Factoring and Eliminating Common Subexpressions in Polynomial Expressions International Conference o

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Factoring and Eliminating Common Subexpressions
in Polynomial ExpressionsInternational
Conference on Computer Aided Design (ICCAD), 2004

• Farzan Fallah
• Advanced CAD Research
• Fujitsu Labs. of America

Anup Hosangadi Ryan Kastner ECE Department, UCSB
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Outline

• Introduction
• Related Work
• Algebraic techniques for redundancy elimination
• Experimental results
• Conclusions

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Introduction

• Embedded system applications need to compute
  polynomial expressions
• Continuous functions can be approximated by
  polynomials to desired degree of accuracy
• Adaptive signal processing (Polynomial filters )
• Polynomial interpolation/extrapolation in
  Computer Graphics
• Encryption

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Introduction

• Multiplications are expensive in Embedded systems
• No good optimization tool for reducing complexity
  of polynomials
• Designers rely on Hand optimized libraries
• Conventional optimization techniques
• CSE, Value numbering not suited for polynomials
• Horner form most popular representation
• anxn a1xn-1 .an-1x a0 (((anx an-1)x
  an-2)x ..a1)x a0
• Not good for multivariate polynomials
• Only a single polynomial expression at a time

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Introduction

• Quartic-spline polynomial (3-D graphics)
• P zu4 4avu3 6bu2v2 4uv3w qv4
• Horner form (from MapleTM)
• P zu4 (4au3 (6bu2 (4uw qv)v)v)v
• (17 multiplications)
• Proposed algebraic method
• d1 v2 d2 d1v
• P u3(uz ad2) d1( qd1 u(wd2 6bu) )
• (11 multiplications)

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Related Work

• Expression Factorization (M.A.Breuer JACM69)
• Allows only one kind of operator at a time
• Symbolic algebra techniques
• (A. Peymandoust, De Micheli DAC01)
• Used for mapping DSP datapaths (polynomials) to
  library elements
• Results depend upon exponential library search
• e.g. a2 b2 (ab)(a-b) iff (ab) or (a b)
  is in library
• Manipulates only one expression at a time

F1 A B C D F2 A P D
gt Extract (A D)
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Motivating Example

• Consider set of expressions
• Naïve implementation 16 multiplications, 4
  additions/subtractions
• Using CSE
• 12 multiplications, 4 additions/subtractions

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Motivating Example

• Using our algebraic techniques
• Total 7 multiplications, 3 additions/subtractions
• Savings of 5 multiplications, 1
  addition/subtraction compared to CSE
• Impossible to obtain such results using
  conventional techniques

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Introduction to algebraic techniques for
redundancy elimination

• Algebraic techniques in multi-level logic
  synthesis (MLLS)
• Decomposition, factoring reduce number of
  literals
• Distill and Condense use Rectangle Covering
  methods
• Polynomial Expressions (Our Technique)
• Factoring, Single term common subexpressions
  reduces number of multiplications
• Multiple term common subexpressions reduces
  number of additions and possibly multiplications
• Key Differences (Generalization to handle higher
  orders)
• Kernelling techniques
• Finding single cube intersections

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Introduction to our technique(Outline)

• Find a subset of all possible subexpressions
  (kernel generation)
• Transformation of Polynomial Expressions
• Problem formulation
• Extract multiple term common subexpressions and
  factors
• Extract single term common factors

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Introduction to our technique

• Terminology
• Literal A variable or a constant eg. a,b,2,3.14
• Cube Product of literals e.g. 3a2b, -2a3b2c
• SOP Sum of cubes e.g. 3a2b 2a3b2c
• Cube-free expression No literal or cube can
  divide all the cubes of the expression
• Kernel A cube free sub-expression of an
  expression, e.g. 3 2abc
• Co-Kernel A cube that is used to divide an
  expression to get a kernel, e.g. a2b

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Introduction to our Technique

• Matrix Representation of Polynomial Expressions
• F x3y xy2z is represented by
• Each row represents a product term
• Each column represents a variable/constant
• Each element (i,j) represents power of variable j
  in term i

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Generation of Kernels (example)

• P1 x3y x2y2z L x,y,z
• Divide by x
• Ft P1/x x2y xy2z

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Generation of Kernels (example)

• Ft P1/x x2y xy2z
• C Biggest Cube dividing all cubes of Ft

/ C
1 1 0
C
xy
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Generation of Kernels (example)

• Obtain Kernel
• F1 Ft/C (x2y xy2z)/(xy) ( x yz)
• Obtain Co-Kernel
• D1 x(xy) x2y
• No kernels within F1. Go back to P1
• P1 x3y x2y2z
• Divide now by next variable y
• Ft x3 x2yz
• C x2
• But (x lt y) e C
• Stop Here, to avoid repeating same kernel Ft/C
  (x yz)
• No more kernels extracted
• Record kernel F1 P1 with co-kernel 1

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Concept of kernels and co-kernels

