Self-similar structure of wire length distribution of random logic

A general scaling theory is proposed to estimate a wire length distribution based on the self-similarity structure of random logic. It is theoretically shown that the d-dimensional wire length distribution denoted by fℓ(d) is of the form fℓ(d) ℓ-γ1(d) with a characteristic exponent γ1(d) = α(d) + 2 − dp for ℓ < ℓcrossover with some crossover length ℓcrossover, where ℓ is a wire length and p is the Rent’s partition exponent. The parameter α(d) is equal to d − 1 and d for serialized and parallel wiring configurations, respectively. For wire lengths larger than ℓcrossover, fℓ(d) ℓ-γ2(d) is obtained with γ2(d) = α(d) + 2. These results are in good agreement with experiments.