Optimal shape function for a bi-directional wire under Elmore delay model

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/spl les/x/spl les/L. Let T/sub DR/ be the right-to-left delay. Let T/sub DL/ be the left-to-right delay. Let T/sub BD/=/spl alpha/T/sub DR/+/spl beta/T/sub DL/ be the total weighted delay where /spl alpha//spl ges/0 and /spl beta//spl ges/0 are given weights such that /spl alpha/+/spl beta/=1. We determine f(x) so that T/sub BD/ is minimized. Our study shows that, if /spl alpha/=/spl beta/, the optimal shape function is f(x)=c, for some constant c; if /spl alpha//spl ne//spl beta/, the optimal shape function can be expressed in terms of the Lambert´s W function as f(x)=-(c/sub f/)/(2c/sub 0/)(1/(W(-ae/sup -bx/)+1)), where c/sub f/ is the unit length fringing capacitance, c/sub 0/ is the unit area capacitance, a and b are constants in terms of the given circuit parameters. If /spl alpha/=0 or /spl beta/=0, our result gives the optimal shape function for a uni-directional wire.