Finding all minimal shapes in a routing channel

We present an algorithm to find a set of all optimal solutions for a channel placement problem. A channel consists of two rows of horizontal line segments (representing components). Each line segment contains some terminals with fixed positions. Sets of terminals, called nets, are to be connected. The relative ordering of line segments in each row is fixed. The line segments can be shifted left or right, which will affect the width needed for routing and the length of the channel. We want to find the tradeoff between channel length and routing width. Since the channel routing problem is NP-complete, we use a lower bound on routing width, called density. The density of a placement is the maximum number of nets crossing each vertical cut. We can increase the total length to minimize the channel density, or minimize the total length by increasing the channel density. The pair (density, total length) is called the shape of a placement. A shape is minimal if a decrease in density would cause an increase in total length, and vice versa. Our algorithm computes all the minimal shapes in O(N⁴) time, where N is the number of nets. This is the first known algorithm for this problem whose running time is polynomial in the number of nets and independent of the length of the channel.