Rectilinear Paths Among Rectilinear Obstacles

We address problems of finding rectilinear paths among rectilinear obstacles subject to certain optimization criteria. They have applications in the fields of motion planning and computer-aided circuit design. The criteria that we consider include minimizing the length of the path, minimizing the number of bends on the path, bounding one of the two factors (length and number of bends) and minimizing the other, and minimizing a given non-decreasing function of both the length and the number of bends. The length measure of a rectilinear path can be the total length of all the segments of the paths or just the sum of lengths of all the vertical or horizontal segments of the path. Besides the various optimization criteria for the paths that we look for, the routing environments may have different constraints. For example, the obstacles can be rectangular or rectilinear, the orientations of the path may be restricted to be horizontal or vertical on different layers in a two-layer interconnection model, or the path can penetrate the obstacles and pay extra costs. Previous results and problem solving approaches are summarized and compared to ours in this dissertation. We obtain more efficient algorithms for finding a shortest rectilinear path between two points, minimum spanning tree of a set of points, and for finding assorted paths taking both length and the number of bends into consideration. We also derive a general problem transformation scheme to convert an instance in two-layer model into an equivalent problem in one-layer model, for which previously obtained algorithms under different objective functions are applicable.

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