Efficient Representation of Interconnection Length Distributions Using Generating Polynomials

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