List-Based Scheduling

courseware

List-Based Scheduling

Sune Fallgaard Nielsen

Informatics and Mathematical ModellingTechnical University of Denmark

Richard Petersens Plads, Building 322DK2800 Lyngby, Denmark

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Overview

Introduction Related work DFG and WPG ASAP, ALAP and Mobility Solution:

Algorithm Selection function Extenstions Time complexity

Results Conclussion

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Introduction

Given a DFG we want to schedule operations to minimize the number of functional units needed (adders, multiplier, ALUs, etc.)

Input to algorithm DFG Number of cycles.(Not less than shortest path in DFG)

Model must include Pipelined operations Chained operations Multicycle operations

+

*

*

*

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Related Work

Other algorithms fix operations one by one Tend to find local minima instead of global minima

ILP(Integer Linear programming) Finds global minima Is an exponential time algorithm and are not usable for

large data-sets

This algorithm can escape local minimals by making several iterations. Tend to find global minimaes (But, no guarantee)

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DFG and WPG

From “Data Flow Graph” to “Weighted Precedence Graph”

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ASAP Scheduling

Minimum number of cycles with no resource limitation

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ALAP Scheduling

Minimum number of cycles with no resource limitation

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Mobility

Mobility tells which steps each operation can be placed

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Algorithm - overview

Starts from ASAP/ALAP Iteratively moves operations based on selection

function Si(j,k). (Must not violate WPG)

If no ”good” moves are possible, ”bad” ones are taken.

Stops when there are no movable operations Select best solution based on cost function from all

solutions explored. (Fx. number of functional units) Repeat the above iteration while the solution

improves

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Algorithm - pseudu

repeat{

Count=1

While movable operations{

select tuple(i, k) with maximum Si(j,k)

move Oi to step k

lock tuple(i, k)

gaincount=change in cost when moving Oi to step k

}

find k that maximizes

if(gainmax>0)

Move operations 1 to k

}until(gainmax<=0)

gainmax i 1

kgain i

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Algorithm - example

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Algorithm - solution

Solution to running example

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Selection Function

In each step the selection function Causes most balanced distribution of operations Reduces number of required functional units.

Expresses the gain by moving Oi from step j to k

Proportional with Maximal density at step j and k Change in Density Gradient(DG)

(Density[i] is the number of functional units at step i)

S i j , k max Density j , Density k CDG

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Selection function

“Change in Density Gradient” (CDG)

CDG equals (Only concerns the 2 affected steps)

2 Number of functional units decreases by 1 0 No change in number of functional units needed -2 Number of functional units increases by 1

CDG DGbefore DG after

CDG Density j Density kDensity j 1 Density k 1

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Selection function

Examples of Selection function calculations

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Extensions

Mutually Exclusive(ME) operations Fx. an if-else statement. Operations that are ME can be

placed in the same step on the same functional unit

Chained operations Are already incoporated into the WPG

Multi-cycle operations Are included in the WPG, but selection function must be

expanded.

Pipelined operations I do not understand the articles interpretation of a

pipeline. Shall be treated as a multi-cycle operation.

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Time complexity

m Number of operations in DFG

S Number of control steps

n Total number of tuples( max=Sm)

After each move update cost O(m*log(m))

This must be done for all tuplesO(nm*log(m))) = (Sm2*log(m))

Practial time complexity (stated by authors)O(n*log(m)) (since n i independant of m)

Number of iterations is assumed independent of problem

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Results – Example 1

Chained operations with the number of cycles less than 8

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Results – Example 2

Pipelined 16-point FIR filter. 2 types of functions units (Multiplier and adder)

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Results – Example 3

5th order elliptic filter with25 additions and 8 multiplications. Multiply is a multicycle operation

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Conclusion

Very fast an effective Can escape local minima Tend to reach optimal solution. (At least for all

experimental results so far) Must faster than than previous approaches. Handles multicycled, chained and pipelined

operations. Possible to use more sophisticated selection

functions.