The 3-D channel routing is a fundamental problem on the physical design of 3-D integrated circuits. The 3-D channel is a 3-D grid G and the terminals are vertices of G located in the top and bottom layers. A net is a set of terminals to be connected. The objective of the 3-D channel routing problem is to connect the terminals in each net with a Steiner tree (wire) in G using as few layers as possible and as short wires as possible in such a way that wires for distinct nets are disjoint. This paper shows that the problem is intractable. We also show that a sparse set of ν 2-terminal nets can be routed in a 3-D channel with O(√ν) layers using wires of length O(√ν).

In this paper, we propose a digital image enlargement method based on a fuzzy technique that improves half-pixel generation, especially for convex and concave signals. The proposed method is a modified version of the image enlargement scheme previously proposed by the authors, which achieves accurate half-pixel interpolation and enlarges the original image by convolution with the Lanczos function. However, the method causes impulse-like artifacts in the enlarged image. In this paper, therefore, we introduce a fuzzy set and fuzzy rule for generating half-pixels to improve the interpolation of convex and concave signals. Experimental results demonstrate that, in terms of image quality, the proposed method shows superior performance compared to bicubic interpolation and our previous method.