PCC Conference on Pervasive Computing and Communications

Transcription

1 PCC Conference on Pervasive Computing and Communications The 2004 International Multiconference in Computer Science & Computer Engineering June 21-24, 2004, Las Vegas, Nevada, USA, Monte Carlo Resort Liquid Schedule Searching Strategies for the Optimization of Collective Network Communications Emin Gabrielyan, Roger D. Hersch Swiss Federal Institute of Technology Lausanne

2 The 2004 International Multiconference in Computer Science & Computer Engineering Conference on Pervasive Computing and Communications (PCC'04) Monte Carlo Resort, Las Vegas, Nevada, USA, June 21-24, 2004 Liquid Schedule Searching Strategies for the Optimization of Collective Network Communications Emin Gabrielyan, Roger D. Hersch Swiss Federal Institute of Technology Lausanne

3 25-transmission request T1 T2 T3 T4 T5 T1 T2 T3 T4 T5 l 1 l 2 l 3 l 4 l 5 l 11 l 12 l 6 l 7 l 8 l 9 l 10 R1 R3 R2 R4 R5 R1 R3 R2 R4 R5

4 Round-robin schedule T1 T2 T3 T4 T5 T1 T2 T3 T4 T5 R1 R2 R3 R4 R5 R1 R2 R3 R4 R5 T1 T2 T3 T4 T5 T1 T2 T3 T4 T5 T1 T2 T3 T4 T5 R1 R2 R3 R4 R5 R1 R2 R3 R4 R5 R1 R2 R3 R4 R5

5 Round-robin Throughput phase 1 phase 3.1 phase 4.1 phase 5 phase 2 phase 3.2 phase 4.2 T roundrobin = Gbps = 3.57Gbps

6 Liquid schedule time frame 2 time frame 2 time frame 2 time frame 2 time frame 2 time frame 2 T liquid = Gbps = 4.16Gbps

7 Transfers and Load of Links X = T1 T2 T3 T4 T5 T1 T2 T3 T4 T bottlenecks R1 R2 R3 R4 R5 The 25 transfer traffic R1 R2 R3 R4 R5 λ( l 1, X) = 5, λ( l 12, X) = 6 Transfers: { l 1, l 6 }, { l 1, l 12, l 9 },

8 Duration of Traffic T1 T2 T3 T4 T5 l 1 l 2 l 3 l 4 l 5 l 11 l 12 l 6 l 7 l 8 l 9 R1 l 10 R2 R3 R4 R5 λ( l 1, X) = 5, λ( l 10, X) = 5 λ( l 11, X) = 5, λ( l 12, X) = 6 Λ( X) = 6 X= {l 1, l 6 }, {l 1, l 7 }, {l 1, l 8 }, {l 1, l 12, l 9 }, {l 1, l 12, l 10 }, {l 2, l 6 }, {l 2, l 7 }, {l 2, l 8 }, {l 2, l 12, l 9 }, {l 2, l 12, l 10 }, {l 3, l 6 }, {l 3, l 7 }, {l 3, l 8 }, {l 3, l 12, l 9 }, {l 3, l 12, l 10 }, {l 4, l 11, l 6 }, {l 4, l 11, l 7 }, {l 4, l 11, l 8 }, {l 4, l 9 }, {l 4, l 10 }, {l 5, l 11, l 6 }, {l 5, l 11, l 7 }, {l 5, l 11, l 8 }, {l 5, l 9 }, {l 5, l 10 }

9 Liquid Throughput X= {l 1, l 6 }, {l 1, l 7 }, {l 1, l 8 }, {l 1, l 12, l 9 }, {l 1, l 12, l 10 }, {l 2, l 6 }, {l 2, l 7 }, {l 2, l 8 }, {l 2, l 12, l 9 }, {l 2, l 12, l 10 }, {l 3, l 6 }, {l 3, l 7 }, {l 3, l 8 }, {l 3, l 12, l 9 }, {l 3, l 12, l 10 }, {l 4, l 11, l 6 }, {l 4, l 11, l 7 }, {l 4, l 11, l 8 }, {l 4, l 9 }, {l 4, l 10 }, {l 5, l 11, l 6 }, {l 5, l 11, l 7 }, {l 5, l 11, l 8 }, {l 5, l 9 }, {l 5, l 10 } the throughput of a single link total number of transfers T liquid #( X) Λ ( X) T 25 = = Gbps = link Gbps traffic s duration (the load of its bottlenecks)

