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Let P be a simple polytope with n − d = 2, where d is the dimension and n is the number of facets. The graph of such a polytope is also called a grid. It is known that the directed random walk along the edges of P terminates after O(log 2 n) steps, if the edges are oriented in a (pseudo-)linear fashion. We prove that the same bound holds for the more general unique sink orientations.

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We show that the pivoting process associated with one line and n points in r-dimensional space may need Ω(log r n) steps in expectation as n → ∞. The only cases for which the bound was known previously were for r ≤ 3. Our lower bound is also valid for the expected number of pivoting steps in the following applications: (1) The Random-Edge simplex algorithm on linear programs with n constraints in d = n − r variables; and (2) the directed random walk on a grid polytope of corank r with n facets.