Independent Tasks Scheduling on Heterogeneous Platforms under Bounded Multi-Port Model Olivier Beaumont, Nicolas Bonichon, Lionel Eyraud-Dubois INRIA Bordeaux Sud-Ouest, CEPAGE, LaBRI Loris Marchal CNRS, ENS Lyon, ROMA, LIP Scheduling in Aussois June 2011
Outline Communication Models 1 Bounded multi-port Model – Divisible Load Scheduling 2 Bounded multi-port Model – Malleable Tasks Scheduling 3 Conclusion 4 Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 2/ 33
Communication Models Outline Communication Models 1 Bounded multi-port Model – Divisible Load Scheduling 2 Bounded multi-port Model – Malleable Tasks Scheduling 3 Conclusion 4 Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 3/ 33
Communication Models Communication Models in the literature No contention One-port ◮ a node can be involved in at most one communication ◮ comes into two flavors (unidirectional or bidirectional) ◮ associated to a topology (physical or at application level) Multi-port ◮ a node can be involved in several communications ◮ provided that incoming and outgoing bandwidths are not exceeded ◮ associated to an overlay network Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 4/ 33
Communication Models Topology vs Coordinate Systems � Topology is more precise ◮ but is not known in general ◮ and tools to discover topology (ENV, AlNEM) are too slow ◮ especially if the churn is high ! � Coordinate systems ◮ embed the nodes into a metric space ◮ i.e. give coordinates to them ◮ and use their coordinates to approximate the available bandwidth (or the latency) between them. ◮ Examples : Vivaldi (2D+1), Sequoia (Trees), PathGuru (traceroute), DMF (SVD), LastMile (bin, bout) Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 5/ 33
Communication Models Comparison of embedding tools [B, Eyraud-Dubois, Won, Europar’11] Extensive comparison of embedding tools On actual PlanetLab data Outgoing bandwidth for a Planetlab node typically looks like this 4 x 10 2.5 bandwidth (kbps) 2 1.5 1 0.5 0 0 50 100 150 200 250 300 350 peer−host ID at first, limited by other nodes incoming bandwidths then by its own outgoing bandwidth right part : nodes in the same local network, bad measurements ? b out ↔ height of the flat area. i Bandwidth ( P i , P j ) = min( b out , b in j ) i Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 6/ 33
Communication Models Comparison of embedding tools (2) 1 0.9 0.8 0.7 0.6 CDF 0.5 0.4 0.3 Last−mile (alpha=0.25) 0.2 Vivaldi Sequoia−15 0.1 Last−mile (alpha=0.25, Random−20) 0 0 2 4 6 8 10 Error error = max( estimated measured , measured estimated ) ( x, y ) : error is less than x for a fraction y of pairs Conclusion : LastMile for estimating Bandwidth ( P i , P j ) is ◮ cheap and robust and decentralized ◮ at least as precise ◮ for PlanetLab dataset (and probably even better for DSL nodes) Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 7/ 33
Bounded multi-port Model – Divisible Load Scheduling Outline Communication Models 1 Bounded multi-port Model – Divisible Load Scheduling 2 Normal Form NP-Completeness Stability Issues and Open Problems Bounded multi-port Model – Malleable Tasks Scheduling 3 Conclusion 4 Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 8/ 33
Bounded multi-port Model – Divisible Load Scheduling Divisible Load One master, holding a large number of identical tasks, P workers Heterogeneity in computing speed and bandwidth Master holding N tasks Worker P i will get a fraction X i = α i × N of these tasks α i is rational , tasks are divisible ⇒ possible to derive analytical solutions (tractability) � � � � � � � � ������ ������ ������ ������ � � � � � � � � � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � � � � � ����� ����� � � � � ������ ������ ����� ����� ������ ������ � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� ����� ����� ����� � � � � � � � � ����� ����� � � � � ����� ����� � � � � � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � � � � � ����� ����� ����� ����� � � � � � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � � � � � ����� ����� � � � � � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � � � � � � � ����� ����� � � ����� ����� � � � � � � � � � � ����� ����� � � � � � � ����� ����� � � � � � � � � ����� ����� ����� � � ����� � � � � � � ����� ����� ����� ����� � � � � � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� ����� ����� � � � � � � � � ����� � � ����� � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � ����� ����� � � � � Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 9/ 33
Bounded multi-port Model – Divisible Load Scheduling Bounded Multi-Port Model P 0 B 0 b 1 b 2 b i b p P 2 P 1 P i P p w 1 w 2 w i w p B 0 : output bandwidth of the master processor. b i : input bandwidth of P i . w i : time to process a unit size task on P i . Use of QoS mechanisms to achieve prescribed bandwidth sharing. ◮ between ( P 0 , P i ) and ( P 0 , P j ) . Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 10/ 33
Bounded multi-port Model – Divisible Load Scheduling Bounded Multi-Port Model (1) t 2 t 3 t 4 t 1 P 1 P 2 B 0 P 4 b 1 P 3 Notations: ◮ b ′ i ( t ) : actual bandwidth used at time t by the communication between P 0 and P i . ◮ t i : the time when processor P i stops communicating. ◮ X i : the fractional number of tasks sent to P i . Constraints: ◮ input bandwidth: ∀ t, b ′ i ( t ) � b i . ◮ output bandwidth: ∀ t, � i b ′ i ( t ) � B 0 . Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 11/ 33
Bounded multi-port Model – Divisible Load Scheduling Bounded Multi-Port Model (2) t 2 t 3 t 4 t 1 P 1 P 2 B 0 P 4 b 1 P 3 Processing constraints : ◮ 1rst part: With a linear cost model X i units of work: ⋆ sent to P i in X i i ( t ) time units � t b ′ ⋆ processed by P i in X i × w i time units ◮ 2nd part: With an affine cost model: code of size S i + data X i units of work ⋆ sent to P i in S i + X i i ( t ) time units � t b ′ ⋆ processed by P i in X i × w i time units Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 12/ 33
Bounded multi-port Model – Divisible Load Scheduling Normal Form Normal Form Definition A schedule is said to be in normal form if 1 all processors are involved in the processing of tasks, 2 all slaves start processing tasks immediately after the end of the communication with the master (at time t i ) and stop processing at time 1, 3 during each time slot ] t i k , t i k +1 ] the bandwidth used by any processor is constant. t 2 t 3 t 4 t 1 P 1 P 2 B 0 P 4 b 1 P 3 Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 13/ 33
Bounded multi-port Model – Divisible Load Scheduling Normal Form Lemma 1 Lemma In an optimal schedule, all processors take part in the computations. Proof: P i Original schedule New schedule Olivier Beaumont (INRIA) Independent Tasks Scheduling under Bounded Multi-port 14/ 33
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