The traditional way of approaching placement problems in computer-aided design (CAD) tools for analog layout is to explore an extremely large search space of feasible or unfeasible placement configurations, where the cells are moved in the chip plane (being even allowed to overlap in possibly illegal ways) by a stochastic optimizer. This paper presents a novel exploration technique for analog placement operating on a subset of tree representations of the layout—called symmetric-feasible, where the typical presence of an arbitrary number of symmetry groups of devices is directly taken into account during the search of the solution space. The computation times exhibited by this novel approach are significantly better than those of the algorithms using the traditional exploration strategy. This superior efficiency is partly due to the use of segment trees, a data structure introduced by Bentley, mainly used in computational geometry.

In order to handle device matching in analog circuits, some pairs of modules are required to be placed symmetrically. This paper addresses this device-level placement problem for analog circuits and our approach can handle symmetry constraint and other placement constraints simultaneously. The problem of placing devices with symmetry constraint has been extensively studied but none of the previous works has considered symmetry constraint with other placement constraints simultaneously. Instead of handling the constraints by having a penalty term in the cost function to penalize violations, a unified method is proposed that, by adjusting the edge weights in a pair of constraint graphs, can try to satisfy all the placement and symmetry constraints simultaneously in a candidate floorplan solution. The maximum distance of the modules in a symmetry group from the corresponding symmetry axis will be minimized in this weight adjusting step, in order to minimize the total packing area. We have compared our method with the most updated results on this problem [2] when there are only symmetry constraints and results show that our approach can give solutions of better quality, in an acceptable amount of run time. We will also demonstrate the effectiveness of our approach in handling different types of constraints simultaneously by testing on data sets with both symmetry and other placement constraints, and the results are very promising.

In order to handle device matching for analog circuits, some pairs of modules need to be placed symmetrically with respect to a common axis. In this paper, we deal with the module placement with symmetry constraints for analog design using the Transitive Closure Graph-Sequence (TCG-S) representation. Since the geometric relationships of modules are transparent to TCG-S and its induced operations, TCG-S has better flexibility than previous works in dealing with symmetry constraints. We first propose the necessary and sufficient conditions of TCG-S for symmetry modules. Then, we propose a polynomialtime packing algorithm for a TCG-S with symmetry constraints. Experimental results show that the TCG-S based algorithm results in the best area utilization. 1

Abstract – This paper presents an improved topological algorithm for device-level analog placement with symmetry constraints. Based on the exploration of symmetric-feasible sequence-pairs [1], the technique employs an efficient model of priority queue [3]. The use of this data structure entails a complexity of O(G·n log log n) for each code evaluation, where n and G are the numbers of devices and symmetry groups, which is better than the complexity of other existent topological placement algorithms supporting symmetry constraints. The computation times exhibited by this approach are significantly better than those of the algorithms using an exploration strategy based on the absolute representation, as well as those of other previous topological algorithms. 1.

Abstract- This paper presents a novel algorithm for device-level analog placement with symmetry constraints. Based on the exploration of symmetric-feasible (S-F) binary tree representations [2] of the layout, the novel approach employs 1-3 deterministic skip lists [7, 10], exhibiting running times at least 20-30 % better than previous (nonslicing) topological algorithms foranalog placement, and significantly better (typically, over 100%) thanmore traditional approaches based on the absolute representation. 1

In this paper, we will present a placement method for analog circuits. We consider both common centroid and 1-D symmetry constraints, which are the two most common types of placement requirements in analog designs. The approach is based on a symmetric feasible condition on the sequence pair representation that can cover completely the set of all placements satisfying the common centroid and 1-D symmetry constraints. This condition is essential for a good searching process to solve the problem effectively. Symmetric placement is an important step to achieve matchings of other electrical properties like delay and temperature variation. We have compared our results with those presented in the most updated previous works. Significant improvements can be obtained by our approach in both common centroid and 1-D symmetry placements, and we are the first who can handle both constraints simultaneously.

With the thermal effect, improper analog placements may degrade circuit performance because the thermal impact from power devices can affect electrical characteristics of the thermally-sensitive devices. There is not much previous work that considers the desired placement configuration between power and thermally-sensitive devices for a better thermal profile to reduce the thermally-induced mismatches. In this paper, we first introduce the properties of a desired thermal profile for better thermal matching of the matched devices. We then propose a thermal-driven analog placement methodology to achieve the desired thermal profile and to consider the best device matching under the thermal profile while satisfying the symmetry and the common-centroid constraints. Experimental results based on real analog circuits show that our approach can achieve the best analog circuit performance/accuracy with the least impact due to the thermal gradient, among existing works. Digital Circuitry DAC

Abstract—To reduce the effect of parasitic mismatches and circuit sensitivity to thermal gradients or process variations for analog circuits, some pairs of modules need to be placed symmetrically with respect to a common axis, and the symmetric modules are preferred to be placed at closest proximity for better electrical properties. Most previous works handle the problem with symmetry constraints by imposing symmetric-feasible conditions in floorplan representations and using cost functions to minimize the distance between symmetric modules. Such approaches are inefficient due to the large search space and cannot guarantee the closest proximity of symmetry modules. In this paper, we present the first linear-time-packing algorithm for the placement with symmetry constraints using the topological floorplan representations. We first introduce the concept of a symmetry island which is formed by modules of the same symmetry group in a single connected placement. Based on this concept and the B ∗-tree representation, we propose automatically symmetric-feasible (ASF) B ∗-trees to directly model the placement of a symmetry island. We then present hierarchical B ∗-trees (HB ∗-trees) which can simultaneously optimize the placement with both symmetry islands and nonsymmetric modules. Unlike the previous works, our approach can place the symmetry modules in a symmetry group in close proximity and significantly reduce the search space based on the symmetry-island formulation. In particular, the packing time for an ASF-B ∗-tree or an HB ∗-tree is the same as that for a plain B ∗-tree (only linear) and much faster than previous works. Experimental results show that our approach achieves the best-published quality and runtime efficiency for analog placement. Index Terms—Analog circuit, floorplanning, physical design, placement. I.

Abstract—In today’s system-on-chip designs, both digital and analog parts of a circuit will be implemented on the same chip. Parasitic mismatch induced by layout will affect circuit performance significantly for analog designs. Consideration of symmetry and common centroid constraints during placement can help to reduce these errors. Besides these two specific types of placement constraints, other constraints, such as alignment, abutment, preplace, and maximum separation, are also essential in circuit placement. In this paper, we will present a placement methodology that can handle all these constraints at the same time. To the best of our knowledge, this is the first piece of work that can handle symmetry constraint, common centroid constraint, and other general placement constraints, simultaneously. Experimental results do confirm the effectiveness and scalability of our approach in solving this mixed constraint-driven placement problem. Index Terms—Analog placement, common centroid constraints, constraint graph, corner block list, sequence pair (SP), symmetry constraints. I.

Abstract—This paper gives an overview of some recent advances in topological approaches to analog layout synthesis and in layout-aware analog sizing. The core issue in these approaches is the modeling of layout constraints for an efficient exploration process. This includes fast checking of constraint compliance, reducing the search space, and quickly relating topological encodings to placements. Sequence-pairs, B*-trees, circuit hierarchy and layout templates are described as advantageous means to tackle these tasks. I.