courseware
Integer Linear Programming
approach to Scheduling
Sune Fallgaard Nielsen
Informatics and Mathematical Modelling
Technical University of Denmark
Richard Petersens Plads, Building 322
DK2800 Lyngby, Denmark
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)
m

ct Mt
i1 i i
n

x i, j M  0
t
k
i1
 Minimize total function
unit cost
 No control step should
contain more than Mtk
function unit of type tk
Oi
Є
FUt
for 1 ≤ j ≤ s, k1 ≤ k ≤
m courseware
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Formulas (3,4)
Li

jSi
 oi can only be scheduled
into a step between Si &
Li
x i . j1
for 1 ≤ i ≤ n
 Ensure the precedence
Lk

relations of DFG will be


preserved
j x k , j -1
 Li
 

 

 

j x i, j 

 
 jS
  jS
i
k

 





for all oi  ok
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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)
m
 ct Mt
i1 i
n

x i, j M  0
t
k
i1
i
minimize
5Mt1 + Mt2
Oi
Є
FUt
k
x 1, 1 x 2, 1 x 6, 1 x 8, 1 Mt  0
1
x 3, 2 x 6, 2 x 7, 2 x 8, 2 Mt  0
1
x 7, 3 x 8, 3 Mt  0
1
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x 10, 1 Mt  0
2
x 9, 2 x 10, 2 x 11, 2 Mt  0
2
x 4, 3 x 9, 3 x 10, 3 x 11, 3 Mt  0
2
x 5, 4 x 9, 4 x 11, 4 Mt  0
2
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Example
 Integer programming formulation (formulas 3,4)
Li
x 1, 1 1

x 2, 1 1
jSi
x 3, 2 1
x i . j1
x 4, 3 1
 Li
  Lk


 


 




j x i, j 
j x k , j -1

 

 jS
  jS

i
k

 



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
O6  O7
x 6, 1 2 x 6, 2 2 x 7, 2 3 x 7, 3 -1
O8  O9
x 8, 1 2 x 8, 2 3 x 8, 3 2 x 9, 2 3 x 9, 3 4 x 9, 4 -1
O10  O11
x 10, 1 2 x 10, 2 3 x 10, 3 2 x 11, 2 3 x 11, 3 4 x 11, 4 -1
x 10, 1 x 10, 2 x 10, 3 1
 11, 3
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x
x
x 11[M-1]
1
Synthesis
, 4High-Level
16
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
 Li
  Lk


 


 
 - di

j x i, j 
j x k , j -1

 

 jS
  jS

 i
  k



for all oi  ok
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Multiple Operations per Cycle
 New precedence relation
 oi => oj
-- oj is the nearest successor of oi
 Li
  Lk


 


 
 0

j x i, j 
j x k , j -1

 

 jS
  jS

i
k

 



 Li
  Lk


 


 


j x i, j 
j x k , j -1

 

 jS
  jS

i
k

 


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for all oi  ok
for all oi => ok
20
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,
otherwise 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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X(oi, oj) = 1
22
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
 Minimize total step
differences
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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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courseware
THE END
Thanks 