Linear Programming-Based Cell Placement With Symmetry Constraints for Analog IC Layout

In recent high-performance analog integrated circuit design, it is often required to place some cells symmetrically to a horizontal or vertical axis. Balasa et al. proposed a method of obtaining the closest placement that satisfies the given symmetry constraints and the topology constraints imposed by a sequence-pair (seq-pair). However, this method has the following defects: 1) Balasa's necessary condition for existence of the cell placement that satisfies the given constraints is incorrect; 2) some cells overlap; 3) the closest placement of satisfying both the symmetry and topology constraints is not always obtained; and 4) there is no explanation of placing cells symmetrically to plural axes. In this paper, we clarify the necessary and sufficient conditions for the existence of the cell placement that satisfies the given symmetry constraints and the topology constraints imposed by a seq-pair, and we propose an efficient method of obtaining, by linear programming, the closest cell placement that satisfies the given constraints. Here, a simple constraint graph is obtained from a seq-pair in order to derive a set of linear constraint expressions. Then, to shorten the running time of linear programming, the number of linear expressions is reduced by substituting the expressions for dependent variables, and the solution is obtained. The effectiveness of the proposed method was shown by computational experiments