Force-Directed Scheduling

courseware

Force-Directed Scheduling

Sune Fallgaard Nielsen

Informatics and Mathematical ModellingTechnical University of Denmark

Richard Petersens Plads, Building 321DK2800 Lyngby, Denmark

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Overview / Agenda

Motivation Introduction Defining problem and model Solution Results Conclussion

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Motivation

The mapping from a behavioral model to a physical Register-Transfer level description is a NP-complete problem.

Therefore an exhaustive comparison on all possible solutions is infeasible.

An algorithm finding an efficient mapping is required.

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Introduction

The paper was written in 1989. The interest on High-Level Languages like VHDL

is increasing at that time. Synthesis tools capable of generating a RTL

realization from a behavioral model is needed.

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Introduction

implementationdesign specification

Physical domain

Mathematical domain

specification

create a model of the physical problem

synthesis

create an alogorithm to solve the problem

implementation

Transform the optimized model back to the physical domain

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Introduction

implementationdesign specification

Physical domain

Mathematical domain

specification

create a model of the physical problem

synthesis

create an alogorithm to solve the problem

implementation

Transform the optimized model back to the physical domain

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Defining problem and model

The problem: For a given function, find an optimal solution. An optimal solution can be constrainted by area-, power-, performance- or flexibility requirements depending on the application.

The model:Scheduling : Determine for each operation the time at which it should be performed without violating any precedence contraints.

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Solution

Common Approach New Approach

List Scheduling Force-Directed Scheduling

Resources given Time contraints given

Minimize delay Minimize required resources

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Solution

Common Approach New Approach

List Scheduling Force-Directed Scheduling

Resources given Time contraints given

Minimize delay Minimize required resources

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Force-Directed Scheduling

The Force-Directed Scheduling approach reduces the amount of:

• Functional Units• Registers• Interconnect

This is achieved by balancing the concurrency of operations to ensure a high utilization of each unit.

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Force-Directed Scheduling

The Force-Directed Scheduling algorithm consists of 3 steps:

1.Determine a time frame of each operation

2.Create a distribution graph

3.Calculate the force (a new metric)

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Scheduling – An example

Solve the differential equationy’’ + 3zy’ + 3y = 0

This can be calculated using this iterative algorithmwhile(z < a) repeat

zl := z + dz;

ul := u – (3 · z · u · dz) – (3 · y · dz);

yl := y + (u · dz);

z := zl;

u := ul;

y := yl;

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Scheduling – An example

Data-Flow and Control-Flow Graphs

ASAP Scheduling ALAP Scheduling

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Scheduling – An example

Data-Flow and Control-Flow Graphs

ASAP Scheduling ALAP Scheduling

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Scheduling – An example

Step 1 : Determine a time frame of each operation

Error in Figure

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Scheduling – An example

Step 2 : Create a distribution graph

DG(1) = 2.833 DG(3) = 0.833DG(2) = 2.333 DG(4) = 0

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Scheduling – An example

b

ti tb

iDGjDGjForce

)1(

)()()(

A metric called force is introduced. The force is used to optimize the utilization of units. A high positive force value indicates a poor utilization.

Step 3 : Calculate the force (a new metric)

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Scheduling – An example

Step 3 : Calculate the force (a new metric)

Fixed

Free

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Scheduling – An example

2

1 2

)()1()1(

i

iDGDGForce

Step 3 : Calculate the force (a new metric)With the operation x’ in control-step 1.

DG(1) = 2.833 DG(3) = 0.833DG(2) = 2.333 DG(4) = 0

25.02

333.2833.2833.2

Poor utilization

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Scheduling – An example

3

2

2

1 2

)()3(

2

)()2()2(

ii

iDGDG

iDGDGForce

Step 3 : Calculate the force (a new metric)With the operation x’ in control-step 2. (x’’ must be in control-step 3)

DG(1) = 2.833 DG(3) = 0.833DG(2) = 2.333 DG(4) = 0

12

333.2833.0833.0

2

333.2833.2333.2

Good utilization

Direct force (calculated as before)Indirect force (on x’’ in control-step 3)

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Scheduling – An example

By repeatedly assigning operations to various control-steps and calculating the force associated with the choice several force values will be available.

The Force-directed scheduling algorithm chooses the assignment with the lowest force value, which also balances the concurrency of operations most efficiently.

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Force-Directed Scheduling

The Force-Directed Scheduling approach reduces the amount of:

• Functional Units• Registers• Interconnect

By introducing Registers and Interconnect as storage operations, the force is calcuted for these as well. The 3 steps of the algorithm are carried out for these operations also.

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Force-Directed Scheduling

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Force-Directed Scheduling

Introducing Registers and Interconnect as operations.

Since the Registers and Interconnect area consumption is reduced in solution b, it might be optimal.

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Force-Directed List Scheduling

List Scheduling Force Directed Scheduling

Resources given Time contraints given

Minimize delay Minimize required resources

If the problem is the other way around, ie. if the resources are

given a Force-Directed List Scheduling algorithm can be applied.

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Force-Directed List Scheduling

The Force-Directed List Scheduling utilizes the strengths of the:

•Force-Directed Scheduling and•List Scheduling

Force-Directed List Scheduling is similar to List Scheduling except force is the priority function, not mobility.

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Force-Directed List Scheduling

List Scheduling Force-Directed List Scheduling

We have 1 adder and 1 multiplier

Mobility is a poor metric Force is a better metric

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Results

Test Application : Fifth-order Elliptic Wave Filter

xp is a pipelined multiplier

Many optimal results, depending on the applicationOne result, ASAP is not optimal.

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Results – The Design Space

The Force-Directed List Scheduling gives better means for exploring the design space.

It offers many results and depending on the application the designer can choose to use more or less resources and see what implications it will have on the delay.

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One more optimization

The Force-Directed List Scheduling is implemented in a system called HAL.

HAL also uses techniques called register merging and multiplexer merging.

These techniques minimizes the cost on registers and interconnections.

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Results

Comparison on various syntesis tools normalized to HAL86 Test Application : Differential Equation

Non-pipelined MultiplierPipelined Multiplier

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Results

Physical unit and interconnection requirements Test Application : Wave Filter

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Conclussions

•The Force-Directed Scheduling and Force-Directed List Scheduling algorithms allows the designer to explore the design space.

•By taking into account the cost of interconnection, registersand multiplexers a more precise algorithm is realized.

•The HAL system, using these algorithms, shows promisingresults compared to other systems.