Finding the Sink Takes Some Time
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
The smallest enclosing ball of balls: combinatorial structure and algorithms
Proceedings of the nineteenth annual symposium on Computational geometry
Linear programming and unique sink orientations
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Violator spaces: structure and algorithms
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Violator spaces: Structure and algorithms
Discrete Applied Mathematics
Finding a Polytope from Its Graph in Polynomial Time
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
A Simple P-Matrix Linear Complementarity Problem for Discounted Games
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
The Holt-Klee condition for oriented matroids
European Journal of Combinatorics
Unique sink orientations of grids
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
Jumping doesn't help in abstract cubes
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
A subexponential lower bound for the random facet algorithm for parity games
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Conflict-Free graph orientations with parity constraints
FUN'12 Proceedings of the 6th international conference on Fun with Algorithms
Families of polytopal digraphs that do not satisfy the shelling property
Computational Geometry: Theory and Applications
Counting unique-sink orientations
Discrete Applied Mathematics
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Suppose we are given (the edge graph of) an n-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 so-called vertex evaluation, where we can access an arbitrary vertex of the cube, for which we obtain the orientations ofthe incident edges.Unique sink orientations occur when the edges of a deformed geometric n-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 e^{2\sqrt n } vertex evaluations have been known [3, 4]. For the general case, a simple randomized (3/2)n procedure exists (without explicit mention in the literature). We present new algorithms, a deterministic O(1.61n) procedure and a randomized O((43/20)n/2) = O(1.47n) 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 structure, and there is ample space for improvement on the bounds given above.