Deterministic algorithms for unique sink orientations of grids

Abstract

We study Unique Sink Orientations (USOs) of grids: Cartesian products of two complete graphs on n vertices, where the edges are oriented in such a way that each subgrid has a unique sink. We consider two different oracle models, the edge query and the vertex query model. An edge query provides the orientation of the queried edge, whereas a vertex query provides the orientation of all edges incident to the queried vertex. We are interested in bounding the number of queries to the oracle needed by an algorithm to find the sink. In the randomized setting, the best known algorithms find the sink using either Θ(n) edge queries, or O(log2n) vertex queries, in expectation. We prove that O(nlog47) edge queries and O(n log n) vertex queries suffice to find the sink in the deterministic setting. A deterministic lower bound for both models is Ω(n). Grid USOs are instances of LP-type problems and violator spaces for which derandomizations of known algorithms remain elusive.