10 Schedules yielding the liquid throughput X= {l 1, l 6 }, {l 1, l 7 }, {l 1, l 8 }, {l 1, l 12, l 9 }, {l 1, l 12, l 10 }, {l 2, l 6 }, {l 2, l 7 }, {l 2, l 8 }, {l 2, l 12, l 9 }, {l 2, l 12, l 10 }, {l 3, l 6 }, {l 3, l 7 }, {l 3, l 8 }, {l 3, l 12, l 9 }, {l 3, l 12, l 10 }, {l 4, l 11, l 6 }, {l 4, l 11, l 7 }, {l 4, l 11, l 8 }, {l 4, l 9 }, {l 4, l 10 }, {l 5, l 11, l 6 }, {l 5, l 11, l 7 }, {l 5, l 11, l 8 }, {l 5, l 9 }, {l 5, l 10 } 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 bottlenecks 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.

11 Swiss-T1 Cluster N07 N09 N08 N06 N10 PR16 PR20 PR19 PR18 PR17 PR15 PR14 N05 N11 PR13 PR12 PR22 PR21 PR11 N04 PR10 PR23 PR09 PR24 PR08 N03 N12 PR25 PR07 PR26 PR05 PR06 N13 N02 PR27 PR04 PR28 PR03 N01 N00 N14 PR29 PR02 PR30 PR01 N15 PR31 PR00 N00 0 PR63 PR32 N16 N31 PR62 PR33 PR61 PR34 PR01 N17 PR60 PR35 N30 PR36 PR59 PR37 PR58 N18 PR38 PR57 N29 PR00 Node Switch Rx Proc Tx Proc PR39 PR56 N19 PR40 PR41 PR55 N28 Routing PR42 PR54 N20 PR43 PR44 PR46 PR52 PR53 PR45 PR47 N21 PR48 PR49 PR50 PR51 N27 N26 Link N22 N23 N25 N24

12 363 Communication Patterns Liquid throughput (MB/s) Number of contributing nodes

13 363 Topology Test-bed Crossbar throughput Liquid throughput Aggregate throughput (MB/s) 0 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)

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

15 Team: a set of mutually non-congesting transfers using all bottlenecks X = {l 1, l 6 }, {l 1, l 7 }, {l 1, l 8 }, {l 1, l 12, l 9 }, {l 1, l 12, l 10 }, {l 2, l 6 }, {l 2, l 7 }, {l 2, l 8 }, {l 2, l 12, l 9 }, {l 2, l 12, l 10 }, {l 3, l 6 }, {l 3, l 7 }, {l 3, l 8 }, {l 3, l 12, l 9 }, {l 3, l 12, l 10 }, {l 4, l 11, l 6 }, {l 4, l 11, l 7 }, {l 4, l 11, l 8 }, {l 4, l 9 }, {l 4, l 10 }, schedule α is liquid α = {l 5, l 11, l 6 }, {l 5, l 11, l 7 }, {l 5, l 11, l 8 }, {l 5, l 9 }, {l 5, l 10 } {{l 1, l 12, l 9 }, {l 1, l 8 }, {l 2, l 7 }, {l 1, l 12, l 10 }, {l 3, l 8 }, {l 2, l 6 }{ {l 2, l 12, l 9 }, }, {l 3, l 6 }, {l 4, l 11, l 6 }, {l 4, l 11, l 7 }, {l 4, l 10 }, {l 5, l 10 }, {l 5, l 9 }, {l 5, l 11, l 7 }, { }{ }{ } {l 1, l 7 }, {l 2, l 8 }, {l 3, l 12, l 9 }, {l 5, l 11, l 6 } }{ } {l 1, l 6 }, {l 2, l 12, l 10 }, {l 3, l 7 }, {l 4, l 11, l 8 },, {l 3, l 12, l 10 }, {l 4, l 9 }, {l 5, l 11, l 8 } load of the bottlenecks number of timeframes #( α) = Λ( X) ( A α) A is a team of X

