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Efficient Representation of Interconnection Length Distributions Using Generating Polynomials

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18 diapositives · 323 Views · Oct 05, 2014

This study explores the efficient enumeration of interconnection length distributions using generating polynomials, highlighting their advantages such as compact representation and ease of calculation. It details the construction and extraction of polynomials, emphasizing their role in analyzing complex network architectures. The presentation includes examples, especially of Manhattan grids, demonstrating practical applications. The authors show that generating polynomials streamline the process of deriving essential properties of interconnections, facilitating advanced computations in network design.

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Efficient Representation of Interconnection Length Distributions Using Generating Polynomials
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  1. 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 SLIP 2000

  2. Outline • Enumerating interconnection length distributions • Advantages of generating polynomials • Construction of generating polynomials • Extraction of the distributions • Examples • Conclusions SLIP 2000

  3. 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 SLIP 2000

  4. 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 SLIP 2000

  5. 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) SLIP 2000

  6. l(p)=8 p n A=B Advantages of Generating Polynomials • Efficient representation • allows easy switching to path-based enumeration • compact representation as rational function • example SLIP 2000

  7. n A=B Advantages (cont.) • Easy to find relevant properties • total number of paths • average length (also higher order moments) • Easy construction of complex polynomials SLIP 2000

  8. A n A n A n __ || B B B ||| 2n A n X B Construction of Polynomials • Composition (adding and subtracting polynomials) SLIP 2000

  9. n n n n A B A C n n n n B C D D Construction of Polynomials (cont.) • Convolution (multiplication of polynomials) • composing paths from “base” paths * || * SLIP 2000

  10. 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-i+1 without effect on result SLIP 2000

  11. 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! SLIP 2000

  12. n A=B A=B Examples • Manhattan grid • convolution of x, y parts • subtract • divide by 2 • extraction = substituting SLIP 2000

  13. k k A C B B n n 2 X || Examples (cont.) • Complicated architectures SLIP 2000

  14. k C E n k C B + || C n n F Examples (cont.) SLIP 2000

  15. k C E n n k + C n * n k k * * F n x Examples (cont.) || SLIP 2000

  16. k C E n + C n k+1 k x * 1 F Examples (cont.) || SLIP 2000

  17. Examples (cont.) • Resulting generating polynomial: • Extraction by simple substitution and calculation of the combinatorial functions: SLIP 2000

  18. 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 SLIP 2000