EFFICIENT LIQUID SCHEDULE EARCH TRATEGIES FOR C file-- ICON 2004, IEEE International Conference On...

-- 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 STRATEGIESFOR COLLECTIVE COMMUNICATIONS

Emin Gabrielyan, Roger D. Hersch

Swiss Federal Institute of Technology - Lausanne


l11

l12

l1

l10

l2 l3 l4 l5

l6 l7 l8 l9

R1 R3R2 R4 R5

T1 T2 T3 T4 T5

R1 R3R2 R4 R5

T1 T2 T3 T4 T5

Example: 25 transmissions to be carried out


Round-robin schedule

R1 R3R2 R4 R5

T1 T2 T3 T4 T5

R1 R3R2 R4 R5

T1 T2 T3 T4 T5

R1 R3R2 R4 R5

T1 T2 T3 T4 T5

R1 R3R2 R4 R5

T1 T2 T3 T4 T5

R1 R3R2 R4 R5

T1 T2 T3 T4 T5


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= =co

ngest

ion

cong

estion


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= =


R1 R3R2 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 R3R2 R4 R5

T1 T2 T3 T4 T55 5 5 5

55 5 5 5

6

6

bottleneck

s

Transfers and Load of Links


l11

l12

l1

l10

l2 l3 l4 l5

l6 l7 l8 l9

R1 R3R2 R4 R5

T1 T2 T3 T4 T5λ 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




X=

traffic’s duration (the load of its bottlenecks)

total number of transfersthe throughput of a single link

Tliquid# X( )Λ X( )------------ Tlink⋅ 25

6------ 1Gbps⋅ 4.17Gbps= = =

Liquid Throughput


Schedules yielding the liquid throughput



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.


0

2

4

56

3

7

1

PR63PR00

PR02

PR04

PR06

PR08PR10

PR12

PR14

PR16

PR18

PR20 PR22 PR24PR26

PR28

PR30

PR32

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PR46

PR48

PR50PR

52PR54

PR56PR58

PR60

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PR51

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PR47

PR45 PR

43 PR41

PR39PR37

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PR25

PR23PR21

PR19

PR17

PR15

PR13PR11PR09PR07PR05

PR03

PR01

N00

N01

N02

N03

N04N05

N06 N07 N08 N09N10

N11

N12N13

N14

N15

N16

N17

N18N19

N20N21

N22N23N24N25N26

N27N28

N29

N30

N31

Swiss-T1 Cluster

Node

Switch

Rx Proc

Tx Proc

Routing

Link

N00

0

PR01

PR00


0200400600800

10001200140016001800

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Number of contributing nodes

Liqu

id th

roug

hput

(MB

/s)

363 Communication Patterns


0

400

800

1200

1600

2000

2400

2800

0 (0

)20

(8)

40 (1

0)60

(11)

80 (1

2)10

0 (1

3)12

0 (1

4)14

0 (1

5)16

0 (1

5)18

0 (1

6)20

0 (1

7)22

0 (1

8)24

0 (1

9)26

0 (2

0)28

0 (2

1)30

0 (2

2)32

0 (2

4)34

0 (2

5)36

0 (3

0)

Topology (contributing nodes)

Agg

rega

te th

roug

hput

(MB/

s)

363 Topology Test-bed

Crossbar throughput

Liquid throughput


0200400600800

10001200140016001800

0 0

064

08

81 0

912

1 1

114

4 1

214

4 1

216

9 1

319

6 1

422

5 1

522

5 1

525

6 1

628

9 1

732

4 1

836

1 1

936

1 1

940

0 2

044

1 2

148

4 2

257

6 2

467

6 2

690

0 3

0

Transfers / Contributing nodes

Thro

ughp

ut (M

B/s)

theoretical liquid measured round-robin

Round-robin 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 =

{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 timeframesload 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


R=

R x= R x=

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

{ }excluder includer

depot


depot


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 of 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.


...

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


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)=5A2 bottlenecks 2 bottlenecks 4 bottlenecks 4 bottlenecks 6 bottlenecks 8 bottlenecks

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

A1,... A1,... A1,...

Additional bottlenecks


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.


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 spaceChoice ℘ 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⊂{ }⊂ original traffic’s teams formed from the reduced traffic

teams of the reduced traffic


0200400600800

100012001400160018002000

0 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-a

ll th

roug

hput

(MB

/s)

liquid throughput carried out according to the liquid schedules

Results


x1,1

x2,1

x3,1 x4,1

x5,1

x1,2

x2,2

x3,2 x4,2

x1,3

x2,3

x3,3 x4,3

x1,4

x2,4

x3,4 x4,4

x5,4

x1,5

x4,5

x5,5

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 R3R2 R4 R5

T1 T2 T3 T4 T5

5

R1 R3R2 R4 R5

T1 T2 T3 T4 T55 5 5 5

55 5 5 5

6

6

bottleneck

s

Congestion Graph


0

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 p

erfo

rman

ce (%

)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%.


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: [email protected]

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