Optimal Algorithms for Running Max and Min Filters on Random Inputs

Given a d-dimensional array and an integer p, the max (or min) filter is the set of maximum (or minimum) elements within a d-dimensional sliding window of edge length p inside the array. The current best algorithm for computing the 1D max (or min) filter, due to Yuan and Atallah [14], uses \(1+o(1)\) comparisons per sample in the worst case. As a direct consequence, the d-dimensional max (or min) filter can be computed in \((1+o(1))d\) comparisons per sample, and the d-dimensional max and min filters can be computed simultaneously using \((2+o(1))d\) comparisons per sample. Both bounds are the best known results for the corresponding problems, on both worst-case inputs and independently and identically distributed (i.i.d.) inputs.