Duplicate-Free State-Space Model for Optimal Task Scheduling Michael - - PowerPoint PPT Presentation

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, cij|Cmax)

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, cij|Cmax)

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, cij|Cmax)

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

1

Background

2

Exhaustive LS State-Space Model

3

AO State-Space Model Overview Allocation Ordering

4

Evaluation

5

Conclusions and Future Work

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

• ne 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, cij|Cmax 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 AO State-Space Model Evaluation Conclusions and Future Work Overview Allocation Ordering

Allocation-Ordering (AO) State-Space Model

Objective: a state-space without duplicates Two distinct sub-problems:

1

allocation

2

• rdering

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 AO State-Space Model Evaluation Conclusions and Future Work Overview Allocation Ordering

Allocation

Goal: allocate each task to a processor For heterogeneous processors:

PV 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 AO State-Space Model Evaluation Conclusions and Future Work Overview Allocation Ordering

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 AO State-Space Model Evaluation Conclusions and Future Work Overview Allocation Ordering

Allocation Bound

Lower bound on schedule length achievable with given (partial) allocation Processor Load fload(s) = maxa∈A

• n∈a

w(n)

• 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 Allocation Ordering

Allocation Bound

Processor Load fload(s) = maxa∈A

• n∈a

w(n)

• Allocated Critical Path

facp(s) = maxn∈V ′ {tla(n) + bl(n)}

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 Allocation Ordering

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 AO State-Space Model Evaluation Conclusions and Future Work Overview Allocation Ordering

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 AO State-Space Model Evaluation Conclusions and Future Work Overview Allocation Ordering

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 AO State-Space Model Evaluation Conclusions and Future Work Overview Allocation Ordering

Ordering Bound

Ordered Load fordered−load(s) = maxp∈P

  tf(p) +

• n∈p∩unordered(s)

w(n)

  

Partially Scheduled Critical Path fscp(s) = maxn∈ordered(s) {eest(n) + bla(n)}

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

Evaluation Setup

AO evaluated by comparison with ELS.

Note, ELS uses more pruning techniques

Almost 500 small task graphs solved by A* search

Types: Fork, Join, Fork-Join, Random, InTree, OutTree, Pipeline, Stencil, Independent, SeriesParallel Communication-to-Computation ratio: 0.1, 1, 2, 10

With 2, 4, and 8 procs Once each for AO and ELS

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

Results

AO better in 75% of cases Median: 3.4 times less states

−4 −2 2 4 log10(ELS/AO)

1 3.4 26.1

States Created

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

Results

Distinct advantage for task graphs with high CCR

0.1 1 2 10 −4 −2 2 4 CCR log10(ELS/AO)

1.1 1.7 3.5 28.2

States Created

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

Results

Average improvement of AO increases with problem difficulty

2 4 6 8 2 4 6 8

States Created

ELS (log10) AO (log10) 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

Summary

Existing ELS model suffers from numerous duplicates Proposed Allocation-Ordering model Already creates less states

Even with less pruning techniques than ELS

Better potential for parallelisation, no "collisions"

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

Current and Future Work

Refinements

Further improved f -value functions. Heuristics for ordering equal f -value states. Strategies for deciding processor order in ordering phase.

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

Current and Future Work

Refinements

Further improved f -value functions. Heuristics for ordering equal f -value states. Strategies for deciding processor order in ordering phase.

New Pruning Techniques

ELS model benefited enormously from specialised pruning techniques. Previous pruning techniques can be adapted to AO (partially done) Entirely new opportunities for pruning are likely to be found.

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

Current and Future Work

Refinements

Further improved f -value functions. Heuristics for ordering equal f -value states. Strategies for deciding processor order in ordering phase.

New Pruning Techniques

ELS model benefited enormously from specialised pruning techniques. Previous pruning techniques can be adapted to AO (partially done) Entirely new opportunities for pruning are likely to be found.

Parallel search formulation

Previous attempts at parallel task scheduling suffered from “collisions”. Without duplicates, this is no longer possible. The AO state-space model may allow much more scalable parallelisation.

Michael Orr and Oliver Sinnen Duplicate-Free State-Space Model for Optimal Task Scheduling