ℑ X ( ) all teams of the traffic X , • To cover the full solution space when - transfer x constructing a liquid schedule an effi- - transfers congesting with x cient technique obtaining the whole set - transfers non-congesting with x of possible teams of a traffic is required. depot { } • We designed an efficient algorithm enu- merating all teams of a traffic traversing R= each team once and only once. • This algorithm obtains each team by excluder includer subsequent partitioning of the set of all teams. • We introduced tri- plets consisting of depot depot { } { } subsets of the traf- fic, representing one- R x = R x = by-one partitions of the set of all teams. excluder excluder includer includer -- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton -- -- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
Liquid schedule search tree → ℘ X ( ) { A 1 A 2 A 3 … A n , , } X = → ℘ X 1 ( ) { , , … } X 1 = X – A 1 = A 1 1 A 1 2 , all teams of X X 1 1 = X 1 – A 1 1 , , ℘ Y ( ) { ∈ ℑ X ( ) ⊂ } = A A Y = – X 1 2 X 1 A 1 2 , , possible steps to the next layer ... → ℘ X 2 ( ) { , , … } X 2 = X – A 2 = A 2 1 A 2 2 , X 2 1 = X 2 – A 2 1 , , X 2 2 = X 2 – A 2 2 , , -- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton -- -- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
Additional bottlenecks A 1 A 1,1 A 1,1,1 A 1,... A 1,... A 1,... 2 bottlenecks 2 bottlenecks 4 bottlenecks 6 bottlenecks 4 bottlenecks 8 bottlenecks A( X )=6 ( X 1 )=5 ( X 1,1 )=4 ( X 1,... )=2 ( X 1,... )=1 ( X 1,... )=3 A A A A A X 1,1 = X 1 - A 1,1 (16 transfers) X 1 = X - A 1 (20 transfers) X (25 transfers) -- 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. -- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton -- -- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
Liquid schedules construction ℑ Y ( ) ⊂ { ∈ ℑ X ( ) ⊂ } teams of the original traffic’s teams formed A A Y reduced traffic from the reduced traffic ℑ full Y ( ) ⊂ ℑ Y ( ) { full teams of the reduced traffic ℘ Y ( ) { ∈ ℑ X ( ) ⊂ } → ℘ Y ( ) ℑ Y ( ) Choice = = A A Y = additionally decreas- ℘ Y ( ) ℑ Y ( ) Choice = = ing the search space without affecting the ℑ full Y solution space ℘ Y ( ) ( ) Choice = = 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). -- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton -- -- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
Results 2000 All-to-all throughput (MB/s) 1800 1600 1400 1200 1000 800 600 400 200 0 0 8 10 12 13 14 15 16 17 18 19 21 22 24 27 Number of contributing nodes for the 363 sub-topologies liquid throughput carried out according to the liquid schedules -- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton -- -- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
Congestion Graph x x 3 x , 1 4 2 , , 1 1 T1 T2 T3 T4 T5 x x 5 1 , 1 , 1 x x 3 x , 2 4 2 , , 2 2 Bold edges represent all conges- tions due to bottleneck links x 1 R1 R2 R3 R4 R5 , 2 The 25 vertices of the graph x represent the 25 transfers x x 3 , 3 4 2 transfers. The edges repre- , , 3 3 sent congestion relations be- x tween transfers, i.e. each 1 , 3 edge represents one or more communication links shared x x 3 x by two transfers. , 4 4 2 , , 4 4 x T2 T3 T4 T5 T1 x 5 1 , 4 5 5 , 5 5 4 5 6 bottlenecks x 4 6 , 5 5 5 5 5 5 x x 5 , 1 5 , 5 R1 R3 R4 R5 R2 -- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton -- -- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
Loss of performance induced by schedules com- puted with a graph colouring heuristic algorithm loss in performance (%) 20 15 10 5 0 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 • 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 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. -- ICON 2004, IEEE International Conference On Networks, November 16-19, 2004, Singapore, Hilton -- -- EFFICIENT LIQUID SCHEDULE SEARCH STRATEGIES FOR COLLECTIVE COMMUNICATIONS --
a network of 100 nodes and 668 links
a network of 100 nodes and 682 links
a network of 100 nodes and 682 links
a network of 100 nodes and 694 links
a network of 100 nodes and 688 links
a network of 100 nodes and 702 links
a network of 100 nodes and 708 links
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a network of 100 nodes and 718 links
a network of 100 nodes and 712 links
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a network of 100 nodes and 684 links
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a network of 100 nodes and 662 links
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a network of 100 nodes and 702 links
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a network of 100 nodes and 678 links
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a network of 100 nodes and 698 links
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a network of 100 nodes and 706 links
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unsolved 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-06 180 nodes, clock 1, layer 12, 17 bottlenecks at load 0.0666666667
unsolved 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-06 180 nodes, clock 2, layer 12, 8 bottlenecks at load 0.09375
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