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≤x≤L. Let T_DR be the right-to-left delay. Let T_DL be the left-to-right delay. Let T_BD = αT_DR+βT_DL be the total weighted delay where α≥0 and β≥0 are given weights such that α+β = 1. We determine f(x) so that T_BD is minimized. Our study shows that, if α = β, the optimal shape function is f(x) = c, for some constant c; if α≠β, the optimal shape function can be expressed in terms of the Lambert's W function as f(x) = -c_f/2c₀(1/W(-ae^-bx)+1), where c_f is the unit length fringing capacitance, c₀ is the unit area capacitance, a and b are constants in terms of the given circuit parameters. If α = 0 or β = 0, our result gives the optimal shape function for a uni-directional wire.