16 I( X), all teams of the traffic X - transfer x - transfers congesting with x - transfers non-congesting with x R= { } depot To cover the full solution space when constructing a liquid schedule an efficient technique obtaining the whole set of possible teams of a traffic is required. We designed an efficient algorithm enumerating all teams of a traffic traversing each team once and only once. { } excluder includer R x = R x = excluder depot includer This algorithm obtains each team by subsequent partitioning of the set of all teams. We introduced triplets consisting of depot subsets of the traffic, representing oneby-one partitions of the set of all teams. { } excluder includer

17 Liquid schedule search tree X ( X) = { A 1, A 2, A 3 A n } X 1 = X A 1 ( X 1 ) = { A 11,, A 12, }... X 11 X 12, = X 1 A 11,, = X 1 A 12, X 2 = X A 2 ( X 2 ) = { A 21,, A 22, } X 21 X 22, = X 2 A 21,, = X 2 A 22, all teams of X ( Y) = { A I( X) A Y} possible steps to the next layer

18 Additional bottlenecks A 1 A 1,1 A 1,1,1 A 1,... A 1,... A 1,... 2 bottlenecks 2 bottlenecks 4 bottlenecks 4 bottlenecks 6 bottlenecks 8 bottlenecks A(X)=6 (X 1 )=5 A A(X 1,1 )=4 A (X 1,... )=3 A (X 1,... )=2 A(X 1,... )=1 X 1,1 = X 1 - A 1,1 (16 transfers) X 1 = X - A 1 (20 transfers) X (25 transfers)

19 Prediction of dead-ends A 1 A 1,1 A 1,1,1 load is 4 2 bottlenecks 2 bottlenecks A(X)=6 A(X 1 )=5 4 bottlenecks A(X 1,1 )=4 16-transfer traffic load is 4 X 1,1 = X 1 - A 1,1 (16 transfers) X 1 = X - A 1 (20 transfers) X (25 transfers)

20 Liquid schedule search optimization teams of the reduced traffic I( Y) { A I( X) A Y} original traffic s teams formed from the reduced traffic X ( X) = { A 1, A 2, A 3 A n } X 1 = X A 1 ( X 1 ) = { A 11,, A 12, }... X 11 X 12, = X 1 A 11,, = X 1 A 12, X 2 = X A 2 ( X 2 ) = { A 21,, A 22, } decreasing the search space without affecting the solution space ( Y) = { A I( X) A Y} ( Y) = I( Y)

21 Liquid schedules construction I full ( Y) I( Y) { full teams of the reduced traffic Choice Choice = ( Y) = I( Y) = ( Y) = I full ( Y) additionally decreasing 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).

22 Results All-to-all throughput (MB/s) Number of contributing nodes for the 363 sub-topologies liquid throughput carried out according to the liquid schedules

23 T1 T2 T3 T4 T5 Congestion Graph x 1,1 x 2,1 x 3,1 x 4,1 x 5,1 x 2,2 x 3,2 x 4,2 R1 R2 R3 R4 R5 The 25 vertices of the graph represent the 25 transfers transfers. The edges represent congestion relations between transfers, i.e. each edge represents one or more communication links shared by two transfers. T1 T2 T3 T4 T R1 bottlenecks R2 R3 R4 R5 x 1,3 x 1,2 x 1,4 x 1,5 x 2,3 x 2,4 x 3,3 x 3,4 x 4,3 x 4,4 x 4,5 x 5,4 x 5,5 Bold edges represent all congestions due to bottleneck links

24 Loss of performance induced by schedules computed with a graph colouring heuristic algorithm loss in performance (%) number of transfers for each of 363 topologies 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%.

25 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 and reassembling. Liquid scheduling is applicable in wormhole, cut-through networks and can be useful in wavelength assignment problem in WDM optical networks. Thank You! Contact: