Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Duplicate-Free State-Space Model for Optimal Task Scheduling Michael Orr and Oliver Sinnen University of Auckland, Department of Electrical and Computer Engineering Dagstuhl 2015 July 6, 2015 Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Abstract Optimally solving scheduling problem ( P | prec , c ij | C max ) Based on state-space search with Branch-and-Bound (A*) State-space model so far: Exhaustive List Scheduling Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Abstract Optimally solving scheduling problem ( P | prec , c ij | C max ) Based on state-space search with Branch-and-Bound (A*) State-space model so far: Exhaustive List Scheduling Good results, but some issues Duplicate states – same schedule appearing multiple times in search need to keeping track of visited states => high memory consumption => difficult to parallelise Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Abstract Optimally solving scheduling problem ( P | prec , c ij | C max ) Based on state-space search with Branch-and-Bound (A*) State-space model so far: Exhaustive List Scheduling Good results, but some issues Duplicate states – same schedule appearing multiple times in search need to keeping track of visited states => high memory consumption => difficult to parallelise Duplicate-free state-space model Task allocation and ordering treated separately Modeling applicable to other combined combinatorial problem Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Overview Background 1 Exhaustive LS State-Space Model 2 AO State-Space Model 3 Overview Allocation Ordering Evaluation 4 Conclusions and Future Work 5 Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Branch-and-Bound Exhaustive search techniques for combinatorial optimisation problems Components State : A partial solution to the problem Solution/search space: all possible states Branch: Create all the children of a partial solution, moving one step closer to a complete solution. Bound: Calculate a lower bound on the quality of any solution that could be reached from this state; often known as f -value. Types of BnB: Depth-first BnB, A*, IDA*, SMA*, RBFS Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Task Scheduling With Communication Delays Scheduling problem P | prec , c ij | C max Task Graph: Models a program distinct computational tasks (with weight) data dependencies (communication) (with weight) Objective: Assign each task to a processor, and give it a start time. Optimal schedule: A schedule with the minimum possible total length. Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Task Scheduling With Communication Delays Constraints Processor Constraint: A processor can only execute one task at any one time. Precedence Constraint: A task can only be executed once all of its predecessors have been completed, and their output data has been communicated to its assigned processor. Local communication is cost free Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Exhaustive List Scheduling A branch-and-bound state-space model. At each step, choose any ready task and schedule it as early as possible on any processor. Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Exhaustive List Scheduling Mature model with many refinements Most obvious weakness: a very large amount of duplicate states. Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Duplicates Processor permutation procs are homogeneous - swapping tasks makes no difference Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model AO State-Space Model Evaluation Conclusions and Future Work Duplicates Independent decision order same scheduling decisions made in different sequence Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Allocation-Ordering (AO) State-Space Model Objective: a state-space without duplicates Two distinct sub-problems: allocation 1 ordering 2 Search tree begins with allocation; once a complete allocation has been found, it becomes the root of an ordering subtree. Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Allocation Goal: allocate each task to a processor For heterogeneous processors: P V possibilities For homogeneous processors: find a partition of the set of tasks. Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Allocation For each task, in a fixed topological order: add to an existing grouping, OR begin a new grouping. Limit number of groupings to number of procs Each allocation that can be reached is entirely unique Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Allocation Bound Lower bound on schedule length achievable with given (partial) allocation Processor Load �� � f l oad ( s ) = max a ∈ A w ( n ) n ∈ a Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Allocation Bound Processor Load �� � f l oad ( s ) = max a ∈ A w ( n ) n ∈ a Allocated Critical Path f acp ( s ) = max n ∈ V ′ { tl a ( n ) + bl ( n ) } Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Ordering Goal: find an ordering for each group of tasks List scheduling approach on a per-processor basis local ready list Global task ordering implicit Reduces factorial ordering complexity! ( x − y )!( y )! < x ! Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Ordering To prevent duplicates, procs considered in a fixed order next proc should be determined only by search depth e.g. one by one, round robin Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Ordering Estimated Earliest Start Time (EEST) can no longer assume that all predecessors of task have been fixed in schedule when task is ordered, EEST of all dependent tasks is updated Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
Background Exhaustive LS State-Space Model Overview AO State-Space Model Allocation Evaluation Ordering Conclusions and Future Work Ordering Bound Ordered Load � f ordered − load ( s ) = max p ∈ P t f ( p ) + w ( n ) n ∈ p ∩ unordered ( s ) Partially Scheduled Critical Path f scp ( s ) = max n ∈ ordered ( s ) { eest ( n ) + bl a ( n ) } Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling
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