Asymptotically optimal algorithms for running max and min filters on random inputs
Author(s)
Poon, Chung Keung
Author(s)
Li, M.
Liang, H.
Liu, S.
Yuan, H.
Date Issued
2018
Publisher
IEEE
Journal
IEEE Transactions on Signal Processing
Volume
66
Issue
13
Start page
3421
End page
3435
Abstract
Given a d-dimensional array of size n d and an integer p, the running 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. This problem is useful in many signal processing applications such as pattern analysis, adaptive signal processing, and morphological analysis. The current best algorithm for computing the one-dimensional (1-D) max (or min) filter, due to the work of [H. Yuan and M. J. Atallah, “Running max/min filters using 1+o(1) comparisons per sample,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 33, no. 12, pp. 2544-2548, Dec. 2011], uses 1+o(1) comparisons per sample in the worst case. As a direct consequence, the d-dimensional max (or min) filter (max and min filters, respectively) can be computed in d+o(1) (2d+o(1), respectively) comparisons per sample. In this paper, we first 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 (on i.i.d. inputs) that gets rid of the dependence on d in the dominating term, with respect to n and p, of the (expected) number of comparisons needed. It is also asymptotically optimal (when d is a fixed constant as n → ∞ and p → ∞). We also consider the dynamic version of the problem of d-dimensional max and min filters simultaneously on i.i.d. inputs where we want to maintain the filters after changes in the input array. We design a linear-sized data structure that stores precomputed information for efficient update using O(p d-1 log 2 p) expected comparisons per update.
SFU Affiliated Publication
Yes
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