• Theorem Two expressions f and g can have a
  multiple term common subexpression iff there are
  2 kernels Kf and Kg having a multiple term
  intersection
• Detection of multiple term common subexpressions
  by intersection of sets of kernels
• Each co-kernel kernel pair represents a
  possible factorization
• e.g. x3y x2y2z x2y(x yz)
• Set of kernels a subset of all possible
  subexpressions

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All Kernels and Co Kernels
Which kernels to choose?
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Kernel Cube Matrix (KCM)

• One row for each Kernel generated
• One column for each distinct kernel cube
• Each non-zero element represents a term

x3y
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Finding Kernel Intersections(Distill Algorithm)

• Each kernel intersection or factor appears as a
  rectangle
• Rectangle Set of rows and columns such that all
  elements are 1
• Value of a rectangle weighted sum of the number
  of operations saved
• Goal Maximum valued rectangular covering of KCM
• Greedy heuristic covering by prime rectangles
• Prime rectangle Rectangle not covered by any
  other rectangle

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Finding Kernel Intersections (Distill Algorithm)

• Formula for Value of a rectangle
• R number of rows
• C number of columns
• M(Ri) of multiplications in row (co-kernel)
  i.
• M(Ci) of multiplications in column
  (kernel-cube) i
• m ratio of weights of multiplication to
  addition
• Value

Formula calculates savings in operation count
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Distill Algorithm
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Distill Algorithm
4xy x2y xyd2 d2 4 x Saves 2
multiplications
Remove covered terms
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Distill Algorithm

• Distill algorithm exits after no more kernel
  intersections can be found

P1 x2yd1 d1 x yz
P2 4d1 xyz d2 4 - x P3 xyd1
Can further optimize by finding single cube
intersections
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Finding single cube intersections (Condense
Algorithm)

• Need an algorithm for finding single term common
  subexpressions
• Consider two single term expressions
• F1 a4b3c
• F2 a2b4c2
• Form Cube Variable Incidence Matrix (CIM)

One row for each product term One column for each
variable
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Finding single cube intersections (Condense
algorithm)

• Each (single term) common subexpression appears
  as a rectangle.
• Rectangle Set of rows and columns where all
  elements are non-zero
• Value of a rectangle is number of multiplications
  saved by selecting it
• C cube corresponding to the rectangle
• Value Rows( (SCi ) -1)
• Maximum valued rectangular covering will give
  minimum number of multiplications
• Use greedy iterative covering by prime rectangles

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Finding single cube intersections (Condense
algorithm)
a4b3c
a2d1
a2b4c2
bc
a2b3c
d1 a2b3c
d2 bc
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Finding single cube intersections (Condense
algorithm)
d3 a2
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Finding single cube intersections (Condense
algorithm)

• Final CIM
• Final Implementation ( 7 multiplications)

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Distill Algorithm

• Distill algorithm exits after no more kernel
  intersections can be found

P1 x2yd1 d1 x yz
P2 4d1 xyz d2 4 - x P3 xyd1
Can further optimize by finding single cube
intersections
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Cube Literal Matrix (Condense Algorithm)
CIM for our example after Distill algorithm
Save 2 multiplications by extracting xy
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Condense Algorithm
Extracting xy
No more favorable cube intersections found
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Final Implementation

• Total 7 multiplications, 3 additions/subtractions
• Savings of 5 multiplications, 1
  addition/subtraction compared to CSE
• Impossible to obtain such results using
  conventional techniques

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Optimization of sin(x)
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Optimization of sin(x)
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Optimization of sin(x)
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Optimization of sin(x)

• Final Implementation
• X xx
• Sin(x) x(1 (-S3 (S5 S7X)X) ) X)
• Total 5 multiplications and 3 additions/subtractio
  ns
• SAME AS GNU C HAND optimized form

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Experimental Setup (Sequential processor)

• Signal processing and multimedia applications
• MP3 decoder, Mesa (graphics), Adaptive filter,
  FFT, FIR
• Taylor series approximation of trigonometric
  functions
• Optimizations on arithmetic subgraphs from
  Dataflow graphs (DFGs)
• Polynomials from computer graphics
• Multivariate polynomial approximation
• Compared number of operations with CSE and Horner
  form
• Estimated savings in clock cycles on ARM core

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Experimental results (comparing number of
operations from different methods)
Average run time 0.45s for our technique
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Experimental results (Improvement over CSE and
Horner)
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Conclusions

• Development of new algebraic technique for
  optimizing polynomial expressions
• Currently used for minimizing number of
  arithmetic operations using greedy rectangular
  covering
• Results better than conventional techniques

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Future Work

• Develop and implement optimal algorithms to
  compare results with our greedy heuristic
• Optimization for delay, energy
• Impact of optimizations on stability

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Thank You

• Questions ??

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• Extra slides

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Finding Kernel Intersections(Distill Algorithm)

• Worst case scenario for Distill algorithm
• Number of prime rectangles exponential in number
  of rows/columns
• Heuristic methods to find best prime rectangle
• In practice polynomial expressions are not so
  large