Abstract-In addition to wirelength, modern placers need to consider various constraints such as preplaced blocks and density. We propose a high-quality analytical placement algorithm considering wirelength, preplaced blocks, and density based on the log-sum-exp wirelength model proposed by Naylor et al. and the multilevel framework. To handle preplaced blocks, we use a two-stage smoothing technique, i.e., Gaussian smoothing followed by level smoothing, to facilitate block spreading during global placement (GP). The density is controlled by white-space reallocation using partitioning and cut-line shifting during GP and cell sliding during detailed placement. We further use the conjugate gradient method with dynamic step-size control to speed up the GP and macro shifting to find better macro positions. Experimental results show that our placer obtains very high-quality results.Index Terms-Legalization (LG), physical design, placement.

A switch module M with W terminals on each side is said to be universal if every set of nets satisfying the dimensional constraint (i.e., the number of nets on each side of M is at most W) is simultaneously routable through M. In this article, we present a class of universal switch modules. Each of our switch modules has 6W switches and switch-module flexibility three (i.e., F S ϭ 3). We prove that no switch module with less than 6W switches can be universal. We also compare our switch modules with those used in the Xilinx XC4000 family FPGAs and the antisymmetric switch modules (with F S ϭ 3) 1 suggested by Rose and Brown [1991]. Although these two kinds of switch modules also have F S ϭ 3 and 6W switches, we show that they are not universal. Based on combinatorial counting techniques, we show that each of our universal switch modules can accommodate up to 25% more routing instances, compared with the XC4000-type switch module of the same size. Experimental results demonstrate that our universal switch modules improve routability at the chip level. Finally, our work also provides a theoretical insight into the important observation by Rose and Brown [1991] (based on extensive experiments) that F S ϭ 3 is often sufficient to provide high routability.

We present in this paper a multilevel floorplanning/placement framework based on the B*-tree representation, called MB*-tree, to handle the floorplanning and packing for large-scale building modules. The MB*-tree adopts a two-stage technique, clustering followed by declustering. The clustering stage iteratively groups a set of modules based on a cost metric guided by area utilization and module connectivity, and at the same time establishes the geometric relations for the newly clustered modules by constructing a corresponding B*-tree for them. The declustering stage iteratively ungroups a set of the previously clustered modules (i.e., perform tree expansion) and then refines the floorplanning/placement solution by using a simulated annealing scheme. In particular, the MB*-tree preserves the geometric relations among modules during declustering, which makes the MB*-tree an ideal data structure for the multilevel floorplanning/placement framework. Experimental results show that the MB*-tree obtains significantly better silicon area and wirelength than previous works. Further, unlike previous works, MB*-tree scales very well as the circuit size increases.

In addition to wirelength, modern placers need to consider various constraints such as preplaced blocks and density. We propose a high-quality analytical placement algorithm considering wirelength, preplaced blocks, and density based on the log-sum-exp wirelength model proposed by Naylor et al. [20] and the multilevel framework. To handle preplaced blocks, we use a two-stage smoothing technique, Gaussian smoothing followed by level smoothing, to facilitate block spreading during global placement. The density is controlled by white-space re-allocation using partitioning and cut-line shifting during global placement and cell sliding during detailed placement. We further use the conjugate gradient method with dynamic step-size control to speed up the global placement and macro shifting to find better macro positions. Experimental results show that our placer obtains the best published results.