5 Conclusion and Future Directions

We have introduced fast and eeective k-way partitioning solutions for the two regimes k n and k = (n). Our contributions include a new \partitioning-speciic" net model used in constructing the Laplacian of the netlist, the use of fast Lanczos implementations to compute the d-dimensional geometric embedding eeciently, and a careful choice of geometric clustering objectives and algorithms. We believe that the spectral geometric embedding preserves important graph properties with respect to partitioning and clustering; moreover, the eeciency of geometric clustering opens the door for new heuristics. We are pursuing several directions for future research. Since eigenvectors of the smaller eigenval-ues correspond to better placements, scaling heuris-tics may be useful for perturbing the eigenvector-based embedding, e.g., we may compute the ratio of the i th-smallest eigenvalue i to the smallest eigenvalue 2 , then reduce the components of the i th eigenvector by this ratio. We believe that our studies of diierent net models and clustering algorithms represent promising directions. For example, the recent thesis of Zien 23] reports work of Chan, Schlag, and Zien which achieves excellent Scaled Cost results by combining a net model due to Frankle, a \partial KC" approach, and use of netlist information. This latest method of Chan et al. uses a variant of KC to make \easy" assignments to clusters; netlist connectivity information is then used to resolve more diicult points. By contrast , our present method relies only on the geometric embedding. Finally, we hope to improve the running time of AGG since this method gives results uniformly superior to those of KC. Currently, the O(n 2) method of 1] for general graphs is also the most eecient method known for geometric instances. 6 Acknowledgements We are very grateful to the authors of 4] for providing early access to their works. References 1] J. P. Benz ecri, \Construction d'une Classiication Ascen-dante Hi erarchique par la Rechereche en Chaine des Voisins R eciproques", in Les Cahiers de l'Analyse des Donn ees