Abstract
The definition of antipower introduced by Fici et al. (ICALP 2016) captures the notion of being the opposite of a power: a sequence of k pairwise distinct blocks of the same length. Recently, Alamro et al. (CPM 2019) defined a string to have an antiperiod if it is a prefix of an antipower, and gave complexity bounds for the offline computation of the minimum antiperiod and all the antiperiods of a word. In this paper, we address the same problems in the online setting. Our solutions rely on new arrays that compactly and incrementally store antiperiods and antipowers as the word grows, obtaining in the process this information for all the word’s prefixes. We show how to compute those arrays online in \(O(n\log n)\) space, \(O(n\log n)\) time, and \(o(n^\epsilon )\) delay per character, for any constant \(\epsilon >0\). Running times are worst-case and hold with high probability. We also discuss more space-efficient solutions returning the correct result with high probability, and small data structures to support random access to those arrays.
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Notes
- 1.
We remark that a word may be a power/antipower for different orders, even though in some cases [1] the focus is on the smallest such order.
- 2.
Using their interface, factor comparisons can be achieved by extracting (splitting) the factors in logarithmic time, then comparing them in constant time (or, alternatively, by navigating their grammar in logarithmic time without performing splits).
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Acknowledgements
The authors would like to thank Roberto Grossi for useful conversations on the topic. GR, NPi are partially, and DG, VG, NPr are supported by the project MIUR-SIR CMACBioSeq (“Combinatorial methods for analysis and compression of biological sequences”) grant n. RBSI146R5L.
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Alzamel, M. et al. (2019). Online Algorithms on Antipowers and Antiperiods. In: Brisaboa, N., Puglisi, S. (eds) String Processing and Information Retrieval. SPIRE 2019. Lecture Notes in Computer Science(), vol 11811. Springer, Cham. https://doi.org/10.1007/978-3-030-32686-9_13
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