Detecting Approximate Periodic Patterns

Given ɛ ∈ [0, 1), the ɛ-Relative Error Periodic Pattern Problem (REPP) is the following: INPUT: An n-long sequence S of numbers s_i ∈ N in increasing order. OUTPUT: The longest ɛ-relative error periodic pattern, i.e., the longest subsequence s_i1,s_i2,…,s_ik of S, for which there exists a number p such that the absolute difference between any two consecutive numbers in the subsequence is at least p and at most p(1 + ɛ). The best known algorithm for this problem has O(n³ ) time complexity. This bound is too high for large inputs in practice. In this paper we give a new algorithm for finding the longest ɛ-relative error periodic pattern (the REPP problem). Our method is based on a transformation of the input sequence into a different representation: the ɛ-active maximal intervals list L, defined in this paper. We show that the transformation of S to the list L can be done efficiently (quadratic in n and linear in the size of L) and prove that our algorithm is linear in the size of L. This enables us to prove that our algorithm works in sub-cubic time on inputs for which the best known algorithm works in O(n³ ) time. Moreover, though it may happen that our algorithm would still be cubic, it is never worse than the known O(n³ )-algorithm and in many situations its complexity is O(n² )time.