Efficient Representation of Interconnection Length Distributions Using Generating Polynomials

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Efficient Representation of Interconnection
Length Distributions Using Generating Polynomials

• D. Stroobandt (Ghent University)
• H. Van Marck (Flanders Language Valley)
• Supported by an IUAP research program on optical
  computing of the Belgian Government and the Fund
  for Scientific Research, Flanders

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Outline

• Enumerating interconnection length distributions
• Advantages of generating polynomials
• Construction of generating polynomials
• Extraction of the distributions
• Examples
• Conclusions

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Enumerating Interconnection Length Distributions

• Distributions contain two parts
• site density function and probability distribution

all possibilities
probability of occurrence
requires enumeration
shorter wires more probable
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Enumerating Interconnection Length Distributions
(cont.)

• Simple Manhattan grids not so difficult
• just start counting
• more clever use convolution
• But what with...?
• anisotropic grids
• partial grids

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Generating Polynomials

• Site function (discrete distribution f(l))
  describes, for each length l, the number of pairs
  between all cells of a set A and a set B, a
  distance l apart (enumeration problem)
• Two ways of reducing calculation effort
• using generating polynomials
• using symmetry in the topology of the
  architecture
• Generating polynomial moment-generating
  polynomial function of f(l) (Z-transform)

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Advantages of Generating Polynomials

• Efficient representation
• allows easy switching to path-based enumeration
• compact representation
• as rational function
• example

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Advantages (cont.)

• Easy to find relevant properties
• total number of paths
• average length (also higher order moments)
• Easy construction of complex polynomials

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Construction of Polynomials

• Composition (adding and subtracting polynomials)

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Construction of Polynomials (cont.)

• Convolution (multiplication of polynomials)
• composing paths from base paths

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Extraction of Distributions

• Construction of polynomials much easier than
  construction of distributions but how to extract
  distributions from polynomials?
• Much simpler than general Z-transform
• Theorem
• Quotient term important, remainder vanishes
• Note summation bound to be chosen between n-1
  and n-i1 without effect on result

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Extraction of Distributions (cont.)

• Simple substitution of terms by summation of
  combinatorial functions (with few factors)
• The different ranges of the distribution
  naturally follow from this!

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Examples

• Manhattan grid
• convolution of x, y parts
• subtract
• divide by 2
• extraction substituting

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Examples (cont.)

• Complicated architectures

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Examples (cont.)
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Examples (cont.)

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Examples (cont.)

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Examples (cont.)

• Resulting generating polynomial
• Extraction by simple substitution and calculation
  of the combinatorial functions

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Conclusions

• Generating polynomials make enumeration easier
• more efficient representation (1 equation, not 5)
• easy to obtain characteristic parameters
• construction facilitated by using symmetry
  (composition, convolution easy with polynomials)
• extraction by substitutions of terms, can be
  automated by symbolic calculator tools!
• Same technique can be used for calculating
  cell-to-I/O-pad lengths
• Enumeration viable for complex architectures