Optimal shape function for a bidirectional wire under Elmore delay model

In this paper, we determine the optimal shape function for a bidirectional wire under the Elmore delay model. Given a bidirectional 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, 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 unidirectional wire