Optimal wire-sizing function under the Elmore delay model with bounded wire sizes

In this brief, we develop the optimal wire-sizing functions under the Elmore delay model with bounded wire sizes. 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) of 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.