4.1 Comparing Cost and Radius

Analysis of Elmore delay in RC trees suggests that low-delay routing trees should trade oo cost and radius according to net size and interconnect technology. Previous approaches 2, 3, 8, 13] begin with a depth-rst traversal of T M and insert shortest paths as needed to maintain a prescribed radius bound. In contrast, our ALG1 (and ALG2) directly combine the recurrences for Prim's MST algorithm and Dijkstra's SPT algorithm. This more natural tradeoo signiicantly improves over the cost-radius performance of BRBC 3, 8] and KRY 13]. Simulation results show that ALG1 yields routing trees with less maximum and average delay than ALG2, KRY or BRBC in both IC and MCM interconnect technologies; improvements over the \provably good" BRBC approach are particularly substantial. Our delay reductions over xed constructions are also substantial (such constructions include standard MST routing and heuristic minimum Steiner tree routing as well as the recent Steiner arborescence approach of 9]). It is therefore of interest to pursue integration of ALG1 within existing performance-driven global routers. 6 Acknowledgements We thank Professor Jee Salowe, Professor Samir Khuller, and Dr. Neal Young for illuminating discussions of the shallow-light literature and their works in this area. We also thank the authors of 25] for use of their simulator code, and Professor Wayne W.-M. Dai for releasing MCM simulation parameters for our study. The anonymous referees gave many helpful comments on the original draft of this work. Table 4: Maximum source-sink delay and average source-sink delay in the best tree for each algorithm, after edge-overlapping is used to induce a Steiner routing. Values are given as a ratio to corresponding MST delay values, averaged over 250 random instances. Numbers in parentheses give average best parameter value for each algorithm. similarly compare our constructions against the standard minimum Steiner tree heuristic of edge-overlapping a minimum spanning tree (this is identical to ALG1-Steiner with xed c = 0). For this standard technique using MCM parameters, the ratios of average sink delays to corresponding MST delays are 0.802 (4 sinks), 0.808 (8 sinks), 0.818 (16 sinks). By contrast, a single ALG1 execution using the xed, best c value from Table 3 will attain ratios of 0.788, 0.590, and 0.461 respectively. When edge-ipping is added in ALG1-Steiner and we use the best c values from Table 4, the average-delay ratios are 0.756, 0.586, and 0.468 respectively. (Delay ratios obtained by considering the family of ALG1 trees …