Unique Sink Orientations of Cubes

Suppose we are given (the edge graph of) an dimensional hypercube with its edges oriented so that every face has a unique sink. Such an orientation is called a unique sink orientation, and we are interested in finding the unique sink of the whole cube, when the orientation is given implicitly. The basic operation available is the socalled vertex evaluation, where we can access an arbitrary vertex of the cube, for which we obtain the orientations of the incident edges. Unique sink orientations occur when the edges of a deformed geometric -dimensional cube (i.e., a polytope with the combinatorial structure of a cube) are oriented according to some generic linear function. These orientations are easily seen to be acyclic. The main motivation for studying unique sink orientations are certain linear complementarity problems, which allow this combinatorial abstraction (due to Alan Stickney and Layne Watson), where orientations with cycles can arise. Similarly, some quadratic optimization problems, like computing the smallest enclosing ball of a finite point set, can be formulated as finding a sink in a unique sink orientation (with cycles possible). For acyclic unique sink orientations, randomized procedures due to Bernd Gärtner with an expected number of at most vertex evaluations have been known [3, 4]. For the general case, a simple randomized procedure exists (without explicit mention in the literature). We present new algorithms, a deterministic procedure and a randomized procedure for unique sink orientations. An interesting aspect of these algorithms is that they do not proceed on a path to the sink (in a simplex-like fashion), but they exploit the potential of random access (in the sense of arbitrary access) to any vertex of the cube. We consider this feature the main contribution of the paper. We believe that unique sink orientations have a rich Supported by the joint Berlin/Zurich graduate program Combinatorics, Geometry, and Computation (CGC), financed by German Science Foundation (DFG) and ETH Zurich, and by NSF grant DMS 99-70270. structure, and there is ample space for improvement on the bounds given above.