Finding the Sink Takes Some Time: An Almost Quadratic Lower Bound for Finding the Sink of Unique Sink Oriented Cubes

Abstract

We give a worst-case Ω(n 2/log n) lower bound on the number of vertex evaluations a deterministic algorithm needs to perform in order to find the (unique) sink of a unique sink oriented n-dimensional cube. We consider the problem in the vertex-oracle model, introduced in [18]. In this model one can access the orientation implicitly, in each vertex evaluation an oracle discloses the orientation of the edges incident to the queried vertex. An important feature of the model is that the access is indeed arbitrary, the algorithm does not have to proceed on a directed path in a simplex-like fashion, but could “jump around”. Our result is the first superlinear lower bound on the problem. The strategy we describe works even for acyclic orientations. We also give improved lower bounds for small values of n and fast algorithms in a couple of important special classes of orientations to demonstrate the difficulty of the lower bound problem.