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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- b that can also be determined in 0(1) time. Our wire-sizing formula can be iteratively applied to optimally size the wire segments in a routing tree. 1 Introduction As VLSI technology continues to scale down, interconnect delay has become the dominant factor in deep submicron designs. As a result, wire-sizing plays an important role in achieving desirable circuit performance.

We have, f is an optimal constrained wire-sizing function. The above method always requires the computation of all six functions in F. With the help of additional analysis, we can speed up the procedure. Table 2 shows a set of six feasibility conditions {fOA, tB, PC, CAB, SCBC, SCABC} on L. Let F = {A, B, C, AB, CB, ABC}. 313 c~,RdU+,ro, Table 1: Definitions of the wire-sizing functions CcL _ fA, fB, fc, fAB, fBC, wire r - (PARB 0 S BC (PABC ~ and fABc. ienghr BC AB A B R d / C L C ratio Figure 7: Relationships among the six types of functions with respect to C and Rd n 0 in In d' U > max IRdln ~jRd L2< (1 + In L) C'L -I L> mar{L - C ma'C{(1 + (I + InLL = L2 According to Theorem 6, we only need to check the six feasibility conditions.

We now define six wire-sizing functions fBC, fABC ae-b L be an optimal constrained wire-sizing function. We have, f is decreasing on [0,C]. According to Theorem 4, the optimal wire-sizing function f, similar to the one shown in Figure 5, consists of (at most) three parts. The first part is f(r) = U, the middle part is a decreasing function, and the last part is f(r) = L. The three parts of f(x) partition W into three wire segments, A, B, and C, where A has width U, C has width L, and B is defined by the middle part of f(r).

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