Rectangle

Abstract: The earliest and the most critical stage in VLSI layout design is the placement. The background is the rectangle packing problem: given a set of rectangular modules of arbitrary sizes, place them without overlap on a plane within a rectangle of minimum area. Since the variety of the packing is uncountably infinite, the key issue for successful optimization is the introduction of a finite solution space which includes an optimal solution. This paper proposes such a solution space where each packing is represented by a pair of module name sequences, called a sequence-pair. Searching this space by simulated annealing, hundreds of modules have been packed efficiently as demonstrated. For applications to VLSI layout, we attack the biggest MCNC benchmark ami49 with a conventional wiring area estimation method, and obtain a highly promising placement.

Abstract: Given an image and an aligned depth map of an object, our goal is to estimate the full 7-dimensional gripper configuration—its 3D location, 3D orientation and the gripper opening width. Recently, learning algorithms have been successfully applied to grasp novel objects—ones not seen by the robot before. While these approaches use low-dimensional representations such as a ‘grasping point’ or a ‘pair of points’ that are perhaps easier to learn, they only partly represent the gripper configuration and hence are sub-optimal. We propose to learn a new ‘grasping rectangle’ representation: an oriented rectangle in the image plane. It takes into account the location, the orientation as well as the gripper opening width. However, inference with such a representation is computationally expensive. In this work, we present a two step process in which the first step prunes the search space efficiently using certain features that are fast to compute. For the remaining few cases, the second step uses advanced features to accurately select a good grasp. In our extensive experiments, we show that our robot successfully uses our algorithm to pick up a variety of novel objects.

Abstract: The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.

Abstract: This paper describes a method for finding the rectangle of minimum area in which a given arbitrary plane curve can be contained. The method is of interest in certain packing and optimum layout problems. It consists of first determining the minimal-perimeter convex polygon that encloses the given curve and then selecting the rectangle of minimum area capable of containing this polygon. Three theorems are introduced to show that one side of the minimum-area rectangle must be collinear with an edge of the enclosed polygon and that the minimum-area encasing rectangle for the convex polygon is also the minimum-area rectangle for the curve.