Minimum Partitioning Simple Rectilinear polygons in O(nloglogen)-Time

Abstract

The minimum rectangular partition problem for a simple rectilinear polygon is to partition the interior of a simple rectilinear polygon into minimum number of rectangles. This problem is related to VLSI mask generation. A VLSI mask is usually a piece of glass with a figure engraved on it. The engraved figure can be viewed as a rectilinear polygon on the digitized plane [OHTS82]. In order to engrave the figure on the VLSI mask, a pattern generator is often used. A traditional pattern generator has a rectangular opening for exposure, which exposes rectangles onto the mask. Therefore, the engraved figure has to be decomposed into rectangles such that the pattern generator can expose each of these rectangles. The number of rectangles will determine the time required for mask generation. Therefore, decomposing a rectilinear polygon into minimum number of rectangles is an important problem for optimal automated VLSI mask fabrication. The decomposition can be classified into two types depending on the resulted rectangles. If the resulted rectangles can not overlap with each other, then the decomposition is a partition. If the resulted rectangles overlap with each other, then the decomposition is a çygr. Both partitioning approach and covering approach for VLSI mask generation have been discussed in previous researches such as [LIPS79, OHTS82, GOUR83, FERR84, 1MA186, NA11A881 for partitioning problems and [CHAI81, HEGE82, FRAN84r for covering problems. In this paper, we shall only consider the partitioning problem for simple rectilinear polygons. The time complexity of our approach is O(nloglogn). The partition problem for convex rectilinear polygons or vertically (horizontally) convex polygons can be solved in linear time which is optimal. As for a rectilinear polygon with holes, we prove that O(nlogn) is a lower bound, though, as far as we know, there is no algorithm achieve this bound.