3.4.4 Decomposition Theorem for T
نویسندگان
چکیده
Table 2: Percent a b o ve optimum of Elmore delay to a single critical sink and wire length for three Steiner tree constructions cost comparison is with 1-Steiner. Averages were taken over 200 random nets for each net size. 4.2 Elmore-Optimality of Generic" SERT Algorithm The counterexample in Section 3.3 showing that BB-SORT is not always optimal was carefully constructed by hand; even then, BB-SORT w as only 0.06 above optimal. Thus, we believe that BB-SORT is within one percent of optimal in essentially all cases. In Table 3 we compare SERT and 1-Steiner with the SORT" trees of BB-SORT. It appears that the SERT constructions are very nearly optimal: the worst case occurs for IC2 and IC3 for jN j = 9, where SERT delays are only 3.9 above those of BB-SORT. Table 3: Percent a b o ve o p t i m um of maximum sink Elmore delay and wire length for three Steiner tree constructions cost comparison is with 1-Steiner. 200 random nets are used for each net size. 5 Conclusions Two main theoretical results show that the BB-SORT-C branch-and-bound method can be used to nd Steiner trees that are optimal for any linear combination of sink Elmore delays. Our rst result is a generalization of Hanan's theorem 11 to Elmore delay. W e then establish a new decomposition theorem for optimal Elmore-delay trees. When the objective is to minimize the maximum Elmore delay in a net, we give a c o u n ter-example for which our BB-SORT d o e s not return the optimal tree. Nevertheless, we believe that BB-SORT will almost always return a tree well within one percent of optimal. BB-SORT-C and BB-SORT m a y be used for routing small nets; a more far-reaching implication of our results lies in delineating the achievable space of performance-driven routing solutions. Our simulations for the SERT-C heuristic of 2 indicate that it is within 5 of optimal on average for 5-pin nets and within 16 on average for 9-pin nets. The generic" SERT constructions appear to be even closer to optimal within 1.5 for jN j =5 and 4 for jN j=9. 6 Acknowledgements We are grateful to Mr. Ashok Vittal of UC Santa Barbara for helpful comments on an earlier draft. Part of this work was performed during a sabbatical visit to UC Berkeley; support from NSF MIP-9117328 and the hospitality …
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