SLIP 2000April 8, 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 2000April 8, Outline Enumerating interconnection length distributions Advantages of generating polynomials Construction of generating polynomials Extraction of the distributions Examples Conclusions

SLIP 2000April 8, Enumerating Interconnection Length Distributions Distributions contain two parts: site density function and probability distribution all possibilities requires enumeration probability of occurrence shorter wires more probable

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

SLIP 2000April 8, Advantages (cont.) Easy to find relevant properties –total number of paths –average length (also higher order moments) Easy construction of complex polynomials n A=B

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

SLIP 2000April 8, Construction of Polynomials (cont.) Convolution (multiplication of polynomials) –composing paths from “base” paths * BA nn C A nn || D nn C * B nn D

SLIP 2000April 8, 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 2000April 8, 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 2000April 8, A=B Examples Manhattan grid –convolution of x, y parts –subtract –divide by 2 –extraction = substituting n

SLIP 2000April 8, B Examples (cont.) Complicated architectures A k n || 2 X C n B k

SLIP 2000April 8, Examples (cont.) C n B k k || C n E C n F +

SLIP 2000April 8, C n E k C n F + Examples (cont.) || n n k * n * k x * k

SLIP 2000April 8, Examples (cont.) || C n E k C n F + * k 1 k+1 x

SLIP 2000April 8, Examples (cont.) Resulting generating polynomial: Extraction by simple substitution and calculation of the combinatorial functions:

SLIP 2000April 8, 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