efficient liquid schedule earch trategies for c file-- icon 2004, ieee international conference on...
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-- 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
-- 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 R3R2 R4 R5
T1 T2 T3 T4 T5
R1 R3R2 R4 R5
T1 T2 T3 T4 T5
Example: 25 transmissions to be carried out
-- 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 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
-- 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= =co
ngest
ion
cong
estion
-- 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= =
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
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
-- 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 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
-- 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 transfersthe throughput of a single link
Tliquid# X( )Λ X( )------------ Tlink⋅ 25
6------ 1Gbps⋅ 4.17Gbps= = =
Liquid Throughput
-- 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.
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
0
2
4
56
3
7
1
PR63PR00
PR02
PR04
PR06
PR08PR10
PR12
PR14
PR16
PR18
PR20 PR22 PR24PR26
PR28
PR30
PR32
PR34
PR36
PR38
PR40
PR42
PR44
PR46
PR48
PR50PR
52PR54
PR56PR58
PR60
PR62
PR61
PR59
PR57
PR55
PR53
PR51
PR49
PR47
PR45 PR
43 PR41
PR39PR37
PR35
PR33
PR31
PR29
PR27
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
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
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
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
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
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
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
-- 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 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
-- 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 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.
-- 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
-- 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)=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
-- 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.
-- 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 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
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
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
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
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
-- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton ---- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
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%.
-- 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: [email protected]