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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