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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

slide-2
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

slide-4
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
slide-5
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

slide-12
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

( ) 2 ( 8 ) 4 ( 1 ) 6 ( 1 1 ) 8 ( 1 2 ) 1 ( 1 3 ) 1 2 ( 1 4 ) 1 4 ( 1 5 ) 1 6 ( 1 5 ) 1 8 ( 1 6 ) 2 ( 1 7 ) 2 2 ( 1 8 ) 2 4 ( 1 9 ) 2 6 ( 2 ) 2 8 ( 2 1 ) 3 ( 2 2 ) 3 2 ( 2 4 ) 3 4 ( 2 5 ) 3 6 ( 3 )

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

6 4 8 8 1 9 1 2 1 1 1 1 4 4 1 2 1 4 4 1 2 1 6 9 1 3 1 9 6 1 4 2 2 5 1 5 2 2 5 1 5 2 5 6 1 6 2 8 9 1 7 3 2 4 1 8 3 6 1 1 9 3 6 1 1 9 4 2 4 4 1 2 1 4 8 4 2 2 5 7 6 2 4 6 7 6 2 6 9 3

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 additionally 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 and Future work

  • 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 sched-

ule search space without affecting the solution space.

  • Our method may be applied when high QoS and efficient bandwidth usage of a

media is required for continuous streaming applications such as video and voice.

  • Liquid scheduling is applicable for TDM wireless networks, optical networks or

low latency wormhole/cut-through networks, such as Myrinet. Streams can be transmitted from edge to edge in large chunks, but global synchronization in the network is required.

  • Future work: fault-tolerance of transmission by spacial diversification of routing
  • paths. Example of an underlying network: a wireless ad-hoc mobile network

seeking to provide end-to-node streaming services, such as packetized voice.

