Geometric quadrisection in linear time, with application to VLSI placement

AbstractWe consider the following problem: given a set of points in the plane, each with a weight, and capacities of the four quadrants, assign each point to one of the quadrants such that the total weight of points assigned to a quadrant does not exceed its capacity, and the total distance is minimized.This problem is most important in placement of VLSI circuits and is likely to have other applications. It is NP-hard, but the fractional relaxation always has an optimal solution which is “almost” integral. Hence for large instances, it suffices to solve the fractional relaxation. The main result of this paper is a linear-time algorithm for this relaxation. It is based on a structure theorem describing optimal solutions by so-called “American maps” and makes sophisticated use of binary search techniques and weighted median computations.This algorithm is a main subroutine of a VLSI placement tool that is used for the design of many of the most complex chips

Flow-based Partitioning and Fast Global Placement in Chip Design

VLSI placement is one of the major steps in the chip design process and an interesting subject of research in industry and academia. Recent chips consist of several millions of circuits connected by millions of nets. The classical placement objective of finding positions for circuits and minimizing netlength among them is an ongoing issue in optimization of chip performance. The increasing instance sizes, the tightness of timing and routability constraints impose a real challenge to the design flows and the designers, which often cannot be addressed properly without considering them explicitly within the placement. Many of the complex design methodologies follow an iterative approach, using placement several times in this process. Thus, placement runtime has a severe impact on the turnaround time in chip development. The major contributios of this thesis deal with the global placement, a common relaxation of the placement problem, which computes rough positions of the circuits minimizing the total length of wires to interconnect the. Based on the idea of subsequent quadratic netlength minimization and partitioning, as in BonnPlace [BrennerStruzynaVygen:2008], we present several new algorithms, generalized data structures and a completely new implementation of this top-down placement scheme. We introduce and formalize the concept of movebounds which are position constraints on subsets of cells. Movebounds, which can be regarded as mandatory or soft constraints, provide a mechanism to explicitly incorporate movement constraints to the placement which result from issues of timing, power and routability. With inclusive movebounds, such restrictions can be assigned to groups of circuits without any influence to other placeable objects. The other constraints, namely the exclusive movebounds, are of particular interest for semi-hierarchical approaches, as they can be used to obtain a flat view of the design and prevent cells from being placed into hierarchy units. Both provide a toolbox to the designer and allow the control of particular circuit sets without netlist manipulations. We also present a top-down partitioning scheme and extend the legalization algorithm of [BrennerVygen:2004] to be able to deal with millions of cells and dozens of movebounds efficiently. The presented algorithm can handle different types of overlapping movebounds, even in legalization, and produces significantly better results than a modern industrial tool. We present a novel partitioning algorithm for global placement. Unlike previous iterative and recursive approaches, the new method provides a global view of the problem using a novel MinCostFlow model with extremely fast and highly parallelizable local realization steps. The new flow-based partitioning can address density targets much more accurately and lowers the risk of density violations. The presented MinCostFlow model does not depend on the number of cells, making it highly interesting for large and huge designs. Moreover, the embedded flow structure responds to the chip's floorplan much better than the classical global partitioning approach. Another significant advantage of this algorithm is the fact that it can be applied to any initial placement and guarantees a feasible (fractional) solution (if one exists), improving the tool's reliability, even with movebounds and starting from placements with significant density violations. Using this method we can extend the congestion-driven placement to a combined movement, density adjustment, and cell size inflation approach. This method is able to handle movebounds and guarantees to resolve density overloads properly. Flow-based partitioning creates the opportunity of applying local, density unaware, optimization steps within global placement and allows it to break the strict recursive structure of levels and save runtime. The extended flexibility and runtime improvement are not the only advantages. The proposed flow realization, which is a combination of local quadratic programs and local partitioning, does not only yield a runtime improvement, but also seems to merge connectivity information to partitioning in a much better way than the old recursive partitioning approach. The new flow-based partitioning helps to significantly improve the results of our placement also in terms of netlength. We provide fast data structures for hierarchically clustered netlists and extend the net models Clique and Star to be applied within the clustered netlists efficiently. We show how shared-memory parallelization can be used for speeding up various routines in placement, without the loss of repeatability. In addition, we commit ourselves to the clustering problem, finding circuit groups which should be placed in the vicinity of each other. In order to provide global information for a fast bottom-up clustering, we propose to incorporate connectivity information using random walks. To this end, we show how the hitting times can be efficiently retrieved from large netlist hypergraphs. Due to the proposed model, parallel computation on sparse, shared-memory matrices can be used for computing hitting times to several targets simultaneously. Combined with a bottom-up clustering, even our preliminary approach significantly outperforms the popular BestChoice} algorithm [Nam et al. 2005]. We conclude this thesis by providing several experimental results on a large testbed of real-world chips and benchmarks demonstrating the performance of our tool. Without movebounds, our tool performs as good as a state-of-the-art force directed placer, but is more than 5x faster. We achieve the same speedup over the old BonnPlace, but produce significantly better results, on average more than 8%. With movebounds, our placements are more than 30% shorter compairing to the force-directed placer and our tool is 9x-20x faster. Our tool also produces the best results on the latest ISPD 2006 placement benchmarks

Timing-Driven Macro Placement

Placement is an important step in the process of finding physical layouts for electronic computer chips. The basic task during placement is to arrange the building blocks of the chip, the circuits, disjointly within a given chip area. Furthermore, such positions should result in short circuit interconnections which can be routed easily and which ensure all signals arrive in time. This dissertation mostly focuses on macros, the largest circuits on a chip. In order to optimize timing characteristics during macro placement, we propose a new optimistic timing model based on geometric distance constraints. This model can be computed and evaluated efficiently in order to predict timing traits accurately in practice. Packing rectangles disjointly remains strongly NP-hard under slack maximization in our timing model. Despite of this we develop an exact, linear time algorithm for special cases. The proposed timing model is incorporated into BonnMacro, the macro placement component of the BonnTools physical design optimization suite developed at the Research Institute for Discrete Mathematics. Using efficient formulations as mixed-integer programs we can legalize macros locally while optimizing timing. This results in the first timing-aware macro placement tool. In addition, we provide multiple enhancements for the partitioning-based standard circuit placement algorithm BonnPlace. We find a model of partitioning as minimum-cost flow problem that is provably as small as possible using which we can avoid running time intensive instances. Moreover we propose the new global placement flow Self-Stabilizing BonnPlace. This approach combines BonnPlace with a force-directed placement framework. It provides the flexibility to optimize the two involved objectives, routability and timing, directly during placement. The performance of our placement tools is confirmed on a large variety of academic benchmarks as well as real-world designs provided by our industrial partner IBM. We reduce running time of partitioning significantly and demonstrate that Self-Stabilizing BonnPlace finds easily routable placements for challenging designs – even when simultaneously optimizing timing objectives. BonnMacro and Self-Stabilizing BonnPlace can be combined to the first timing-driven mixed-size placement flow. This combination often finds placements with competitive timing traits and even outperforms solutions that have been determined manually by experienced designers