Geometry of Arrays: Mathematics of Arrays and ycalculus

1
Geometry of Arrays Mathematics of Arrays and
y-calculus

• Lenore R. Mullin
• Computer Science Department
• College of Computing and Information University
  at Albany, State University of New York

2
Overview

• Mathematics of Arrays (MoA) an algebraic system
  for dealing with arbitrary multi-dimensional
  arrays
• y-calculus reducing a complicated set of array
  operations to direct indexing of the original
  array
• Elimination of temporary arrays significant
  enhancement in speed and reduction of memory
  requirements
• Example block decomposition lifts (raises) the
  dimension of the a array

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Reshape Operator

2-dimensional
Becomes 4-dimensional
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Block Decomposition
2-dimensional
Viewed as 4-dimensional
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Array shapes

• Shape operator r returns a vector containing the
  lengths of each dimension
• First vector (two-dimensional)
• Second vector (four-dimensional)

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Transpose operator

• Transpose operator f permutes the dimensions

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y-operator

• The y operator extracts a component of the array
• Full index extracts an element
• Partial index extracts a sub array

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Reshape operator

• The process of lifting the dimension is carried
  out with the reshape operator
• We write
• And

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Hypercube Representation

• The arrays and are examples of
  hypercubes
• Often array operations simplify in the hypercube
  representation
• In a hypercube all dimensions have length 2
• For an arbitrary array B we write

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Product operator

• The product operator p returns an integer equal
  to the product of the elements of a vector
• The total number of elements in A is
• Number of hypercube dimensions

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Composing operations

• The hypercube representation of A is thus
  written
• Likewise
• Lastly

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Big Picture y-calculus

• Multiple operations acting on original array
  (previous slide)
• Each operation defined in terms of its action on
  the indices of the array
• By applying all operator definitions we
  y-reduce the composite operations
• Final result prescription (operational normal
  form (ONF)) for indexing operations on original
  array

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y-reduction

• By selecting the th component of the
  two-dimensional array
• We obtain the ONF for all 0ltilt2,0ltjlt2
• (20)(20)(20) (20)0, (20)(20)(20)
  (21)1,
• (20)(21)(20) (20)4, (20)(21)(20)
  (21)5
• NOTE a is the row major layout of A.

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Temporal Index Composition

• Usual methods Pipeline multiple temporal
  operations
• LU followed by QR for example
• Decompose LU
• Map to Processors
• Execute
• Start the pipeline
• Set up for QR
• Decompose
• Map to processors
• Execute
• Feed pipeline

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Temporal Index CompositionLU Decomposition
3
2
3
2
1
3
1
2
3
2
T0
T1
T2
1
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LU to QR(or anything else)

• Think of it as indexing
• Temporal index 0 1 2
• Then again with QR temporal index 0 1 2
• Then again with something else index 0 1 2
• Formulate temporal index as an array
• index like processors or anything else
• Consequently temporal indices can be
• Transposed to produce
• 0 0 0
• 1 1 1
• 2 2 2

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3
3
T0
3
3
2
2
T1
2
2
T2
1
1
1
1
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Temporal Index Composition

• Without transpose Data moves around connecting
• networks.
• Multiple decompositions and communication
• Filling and emptying pipeline causes delays
• With transpose
• One decomposition
• Mimimal data flow
• Highest Performance

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Impact efficiency and programming ease

• Efficiency through direct indexing access we
  eliminate temporary arrays allowing BIGGER
  PROBLEMS
• Minimizing operations significant speed
  enhancement
• Programming ease ONF directly translated into
  code (no thinking or hackery)
• Recent FFT cache optimization with design (ONF)
  in handcode built in ONE DAY and it WORKED!!

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Mathematical Reasoning

• Same mathematics used to describe the problem as
  is used to describe the organization of the
  machine
• Machine decomposition over levels of memory,
  processors, networks, FPGAs in terms of array
  dimensions
• Provability two designs leading to the same ONF
  are equivalent!

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Success to date

• This approach applied to many operations
  ubiquitous across science and engineering
  disciplines
• FFT, QR decomposition, LU decomposition
• Digital signal processing RADAR (MIT Lincoln
  Labs)
• Most recent FFT cache optimization factor of
  FOUR speedup over previous records (next talk)

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Ongoing Research

• General-radix, multiple-dimension FFT optimized
  over cache and processors
• Application-specific hardware programming hybrid
  software/hardware (FPGA) calculations
• Grid computing
• Large-scale materials simulations with
  density-functional theory
• Quantum computing and quantum algorithms (Density
  Matrix naturally expressed as a hypercube and
  qubits as indices(talk follows).