Data Shackling

1
Data Shackling

• Locality enhancement of dense numerical linear
  algebra codes
• Traversals along co-ordinate axes
• Data-centric reference for each statement
• No array copying

2
Workloads

• Basic Linear Algebra Subroutines
• Dot Product xTy
• Sapxy a x y
• Matrix vector y A x
• Triangular Solve Lx b
• Matrix Multiplication C C AB
• Matrix Factorizations
• Cholesky
• LU
• QR

3
Why is BLAS approach not good enough?

• What is BLAS?
• Machine specific code for each BLAS
• Matrix factorizations are blocked
• Exposes a BLAS like interface
• ?
• BLAS not portable but matrix codes are
• ?
• Automating the production of block codes not easy
• Too difficult an approach for compilers

4
What is a data shackle?

• One array in the program
• Divided into blocks
• Using parallel equally spaced cutting planes
• Order for visiting blocks of data
• One reference of that array is selected for each
  statement

5
IntuitivelyA Data Shackle

• Specifies the order for touching blocks
• Data centric reference
• Which iterations of each statement are performed
• When that block is touched
• Code is generated to group those iterations
  together

6
Example Matrix Multiplication

• Obtain the data shackle on C
• Divide C into
• 25 x 25 blocks
• Vertical and horizontal cutting planes
• Visited L-R T-B
• C(i,j) is data-centric reference

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Example Contd
8
Discussion on example

• More optimzed code can be produced by folding the
  loop bounds
• Within a block order of instructions preserved
• Cannot say the same for program
• Not the real blocked code
• Low locality for A and B
• Compose shackles .. later

9
Are all shackles legal?

• Definitely not
• Program instructions reordered