A Linear Algorithm for Finding the Sink of Unique Sink Orientations on Grids

An orientation of a grid is called unique sink orientation (USO) if each of its nonempty subgrids has a unique sink. Particularly, the original grid itself has a unique global sink. In this work we investigate the problem of how to find the global sink using minimum number of queries to an oracle. There are two different oracle models: the vertex query model where the orientation of all edges incident to the queried vertex are provided, and the edge query model where the orientation of the queried edge is provided. In the 2-dimensional case, we design an optimal linear deterministic algorithm for the vertex query model and an almost linear deterministic algorithm for the edge query model, previously the best known algorithms run in O(N logN) time for the vertex query model and O(N^1.404) time for the edge query model.