Courseware Integer Linear Programming approach to Scheduling Sune Fallgaard Nielsen Informatics and Mathematical Modelling Technical University of Denmark.

courseware

Integer Linear Programmingapproach to Scheduling

Sune Fallgaard Nielsen

Informatics and Mathematical ModellingTechnical University of Denmark

Richard Petersens Plads, Building 322DK2800 Lyngby, Denmark

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Outline

Introduction

- The new approach & objective

- Automated data path synthesis ILP Formulation Example Generalizations Experimental Results Conclusion

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The New Approach

Solve scheduling problem

1) ASAP -> start time

2) ALAP -> require time

3) ILP (Integer Linear Programming)

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Objective

Fully utilize the hardware resources

i.e. minimize the requirement of function units under a given timing constraint

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Support different kinds of data path

Multicycle operations Multiple operations per cycle Pipelined data paths Mutually exclusive operations Variables’ lifetime consideration

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Automated Data Path Synthesis

Scheduling

Allocation

Tightly interdependent

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Scheduling

** very important

FIX

1) number & types of function units

2) lifetime of variables

3) timing constraints

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The New Approach

1) ASAP

2) ALAP

3) ILP (Integer Linear Programming)

Function Units: Fully utilized minimize maximal no.

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The ILP Formulation

2 Assumptions: Each operation

– 1 cycle propagation delay Consider non-pipelined data path

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Data Flow Graph

n operations s steps oi – each operation 1 ≤ i ≤ n oi oj – precedence relation

oi immediate predecessor of oj

m types of function units

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Si – start time (ASAP) Li – require time (ALAP) Cti – cost of function unit of type ti (FUti) Mti – number of function unit of type ti xi,j – 1: if oi is scheduled into step j

0: otherwise

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Formulas (1,2)

i 1

m

cti

Mti

Minimize total function unit cost

Mtk

0i 1

n

x ,i j

Oi Є FUtkfor 1 ≤ j ≤ s, 1 ≤ k

≤ m

No control step should contain more than Mtk function unit of type tk

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Formulas (3,4)

oi can only be scheduled into a step between Si & Li

for all oi ok

j Si

Li

x . i j 1

j Si

Li

j x ,i j

j Sk

Lk

j x ,k j -1

for 1 ≤ i ≤ n Ensure the precedence

relations of DFG will be preserved

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Example

Available function units:

~ multipliers (FUt1)

~ ALUs (FUt2)

Cost:

~ Ct1 = 5

~ Ct2 = 1

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Example

Integer programming formulation (formulas 1,2)

i 1

m

cti

Mti

minimize 5Mt1 + Mt2

x ,1 1 x ,2 1 x ,6 1 x ,8 1 Mt1

0

x ,3 2 x ,6 2 x ,7 2 x ,8 2 Mt1

0

x ,7 3 x ,8 3 Mt1

0

x ,10 1 Mt2

0

x ,9 2 x ,10 2 x ,11 2 Mt2

0

x ,4 3 x ,9 3 x ,10 3 x ,11 3 Mt2

0

x ,5 4 x ,9 4 x ,11 4 Mt2

0

Mtk

0i 1

n

x ,i j

Oi Є FUtk

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Example

Integer programming formulation (formulas 3,4)

j Si

Li

x . i j 1

j Si

Li

j x ,i j

j Sk

Lk

j x ,k j -1

x ,1 1 1

x ,2 1 1

x ,3 2 1

x ,4 3 1

x ,5 4 1

x ,6 1 x ,6 2 1

x ,7 2 x ,7 3 1

x ,8 1 x ,8 2 x ,8 3 1

x ,9 2 x ,9 3 x ,9 4 1

x ,10 1 x ,10 2 x ,10 3 1

x ,11 2 x ,11 3 x ,11 4 1

x ,6 1 2 x ,6 2 2 x ,7 2 3 x ,7 3 -1

x ,8 1 2 x ,8 2 3 x ,8 3 2 x ,9 2 3 x ,9 3 4 x ,9 4 -1

x ,10 1 2 x ,10 2 3 x ,10 3 2 x ,11 2 3 x ,11 3 4 x ,11 4 -1

O6 O7

O8 O9

O10 O11

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Example

Scheduling result-- optimal this formulation variables x1,1, x2,1,

x3,2, x4,3, x5,4, x7,3, x8,3, x9,4, x10,1 & x11,2

=> 1 2 multipliers &

2 ALUs

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Generalizations

Multicycle operations Multiple operations per cycle Pipelined data paths Mutually exclusive operations Variables’ lifetime consideration

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Multicycle Operations

oi – operation

di – delay

for 1 ≤ j ≤ s, 1 ≤ k ≤ m

for all oi ok

j Si

Li

j x ,i j

j Sk

Lk

j x ,k j -1- di

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Multiple Operations per Cycle

New precedence relation oi => oj

-- oj is the nearest successor of oi

j Si

Li

j x ,i j

j Sk

Lk

j x ,k j -1 for all oi => ok

for all oi ok

j Si

Li

j x ,i j

j Sk

Lk

j x ,k j -10

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Pipelined Data Paths

l: fixed latency (integer multiple of a clock cycle) | si – sj |: integer multiple of l

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Mutually Exclusive Operations

If oi, oj – two mutually exclusive operations, X(oi, oj) = 1otherwise X(oi, oj) = 0

Both scheduled in control step k Count function unit cost as 1, not 2 New 0/1 integer variable yk

-- 0 if xi,k = xj,k = 0-- 1 if otherwise

xi,k + xj,k = yk in constraint (2)

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Variables’ Lifetime Consideration

Function unit cost for both schedules are the same, but fewer number of registers needed in Fig(a)

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Variables’ Lifetime Consideration

SLKi,j – difference between the assigned control steps of oi, oj

(oi oj)

Minimize total step differences

minimize

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Experimental Results

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Experimental Results

Fifth order wave filter Containing 26

addition (1 cycle) &8 multiplication (2 cycles) operations

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Conclusion

Integer Linear Programming formulation (ILP) minimize the function unit cost

Quite acceptable for practical synthesis Always find the optimal solution Different kinds of data path are taken into

account

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THE END

Thanks