Partitioning parameterized 45-degree polygons with constraint programming

Abstract

An algorithm for partitioning parameterized 45-degree polygons into parameterized trapezoids is proposed in this article. The algorithm is based on the plane-sweep technique and can handle polygons with complicated constraints. The input to the algorithm consists of the contour of a parameterized polygon to be partitioned and a set of constraints for parameters of the contour. The algorithm uses horizontal cuts only and generates a number of nonoverlapping trapezoids whose union is the original parameterized polygon. Processing of constraints and coordinates that contain first-order multiple-variable polynomials has been made possible by incorporating the JaCoP constraint programming library. The proposed algorithm has been implemented in Java programming language and can be used as the basis to build the trapezoidal corner stitching data structure for parameterized VLSI layout masks.

Fast partitioning of parameterized 45-degree polygons into parameterized trapezoids

“…As described in [7], a parameterized 45-degree layout can contain many parameterized 45-degree polygons. Additionally, a parameterized 45-degree polygon can contain a number of parameters (also called variables).…”

“…A parameterized 45-degree polygon is defined if two types of information are given; one type of information is the contour of the polygon and the other is a set of linear constraints. Also, in a parameterized 45-degree polygon, x and y coordinates of each vertex must be in the form of linear expressions (called FOMV polynomials in [7]), and each edge must be a vertical, horizontal, or 45-degree line segment. The shape of a parameterized 45-degree polygon can be controlled by a number of parameters.…”

“…In order to partition parameterized 45-degree polygons into parameterized trapezoids, a scanline-based partitioning algorithm has been proposed in [7]. The algorithm makes use of a constraint programming (CP) solver to compare linear expressions.…”

“…Additionally, including parameters in physical layouts enables us to build accurate models of parasitics [5,6]. In order to automatically and systematically build models of layout-induced parasitics, moreover, researchers have proposed the use of parameterized 45-degree layouts, which contain parameterized 45-degree polygons, in an analog design methodology [7]. Since trapezoidal corner stitching data structures [8] allow efficient parasitic extraction from a given physical layout, furthermore, it is desirable to partition parameterized 45-degree polygons into parameterized trapezoids so that parameterized 45-degree layouts can be stored in trapezoidal corner stitching data structures and thus relevant algorithms can be applied on these layouts.…”

“…Although algorithms for partitioning parameterized 45-degree polygons containing complex linear constraints have been proposed in the literature [7,9], executing programs which implement these algorithms can be time-consuming. The major reason is that these algorithms involve invocations of a solver (either a constraint programming solver or a mixed integer linear programming solver) many times before a partitioning process completes.…”