Closed form solution for optimal buffer sizing using the Weierstrass elliptic function

This paper presents a fundamental result on buffer sizing. Given an interconnection wire with n buffers evenly spaced along the wire, we would like to size all buffers such that the Elmore delay is minimized. It is well known that the problem can be solved by an iterative algorithm which sizes one buffer at a time. However, no closed form solution has ever been reported. In this paper, we derive a closed form buffer sizing function f(x) where f(x) gives the optimal buffer size for the buffer at position x. We show that f(x) can be expressed in terms of the Weierstrass elliptic function p(x) and its derivative p'(x)