2 R4 R5 T1 T2 T3 T4 T5

The 25 transfer traffic

X =

λ l1 X , ( ) 5 = …λ l12 X , ( ) 6 = , l1 l6 , { } … l1 l12 l9 , , { } … , ,

Transfers:

5

R1 R3 R2 R4 R5 T1 T2 T3 T4 T5

5 5 5 5 5 5 5 5 5 6 6

b

  • t

t l e n e c k s

Transfers and Load of Links

slide-7
SLIDE 7
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

l11 l12 l1 l10 l2 l3 l4 l5 l6 l7 l8 l9

R1 R3 R2 R4 R5 T1 T2 T3 T4 T5

λ l1 X , ( ) 5 = …λ l10 X , ( ) 5 = , λ l11 X , ( ) 5 = …λ l12 X , ( ) 6 = ,

{l1, l6}, {l1, l7}, {l1, l8}, {l1, l12, l9}, {l1, l12, l10}, {l2, l6}, {l2, l7}, {l2, l8}, {l2, l12, l9}, {l2, l12, l10}, {l3, l6}, {l3, l7}, {l3, l8}, {l3, l12, l9}, {l3, l12, l10}, {l4, l11, l6}, {l4, l11, l7}, {l4, l11, l8}, {l4, l9}, {l4, l10}, {l5, l11, l6}, {l5, l11, l7}, {l5, l11, l8}, {l5, l9}, {l5, l10}

X=

Λ X ( ) 6 =

Duration of Traffic

slide-8
SLIDE 8
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

{l1, l6}, {l1, l7}, {l1, l8}, {l1, l12, l9}, {l1, l12, l10}, {l2, l6}, {l2, l7}, {l2, l8}, {l2, l12, l9}, {l2, l12, l10}, {l3, l6}, {l3, l7}, {l3, l8}, {l3, l12, l9}, {l3, l12, l10}, {l4, l11, l6}, {l4, l11, l7}, {l4, l11, l8}, {l4, l9}, {l4, l10}, {l5, l11, l6}, {l5, l11, l7}, {l5, l11, l8}, {l5, l9}, {l5, l10}

X=

traffic’s duration (the load of its bottlenecks) total number of transfers the throughput of a single link

Tliquid # X ( ) Λ X ( )

  • Tlink

⋅ 25 6

  • 1Gbps

⋅ 4.17Gbps = = =

Liquid Throughput

slide-9
SLIDE 9
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

Schedules yielding the liquid throughput

{l1, l6}, {l1, l7}, {l1, l8}, {l1, l12, l9}, {l1, l12, l10}, {l2, l6}, {l2, l7}, {l2, l8}, {l2, l12, l9}, {l2, l12, l10}, {l3, l6}, {l3, l7}, {l3, l8}, {l3, l12, l9}, {l3, l12, l10}, {l4, l11, l6}, {l4, l11, l7}, {l4, l11, l8}, {l4, l9}, {l4, l10}, {l5, l11, l6}, {l5, l11, l7}, {l5, l11, l8}, {l5, l9}, {l5, l10}

X=

  • Without a right schedule we may have intervals when

the access to the bottleneck links is blocked by other transmissions.

  • Our goal is to schedule the transfers such that all bot-

tlenecks are always kept occupied ensuring that the liquid throughput is obtained.

  • A schedule yielding the liquid throughput we call as a

liquid schedule and our objective is to find a liquid schedule whenever it exists.

slide-10
SLIDE 10
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

2 4 5 6 3 7 1 PR63 PR00 PR02 PR04 PR06 PR08 PR10 PR12 P R 1 4 PR16 PR18 PR20 PR22 PR24 PR26 PR28 P R 3 PR32 PR34 PR36 PR38 PR40 PR42 PR44 P R 4 6 PR48 PR50 PR52 PR54 PR56 PR58 PR60 P R 6 2 PR61 PR59 PR57 PR55 PR53 PR51 P R 4 9 PR47 PR45 PR43 PR41 PR39 PR37 PR35 P R 3 3 PR31 PR29 PR27 PR25 PR23 PR21 PR19 P R 1 7 PR15 PR13 PR11 PR09 PR07 PR05 PR03 P R 1 N00 N01 N02 N03 N04 N 5 N 6 N07 N08 N 9 N10 N11 N12 N 1 3 N 1 4 N15 N16 N 1 7 N18 N19 N20 N 2 1 N 2 2 N23 N24 N 2 5 N26 N27 N28 N29 N30 N 3 1

