To improve immunity against process gradients, a common centroid constraint, in which every pair of capacitors should be placed symmetrically with respect to a common center point, is widely used. The pair of capacitors are derived by dividing some original capacitors into two halves. Xiao et al. proposed a method to obtain a placement which satisfies the common centroid constraints, but this method has a defect. In this paper, we propose a decoding algorithm to obtain a placement which satisfies common centroid constraints.

Symmetry constrains are the constraints that the given cells should be placed symmetrically in design of analog ICs. We use O-tree to represent placements and propose a decoding algorithm which can obtain one of the minimum placements satisfying the constraints. The decoding algorithm uses linear programming, which is too much time consuming. Therefore we propose a graph based method to recognize if there exists no placement satisfying both the given symmetry and O-tree constraints, and use the method before application of linear programming. The effectiveness of the proposed method was shown by computational experiments.