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**Published in:** LIPIcs, Volume 296, 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)

The notions of synchronizing and partitioning sets are recently introduced variants of locally consistent parsings with a great potential in problem-solving. In this paper we propose a deterministic algorithm that constructs for a given readonly string of length n over the alphabet {0,1,…,n^{𝒪(1)}} a variant of a τ-partitioning set with size 𝒪(b) and τ = n/b using 𝒪(b) space and 𝒪(1/(ε)n) time provided b ≥ n^ε, for ε > 0. As a corollary, for b ≥ n^ε and constant ε > 0, we obtain linear time construction algorithms with 𝒪(b) space on top of the string for two major small-space indexes: a sparse suffix tree, which is a compacted trie built on b chosen suffixes of the string, and a longest common extension (LCE) index, which occupies 𝒪(b) space and allows us to compute the longest common prefix for any pair of substrings in 𝒪(n/b) time. For both, the 𝒪(b) construction storage is asymptotically optimal since the tree itself takes 𝒪(b) space and any LCE index with 𝒪(n/b) query time must occupy at least 𝒪(b) space by a known trade-off (at least for b ≥ Ω(n / log n)). In case of arbitrary b ≥ Ω(log² n), we present construction algorithms for the partitioning set, sparse suffix tree, and LCE index with 𝒪(nlog_b n) running time and 𝒪(b) space, thus also improving the state of the art.

Dmitry Kosolobov and Nikita Sivukhin. Construction of Sparse Suffix Trees and LCE Indexes in Optimal Time and Space. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kosolobov_et_al:LIPIcs.CPM.2024.20, author = {Kosolobov, Dmitry and Sivukhin, Nikita}, title = {{Construction of Sparse Suffix Trees and LCE Indexes in Optimal Time and Space}}, booktitle = {35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-326-3}, ISSN = {1868-8969}, year = {2024}, volume = {296}, editor = {Inenaga, Shunsuke and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2024.20}, URN = {urn:nbn:de:0030-drops-201309}, doi = {10.4230/LIPIcs.CPM.2024.20}, annote = {Keywords: (\tau,\delta)-partitioning set, longest common extension, sparse suffix tree} }

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**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

Given d strings over the alphabet {0,1,...,sigma{-}1}, the classical Aho - Corasick data structure allows us to find all occ occurrences of the strings in any text T in O(|T| + occ) time using O(m log m) bits of space, where m is the number of edges in the trie containing the strings. Fix any constant epsilon in (0, 2). We describe a compressed solution for the problem that, provided sigma <=m^delta for a constant delta < 1, works in O(|T| 1/epsilon log(1/epsilon) + occ) time, which is O(|T| + occ) since epsilon is constant, and occupies mH_k + 1.443 m + epsilon m + O(d log m/d) bits of space, for all 0 <= k <= max{0,alpha log_sigma m - 2} simultaneously, where alpha in (0,1) is an arbitrary constant and H_k is the kth-order empirical entropy of the trie. Hence, we reduce the 3.443m term in the space bounds of previously best succinct solutions to (1.443 + epsilon)m, thus solving an open problem posed by Belazzougui. Further, we notice that L = log binom{sigma (m+1)}{m} - O(log(sigma m)) is a worst-case space lower bound for any solution of the problem and, for d = o(m) and constant epsilon, our approach allows to achieve L + epsilon m bits of space, which gives an evidence that, for d = o(m), the space of our data structure is theoretically optimal up to the epsilon m additive term and it is hardly possible to eliminate the term 1.443m. In addition, we refine the space analysis of previous works by proposing a more appropriate definition for H_k. We also simplify the construction for practice adapting the fixed block compression boosting technique, then implement our data structure, and conduct a number of experiments showing that it is comparable to the state of the art in terms of time and is superior in space.

Dmitry Kosolobov and Nikita Sivukhin. Compressed Multiple Pattern Matching. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kosolobov_et_al:LIPIcs.CPM.2019.13, author = {Kosolobov, Dmitry and Sivukhin, Nikita}, title = {{Compressed Multiple Pattern Matching}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.13}, URN = {urn:nbn:de:0030-drops-104847}, doi = {10.4230/LIPIcs.CPM.2019.13}, annote = {Keywords: multiple pattern matching, compressed space, Aho--Corasick automaton} }