Swiss-T1 Cluster

Node Switch Rx Proc Tx Proc Routing Link

N00 PR01 PR00

slide-11
SLIDE 11
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

200 400 600 800 1000 1200 1400 1600 1800 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Number of contributing nodes Liquid throughput (MB/s)

363 Communication Patterns

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SLIDE 12
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

400 800 1200 1600 2000 2400 2800

0 (0) 20 (8) 40 (10) 60 (11) 80 (12) 100 (13) 120 (14) 140 (15) 160 (15) 180 (16) 200 (17) 220 (18) 240 (19) 260 (20) 280 (21) 300 (22) 320 (24) 340 (25) 360 (30)

Topology (contributing nodes) Aggregate throughput (MB/s)

363 Topology Test-bed

Crossbar throughput Liquid throughput

slide-13
SLIDE 13
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

200 400 600 800 1000 1200 1400 1600 1800

0 00 64 08 81 09 121 11 144 12 144 12 169 13 196 14 225 15 225 15 256 16 289 17 324 18 361 19 361 19 400 20 441 21 484 22 576 24 676 26 900 30

Transfers / Contributing nodes Throughput (MB/s) theoretical liquid measured round-robin

Round-robin throughput

slide-14
SLIDE 14
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

{l1, l6}, {l1, l7}, {l1, l8}, {l1, l12, l9}, {l1, l12, l10}, {l2, l6}, {l2, l7}, {l2, l8}, {l2, l12, l9}, {l2, l12, l10}, {l3, l6}, {l3, l7}, {l3, l8}, {l3, l12, l9}, {l3, l12, l10}, {l4, l11, l6}, {l4, l11, l7}, {l4, l11, l8}, {l4, l9}, {l4, l10}, {l5, l11, l6}, {l5, l11, l7}, {l5, l11, l8}, {l5, l9}, {l5, l10}

X =

{l1, l7}, {l2, l8}, {l3, l12, l9}, {l5, l11, l6} {l1, l6}, {l2, l12, l10}, {l3, l7}, {l4, l11, l8} {l3, l12, l10}, {l4, l9}, {l5, l11, l8}

} }

}

{ {

{

, ,

{l1, l12, l9}, {l2, l7}, {l3, l8}, {l4, l11, l6}, {l5, l10} {l1, l12, l10}, {l2, l6}, {l4, l11, l7}, {l5, l9} {l1, l8}, {l2, l12, l9}, {l3, l6}, {l4, l10}, {l5, l11, l7}

}

}{

} {

{

, , ,

α =

number of timeframes load of the bottlenecks

A α ∈ ( ) ∀ ⇔ # α ( ) Λ X ( ) = ⇔ ⇔ schedule α is liquid ⇔ A is a team of X

Team: a set of mutually non-congesting transfers using all bottlenecks

slide-15
SLIDE 15
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

R= R x= R x=

  • transfer x
  • transfers congesting with x
  • transfers non-congesting with x

{ }

excluder includer depot

{ }

excluder includer depot

{ }

excluder includer depot

ℑ X ( ) all teams of the traffic X ,

  • To cover the full solution space when

constructing a liquid schedule an effi- cient technique obtaining the whole set

  • f possible teams of a traffic is required.
  • We designed an efficient algorithm enu-

merating all teams of a traffic traversing each team once and only once.

  • This algorithm obtains each team by

subsequent partitioning of the set of all teams.

  • We introduced tri-

plets consisting of subsets of the traf- fic, representing one- by-one partitions of the set of all teams.

slide-16
SLIDE 16
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

...

X ℘ X ( ) A1 A2 A3…An , , { } = → X1 X A1 – = ℘ X1 ( ) A1 1

,

A1 2

, …

, { } = → X1 1

,

X1 A1 1

,

– = X1 2

,

X1 A1 2

,

– = X2 X A2 – = ℘ X2 ( ) A2 1

,

A2 2

, …

, { } = → X2 1

,

X2 A2 1

,

– = X2 2

,

X2 A2 2

,

– = ℘ Y ( ) A ℑ X ( ) ∈ A Y ⊂ { } =

all teams of X possible steps to the next layer

Liquid schedule search tree

slide-17
SLIDE 17
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

A1,1 A1,1,1 X (25 transfers) X1 = X - A1 (20 transfers) X1,1 = X1 - A1,1 (16 transfers) A1 A(X)=6 (X1)=5 A

2 bottlenecks 2 bottlenecks 4 bottlenecks 4 bottlenecks 6 bottlenecks 8 bottlenecks

(X1,...)=3 A (X1,...)=2 A (X1,...)=1 A (X1,1)=4 A A1,... A1,... A1,...

