Tance and Unit Wire Resistance. 3

The delays at all sink nodes were measured using the two-pole circuit sim-ulator proposed by Zhou et al. 17] and discussed in 3]. This simulator is a computationally eecient code which has produced very accurate results (within a few percent) when tested against SPICE. We consider both the average delay (taken over all sinks) and the worst-case delay (i.e., the latest arrival time of the signal to any sink); all results are normalized to the corresponding values for the MST routing. We make the following observations. 1. While BRBC tends to yield lower tree cost for any xed , if we consider the family of trees over each instance, we see that the AHHK solution will almost always have lower cost for any given tree radius. 2. AHHK yields lower worst-case and average-case signal delay than BRBC (Table 2). As nets become larger, MST radius becomes quite large, and the delay improvements from the AHHK solution become more obvious. 3. While tree cost closely reeects Elmore delay in the IC technology, diierences between the algorithms are small since both will closely approximate the MST for those values of that minimize delay. On the other hand, since tree radius is dominant for MCM interconnects, AHHK is superior in this technology since it gives less cost for a given radius. 4. For each instance of each net size, we recorded the c and values that aaorded lowest signal delay over the entire family of 21 trees generated by each algorithm. The average \best" parame-terization is reported in Table 3, and is useful in guiding the application of AHHK when it is too expensive to generate an entire family of trees. 5 Conclusions Analysis of distributed RC delay shows that performance-driven routing requires tree constructions that can trade oo cost and radius according to interconnect technology and net size. Previous approaches 2, 4, 10] essentially rely on a depth rst traversal of the MST and insert shortest paths as needed to maintain a prescribed radius bound. In contrast, our new AHHK approach directly combines the recurrences for Prim's MST algorithm and Dijk-stra's SPT algorithm. The result is an elegant trade-oo between radius and cost which empirically yields lower-cost trees than the BRBC algorithm 4] for any given tree radius. Simulation results show that the AHHK algorithm constructs routing trees with sig-niicantly less maximum and average delay than the BRBC method 4], in both …