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An orientation of a grid is called unique sink orientation (USO) if each ofits nonempty subgrids has a unique sink. Particularly, the original grid itselfhas a unique global sink. In this work we investigate the problem of how tofind the global sink using minimum number of queries to an oracle. There aretwo different oracle models: the vertex query model where the orientation ofall edges incident to the queried vertex are provided, and the edge query modelwhere the orientation of the queried edge is provided. In the 2-dimensionalcase, we design an optimal linear deterministic algorithm for the vertex querymodel 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 querymodel and O(N^1.404) time for the edge query model.

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