slide-24
SLIDE 24

a network of 100 nodes and 668 links

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SLIDE 25

a network of 100 nodes and 682 links

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SLIDE 26

a network of 100 nodes and 682 links

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SLIDE 27

a network of 100 nodes and 694 links

slide-28
SLIDE 28

a network of 100 nodes and 688 links

slide-29
SLIDE 29

a network of 100 nodes and 702 links

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SLIDE 30

a network of 100 nodes and 708 links

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SLIDE 31

a network of 100 nodes and 702 links

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SLIDE 32

a network of 100 nodes and 708 links

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SLIDE 33

a network of 100 nodes and 708 links

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SLIDE 34

a network of 100 nodes and 714 links

slide-35
SLIDE 35

a network of 100 nodes and 718 links

slide-36
SLIDE 36

a network of 100 nodes and 718 links

slide-37
SLIDE 37

a network of 100 nodes and 712 links

slide-38
SLIDE 38

a network of 100 nodes and 704 links

slide-39
SLIDE 39

a network of 100 nodes and 692 links

slide-40
SLIDE 40

a network of 100 nodes and 678 links

slide-41
SLIDE 41

a network of 100 nodes and 680 links

slide-42
SLIDE 42

a network of 100 nodes and 694 links

slide-43
SLIDE 43

a network of 100 nodes and 682 links

slide-44
SLIDE 44

a network of 100 nodes and 672 links

slide-45
SLIDE 45

a network of 100 nodes and 682 links

slide-46
SLIDE 46

a network of 100 nodes and 680 links

slide-47
SLIDE 47

a network of 100 nodes and 680 links

slide-48
SLIDE 48

a network of 100 nodes and 694 links

slide-49
SLIDE 49

a network of 100 nodes and 696 links

slide-50
SLIDE 50

a network of 100 nodes and 688 links

slide-51
SLIDE 51

a network of 100 nodes and 692 links

slide-52
SLIDE 52

a network of 100 nodes and 706 links

slide-53
SLIDE 53

a network of 100 nodes and 684 links

slide-54
SLIDE 54

a network of 100 nodes and 698 links

slide-55
SLIDE 55

a network of 100 nodes and 686 links

slide-56
SLIDE 56

a network of 100 nodes and 684 links

slide-57
SLIDE 57

a network of 100 nodes and 680 links

slide-58
SLIDE 58

a network of 100 nodes and 668 links

slide-59
SLIDE 59

a network of 100 nodes and 684 links

slide-60
SLIDE 60

a network of 100 nodes and 678 links

slide-61
SLIDE 61

a network of 100 nodes and 680 links

slide-62
SLIDE 62

a network of 100 nodes and 678 links

slide-63
SLIDE 63

a network of 100 nodes and 672 links

slide-64
SLIDE 64

a network of 100 nodes and 680 links

slide-65
SLIDE 65

a network of 100 nodes and 674 links

slide-66
SLIDE 66

a network of 100 nodes and 662 links

slide-67
SLIDE 67

a network of 100 nodes and 662 links

slide-68
SLIDE 68

a network of 100 nodes and 666 links

slide-69
SLIDE 69

a network of 100 nodes and 670 links

slide-70
SLIDE 70

a network of 100 nodes and 684 links

slide-71
SLIDE 71

a network of 100 nodes and 702 links

slide-72
SLIDE 72

a network of 100 nodes and 698 links

slide-73
SLIDE 73

a network of 100 nodes and 692 links

slide-74
SLIDE 74

a network of 100 nodes and 678 links

slide-75
SLIDE 75

a network of 100 nodes and 690 links

slide-76
SLIDE 76

a network of 100 nodes and 698 links

slide-77
SLIDE 77

a network of 100 nodes and 700 links

slide-78
SLIDE 78

a network of 100 nodes and 680 links

slide-79
SLIDE 79

a network of 100 nodes and 672 links

slide-80
SLIDE 80

a network of 100 nodes and 678 links

slide-81
SLIDE 81

a network of 100 nodes and 670 links

slide-82
SLIDE 82

a network of 100 nodes and 666 links

slide-83
SLIDE 83

a network of 100 nodes and 676 links

slide-84
SLIDE 84

a network of 100 nodes and 694 links

slide-85
SLIDE 85

a network of 100 nodes and 710 links

slide-86
SLIDE 86

a network of 100 nodes and 710 links

slide-87
SLIDE 87

a network of 100 nodes and 698 links

slide-88
SLIDE 88

a network of 100 nodes and 704 links

slide-89
SLIDE 89

a network of 100 nodes and 702 links

slide-90
SLIDE 90

a network of 100 nodes and 702 links

slide-91
SLIDE 91

a network of 100 nodes and 708 links

slide-92
SLIDE 92

a network of 100 nodes and 708 links

slide-93
SLIDE 93

a network of 100 nodes and 710 links

slide-94
SLIDE 94

a network of 100 nodes and 702 links

slide-95
SLIDE 95

a network of 100 nodes and 706 links

slide-96
SLIDE 96

a network of 100 nodes and 718 links

slide-97
SLIDE 97

a network of 100 nodes and 728 links

slide-98
SLIDE 98

a network of 100 nodes and 736 links

slide-99
SLIDE 99

180 nodes, clock 1, layer 12, 17 bottlenecks at load 0.0666666667 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-100
SLIDE 100

180 nodes, clock 2, layer 12, 8 bottlenecks at load 0.09375 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-101
SLIDE 101

180 nodes, clock 3, layer 12, 12 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-102
SLIDE 102

180 nodes, clock 4, layer 12, 7 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-103
SLIDE 103

180 nodes, clock 5, layer 12, 10 bottlenecks at load 0.08 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-104
SLIDE 104

180 nodes, clock 6, layer 12, 5 bottlenecks at load 0.1086419753 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-105
SLIDE 105

180 nodes, clock 7, layer 0, 0 bottlenecks at load N/A 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-106
SLIDE 106

180 nodes, clock 8, layer 12, 24 bottlenecks at load 0.0555555556 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-107
SLIDE 107

180 nodes, clock 9, layer 12, 5 bottlenecks at load 0.0653333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-108
SLIDE 108

180 nodes, clock 10, layer 12, 4 bottlenecks at load 0.059375 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-109
SLIDE 109

180 nodes, clock 11, layer 12, 2 bottlenecks at load 0.0487179487 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-110
SLIDE 110

180 nodes, clock 12, layer 12, 2 bottlenecks at load 0.08 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-111
SLIDE 111

180 nodes, clock 13, layer 12, 7 bottlenecks at load 0.03125 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-112
SLIDE 112

180 nodes, clock 14, layer 12, 2 bottlenecks at load 0.0555555556 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-113
SLIDE 113

