LAYOUT OF AN ARBITRARY PERMUTATION IN A MINIMAL RIGHT TRIANGLE AREA

Abstract

In VLSI layout of interconnection networks, routing two-point nets in some restricted area is one of the central operations. The main aim is usually minimization of the layout area, while reducing the number of wire bends is also very useful. In this paper, we consider connecting a set of N inputs on a line to a set of N outputs on a perpendicular line inside a right triangle shaped area, where the order of the outputs is a given permutation of the order of the corresponding inputs. Such triangles were used, for example, by Dinitz, Even, and Artishchev-Zapolotsky for an optimal layout of the Butterfly network. That layout was of a particular permutation, while here we solve the problem for an arbitrary permutation case. We show two layouts in an optimal area of ½ N2 + o(N2), with O (N) bends each. We prove that the first layout requires the absolutely minimum area and yields the irreducible number of bends, while containing knock-knees. The second one eliminates knock-knees, still keeping a constant number, 7, of bends per connection. As well, we prove a lower bound of 3N - o(N) for the number of bends in the worst case layout in an optimal area of ½ N2 + o(N2).