Integer Linear Programming approach to Scheduling

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Integer Linear Programmingapproach to Scheduling

• Sune Fallgaard Nielsen
• Informatics and Mathematical Modelling
• Technical University of Denmark
• Richard Petersens Plads, Building 322
• DK2800 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 -gt start time
• 2) ALAP -gt 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)

• Minimize total function unit cost

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

for 1 j s, 1 k m
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Formulas (3,4)

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

for 1 i n

• Ensure the precedence relations of DFG will be
  preserved

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)

minimize 5Mt1 Mt2
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Example

• Integer programming formulation (formulas 3,4)

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 gt 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
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Multiple Operations per Cycle

• New precedence relation
• oi gt oj
• -- oj is the nearest successor of oi

for all oi ? ok
for all oi gt ok
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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) 1
• 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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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 ?