180 nodes, clock 15, layer 12, 3 bottlenecks at load 0.049382716 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-114
SLIDE 114

180 nodes, clock 16, layer 12, 12 bottlenecks at load 0.1 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-115
SLIDE 115

180 nodes, clock 17, layer 12, 2 bottlenecks at load 0.0729166667 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-116
SLIDE 116

180 nodes, clock 18, layer 12, 4 bottlenecks at load 0.075 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-117
SLIDE 117

180 nodes, clock 19, layer 12, 8 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-118
SLIDE 118

180 nodes, clock 20, layer 12, 5 bottlenecks at load 0.0916666667 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-119
SLIDE 119

180 nodes, clock 21, layer 12, 17 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-120
SLIDE 120

180 nodes, clock 22, layer 12, 7 bottlenecks at load 0.09375 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-121
SLIDE 121

180 nodes, clock 23, layer 12, 3 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-122
SLIDE 122

180 nodes, clock 24, layer 12, 5 bottlenecks at load 0.1 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-123
SLIDE 123

180 nodes, clock 25, layer 12, 8 bottlenecks at load 0.0526406036 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-124
SLIDE 124

180 nodes, clock 26, layer 12, 3 bottlenecks at load 0.1166666667 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-125
SLIDE 125

180 nodes, clock 27, layer 12, 4 bottlenecks at load 0.0729166667 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-126
SLIDE 126

180 nodes, clock 28, layer 12, 2 bottlenecks at load 0.0933333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-127
SLIDE 127

180 nodes, clock 29, layer 12, 4 bottlenecks at load 0.09375 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-128
SLIDE 128

180 nodes, clock 30, layer 12, 5 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-129
SLIDE 129

180 nodes, clock 31, layer 12, 2 bottlenecks at load 0.1 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-130
SLIDE 130

180 nodes, clock 32, layer 12, 4 bottlenecks at load 0.0740740741 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-131
SLIDE 131

180 nodes, clock 33, layer 12, 4 bottlenecks at load 0.0740740741 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-132
SLIDE 132

180 nodes, clock 34, layer 0, 0 bottlenecks at load N/A 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-133
SLIDE 133

180 nodes, clock 35, layer 12, 2 bottlenecks at load 0.0722222222 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-134
SLIDE 134

180 nodes, clock 36, layer 12, 1 bottlenecks at load 0.0952380952 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-135
SLIDE 135

180 nodes, clock 37, layer 12, 4 bottlenecks at load 0.0615079365 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-136
SLIDE 136

180 nodes, clock 38, layer 12, 1 bottlenecks at load 0.0902777778 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-137
SLIDE 137

180 nodes, clock 39, layer 12, 4 bottlenecks at load 0.1071428571 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-138
SLIDE 138

180 nodes, clock 40, layer 12, 2 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-139
SLIDE 139

180 nodes, clock 41, layer 0, 0 bottlenecks at load N/A 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-140
SLIDE 140

180 nodes, clock 42, layer 0, 0 bottlenecks at load N/A 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-141
SLIDE 141

180 nodes, clock 43, layer 12, 3 bottlenecks at load 0.0833333333 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-142
SLIDE 142

180 nodes, clock 44, layer 12, 8 bottlenecks at load 0.0653935185 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-143
SLIDE 143

180 nodes, clock 45, layer 12, 5 bottlenecks at load 0.086 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-144
SLIDE 144

180 nodes, clock 46, layer 12, 2 bottlenecks at load 0.0785714286 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-145
SLIDE 145

180 nodes, clock 47, layer 12, 7 bottlenecks at load 0.0892857143 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-146
SLIDE 146

180 nodes, clock 48, layer 12, 2 bottlenecks at load 0.0666666667 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-147
SLIDE 147

180 nodes, clock 49, layer 12, 3 bottlenecks at load 0.0808080808 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-148
SLIDE 148

180 nodes, clock 50, layer 12, 4 bottlenecks at load 0.075 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 8e-05 6e-05 4e-05 2e-05 1e-05 8e-06 6e-06 4e-06 2e-06 1e-06 unsolved

slide-149
SLIDE 149
  • - ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton --
  • - EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --

Thank you!

Emin.Gabrielyan@epfl.ch