List-Based Scheduling

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List-Based Scheduling
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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
• Count1
• While movable operations
• select tuple(i, k) with maximum Si(j,k)
• move Oi to step k
• lock tuple(i, k)
• gaincountchange in cost when moving Oi to step k
• find k that maximizes
• if(gainmaxgt0)
• Move operations 1 to k
• until(gainmaxlt0)

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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)
• (Densityi is the number of functional units at
  step i)

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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

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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( maxSm)
• After each move update cost
• O(mlog(m))
• This must be done for all tuples
• O(nmlog(m))) (Sm2log(m))
• Practial time complexity (stated by authors)
• O(nlog(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.