Given ɛ ∈ [0, 1), the ɛ-Relative Error Periodic Pattern Problem (REPP) is the following: INPUT: An n-long sequence S of numbers si ∈ N in increasing order. OUTPUT: The longest ɛ-relative error periodic pattern, i.e., the longest subsequence si1,si2,…,sik 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.
|Title of host publication||Design and Analysis of Algorithms - 1st Mediterranean Conference on Algorithms, MedAlg 2012, Proceedings|
|Editors||Guy Even, Dror Rawitz|
|Publisher||Springer Science and Business Media Deutschland GmbH|
|Number of pages||12|
|State||Published - 2012|
|Event||1st Mediterranean Conference on Algorithms, MedAlg 2012 - Kibbutz Ein Gedi, Israel|
Duration: 3 Dec 2012 → 5 Dec 2012
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||1st Mediterranean Conference on Algorithms, MedAlg 2012|
|City||Kibbutz Ein Gedi|
|Period||3/12/12 → 5/12/12|
Bibliographical noteFunding Information:
Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217.?? Partly supported by BSF grant 2008217.Partly supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 347/09, Yahoo, Grant No. 2008217 from the United StatesIsrael Binational Science Foundation (BSF) and DFG.Partly supported by ISF grant 347/09.
© Springer-Verlag Berlin Heidelberg 2012.
ASJC Scopus subject areas
- Theoretical Computer Science
- Computer Science (all)