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Compressed property suffix trees. (English) Zbl 1435.68399
Summary: Property matching is a biologically motivated problem where the task is to find those occurrences of an online pattern $$P$$ in a string text $$T$$ (of size $$n$$), such that the matched text part satisfies some conceptual property. The property of a string is a set $$\pi$$ of (possibly overlapping) intervals $$\{(s_1,f_1)$$, $$(s_2,f_2),\dots\}$$ corresponding to the part of text and an occurrence of a pattern $$P=T[i,\dots,(i+| P|-1)]$$ is a valid output only if $$T[i,\dots,(i+| P|-1)]$$ is completely contained in at least one interval $$(s_j,f_j)\in\pi$$. The indexing version of this problem was introduced by A. Amir et al. [Theor. Comput. Sci. 395, No. 2–3, 298–310 (2008; Zbl 1142.68066)], where the text is preprocessed in $$O(n\log \sigma+n\log\log n)$$ time and an $$O(n\log n)$$ bits index, named Property Suffix Tree (PST) is maintained. PST can perform property matching in $$O(| P|\log\sigma+\mathrm{occ}_\pi)$$ time, where $$\mathrm{occ}_\pi$$ is the number of occurrences of $$P$$ in $$T$$ satisfying the property. T. Kopelowitz [Lect. Notes Comput. Sci. 6129, 63–75 (2010; Zbl 1286.68528)] considered the dynamic version of this problem where intervals can be added or deleted. However, all these indexes take space linear to the size of text $$(O(n\log n)$$ bits), which can be much more than the size of the text $$(n\log\sigma$$ bits). In this paper, we propose the first index for property matching occupying space close to the entropy compressed space requirement of the text. Our compressed index takes $$|\mathrm{CSA}|+n(2+\epsilon+o(1))$$ bits space and performs query answering in $$O(t(| P|)+\frac{1}{\epsilon}(1+\mathrm{occ}_\pi)t_{\mathrm{SA}})$$ time, where $$|\mathrm{CSA}|$$ is the size of compressed suffix array of $$T$$, $$t(| P|)$$ be the time for searching a pattern of length $$| P|$$ in CSA, $$t_{\mathrm{SA}}$$ is the time for computing the suffix array value and $$\epsilon>0$$ is a constant. We also introduce a dynamic index, which takes $$|\mathrm{CSA}|+O(n+|\pi|\log n)$$ bits space and performs query answering in $$O(t(| P|)+(1+\mathrm{occ}_\pi)\log n(t_{\mathrm{SA}}+\log n/\log\log n))$$ time and can update (insert/delete) an interval $$(s,f)$$ in $$O((f-s)(\log n+t_{\mathrm{SA}}))$$ time.

##### MSC:
 68W32 Algorithms on strings 68P05 Data structures 68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
##### Keywords:
property matching; suffix trees; property suffix tress
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##### References:
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