Additional bottlenecks

slide-18
SLIDE 18
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

Prediction of dead-ends and search optimization

  • When a team of transfers is carried out - for the remaining traffic we

have the same bottleneck links as before - with possibly new addition- ally emerged bottleneck links.

  • Considering new bottleneck links (at every step of construction) in the

choice of the further teams substantially reduces the search space.

  • Team is a collection of simultaneous transmissions using all bottle-

necks of the network. Teams are full if they congest with all other transmissions of the traffic.

  • Limiting our choice with only full teams reduces the search space

without affecting the solution space.

slide-19
SLIDE 19
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

For more than 90% of the test-bed topologies construction of a global liquid schedule is completed in a fraction of a second (less than 0.1s).

additionally decreas- ing the search space without affecting the solution space

Choice ℘ Y ( ) ℑfull Y ( ) = = ℑfull Y ( ) ℑ Y ( ) ⊂

full teams of the reduced traffic

{

Liquid schedules construction

Choice ℘ Y ( ) ℑ Y ( ) = = Choice ℘ Y ( ) A ℑ X ( ) ∈ A Y ⊂ { } = ℘ Y ( ) ℑ Y ( ) = → = ℑ Y ( ) A ℑ X ( ) ∈ A Y ⊂ { } ⊂

  • riginal traffic’s teams formed

from the reduced traffic teams of the reduced traffic

slide-20
SLIDE 20
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

200 400 600 800 1000 1200 1400 1600 1800 2000 8 10 12 13 14 15 16 17 18 19 21 22 24 27 Number of contributing nodes for the 363 sub-topologies All-to-all throughput (MB/s) liquid throughput carried out according to the liquid schedules

Results

slide-21
SLIDE 21
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

x

1 , 1

x

2 , 1

x

3 , 1

x

4 , 1

x

5 , 1

x

1 , 2

x

2 , 2

x

3 , 2

x

4 , 2

x

1 , 3

x

2 , 3

x

3 , 3

x

4 , 3

x

1 , 4

x

2 , 4

x

3 , 4

x

4 , 4

x

5 , 4

x

1 , 5

x

4 , 5

x

5 , 5

The 25 vertices of the graph represent the 25 transfers

  • transfers. The edges repre-

sent congestion relations be- tween transfers, i.e. each edge represents one or more communication links shared by two transfers.

Bold edges represent all conges- tions due to bottleneck links

R1 R3 R2 R4 R5 T1 T2 T3 T4 T5

5

R1 R3 R2 R4 R5 T1 T2 T3 T4 T5

5 5 5 5 5 5 5 5 5 6 6

bottlenecks

Congestion Graph

slide-22
SLIDE 22
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

5 10 15 20 1 49 64 81 100 100 121 144 144 144 169 169 196 196 225 225 225 256 256 289 289 324 324 324 361 361 400 400 441 484 484 529 576 576 676 729 961 number of transfers for each of 363 topologies loss in performance (%)

Loss of performance induced by schedules com- puted with a graph colouring heuristic algorithm

  • For 74% of the topologies Dsatur algorithm does not induce a loss of performance.
  • For 18% of topologies, the performance loss is bellow 10%.
  • For 8% of topologies, the loss of performance is between 10% and 20%.
slide-23
SLIDE 23
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

Conclusion

  • Data exchanges relying on the liquid schedules may be carried out several

times faster compared with topology-unaware schedules.

  • Thanks to introduced theoretical model we considerably reduce the liquid

schedule search space without affecting the solution space.

  • Our method may be applied to applications requiring efficiency in concurrent

continuous transmissions, such as video and voice traffic management, high energy physics data acquisition.

  • Liquid scheduling is applicable in wormhole, cut-through and WDM optical

networks.

Thank You!

Contact: Emin.Gabrielyan@epfl.ch