Fast algorithms for the maximum convolution problem

Abstract

We describe two algorithms for solving the maximum convolution problem, i.e. the calculation of c k : = max { a k− i + b i ∥ 0 ⩽ i ⩽ n - 1} for all k with respect to given sequences ( a 0…, a n−1 ), ( b 0,…, b n−1 ), of real numbers. Our first algorithm with expected running time O ( n log n) is mainly of theoretical interest while our second algorithm allows a simpler, more practicable implementation and showed quite fast performance in numerical experiments.