Optimal shape function for a bidirectional wire under Elmore delay model

In this paper we determine the optimal shape function for a bidirectional w ire under the Elmore delay model. Given a bidirectional wire of length L, l et f(x) be the width of the wire at position st 0 less than or equal to x l ess than or equal to L, Let T-DR be the right-to-left delay. Let T-DL be th e left-to-right delay. Let T-BD = alpha T-DR + 3T(DL) be the total weighted delay where a greater than or equal to 0 and 3 greater than or equal to 0 are given weights such that alpha+beta = 1. We determine f(x) so that TBD i s minimized. Our study shows that, if alpha not equal beta, the optimal sha pe function is f(x) = c, for some constant c; if alpha f beta, the optimal shape function can be expressed in terms of the Lambert's W function as f(x ) = -c(f)/2c(0)((1/W(-ac(-bx)))+1), where c(f) is the unit length fringing capacitance, co is the unit area capacitance, a and b are constants in term s of the given circuit parameters. If alpha = 0 or beta = 0, our result gi ies the optimal shape function for a unidirectional wire.