Optimal wire-sizing formula under the Elmore delay model

In this paper, we consider non-uniform wire-sizing. Given a wire segment of length ℓ, let f(x) be the width of the wire at position x, 0≤x≤ℓ. We show that the optimal wire-sizing function that minimizes the Elmore delay through the wire is f(x) = ae^-bx, where a>0 and b>0 are constants that can be computed in O(1) time. In the case where lower bound (L>0) and upper bound (U>0) on the wire widths are given, we show that the optimal wire-sizing function f(x) is a truncated version of ae^-bx that can also be determined in O(1) time. Our wire-sizing formula can be iteratively applied to optimally size the wire segments in a routing tree.