Rotationally optimal spanning and Steiner trees in uniform orientation metrics

We consider the problem of finding a minimum spanning and Steiner tree for a set of points in the plane where the orientations of edge segments are restricted to uniformly distributed orientations, , and where the coordinate system can be rotated around the origin by an arbitrary angle. The most important cases with applications in VLSI design arise when or . In the former, so-called rectilinear case, the edge segments have to be parallel to one of the coordinate axes, and in the latter, so-called octilinear case, the edge segments have to be parallel to one of the coordinate axes or to one of the lines making with the coordinate axes (so-called diagonals). cd ansAs the coordinate system is rotated — while the points remain stationary — the length and indeed the topology of the minimum spanning or Steiner tree changes. We suggest a straightforward polynomial-time algorithm to solve the rotational minimum spanning tree problem. We also give a simple algorithm to solve the rectilinear Steiner tree problem in the rotational setting, and a finite time algorithm for the general Steiner tree problem with uniform orientations. Finally, we provide some computational results indicating the average savings for different values of and both for spanning and Steiner trees.