Shaping a VLSI wire to minimize Elmore delay with consideration of coupling capacitance

In this paper, by using calculus of variations, we determine the optimal shape for a wire under the Elmore delay model. Coupling capacitance has been taken into consideration explicitly by treating it as another source of grounded capacitance. Given two wires in parallel, one has uniform width and the other has non-uniform width whose shape is described by a function f(x). Let T_D be the delay through the non-uniform wire. We determine f(x) such that T_D is minimized. We also extend our study to the case where a non-uniform wire has two neighboring wires. Our study shows that the optimal shape function satisfies an integral equation. Numerical methods are employed to solve the corresponding differential equation and carry out the integration. We provide an efficient algorithm to find the optimal solution. Experiments show that it only takes several iterations to get the optimal results by using our algorithm. Our experiments also show that the wire delay T_D is a convex function of the wire width at the driver end.