Optimal Algorithms for Running Max and Min Filters on Random Inputs

Abstract

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.

In this paper, we present an algorithm for computing d-dimensional max and min filters simultaneously on i.i.d. inputs that uses $1.5+o(1)$ expected comparisons per sample. This is the first algorithm for d-dimensional max and min filters (on i.i.d. inputs) that gets rid of the dependence on d in (the dominating term of) the number of comparisons needed. It is also asymptotically optimal. In particular, for the 1D case, our algorithm improves the previous best upper bound of $2+o(1)$ to $1.5+o(1)$. As a by-product of our algorithm, we can also compute the d-dimensional max (or min) filter on i.i.d. inputs using $1+o(1)$ expected comparisons per sample, which matches the bound for the 1D case.

This work was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 124411].