optimal shape function for a bi-directional wire under elmore delay model
In this paper, we determine the optimal shape function for a bi-directional wire under the Elmore delay model. Given a bi-directional wire of length L, let f(x) be the width of the wire at position x, 0\leq x \leq L. Let T_{DR} be the right-to-left delay. Let T_{DL} be the left-to-right delay. Let T_{BD}=\alpha T_{DR}+\beta T_{DL} be the total weighted delay where \alpha\geq 0 and \beta\geq 0 are given weights such that \alpha+\beta=1. We determine f(x) so that T_{BD} is minimized. Our study shows that, if \alpha=\beta, the optimal shape function is f(x)=c, for some constant c; if \alpha\neq \beta, the optimal shape function can be expressed in terms of the Lambert's W function as f(x)=-\frac{c_f}{2c_0}(\frac{1}{W(-ae^{-bx})}+1), where c_f is the unit length fringing capacitance, c_0 is the unit area capacitance, a and b are constants in terms of the given circuit parameters. If \alpha=0 or \beta=0, our result gives the optimal shape function for a uni-directional wire.
optimal shape optimality shape functional bi-directional wire elmore delay delay model paper posit circuits direct