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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Differential privacy is the de facto privacy standard in data analysis and widely researched in various application areas. On the other hand, analyzing sequences, or strings, is essential to many modern data analysis tasks, and those data often include highly sensitive personal data. While the problem of sanitizing sequential data to protect privacy has received growing attention, there is a surprising lack of theoretical studies of algorithms analyzing sequential data that preserve differential privacy while giving provable guarantees on the accuracy of such an algorithm. The goal of this paper is to initiate such a study.
Specifically, in this paper, we consider the k-approximate pattern matching problem under differential privacy, where the goal is to report or count all substrings of a given string S which have a Hamming distance at most k to a pattern P, or decide whether such a substring exists. In our definition of privacy, individual positions of the string S are protected. To be able to answer queries under differential privacy, we allow some slack on k, i.e. we allow reporting or counting substrings of S with a distance at most (1+γ)k+α to P, for a multiplicative error γ and an additive error α. We analyze which values of α and γ are necessary or sufficient to solve the k-approximate pattern matching problem while satisfying ε-differential privacy. Let n denote the length of S. We give
- an ε-differentially private algorithm with an additive error of O(ε^{-1}log n) and no multiplicative error for the existence variant;
- an ε-differentially private algorithm with an additive error O(ε^{-1}max(k,log n)⋅log n) for the counting variant;
- an ε-differentially private algorithm with an additive error of O(ε^{-1}log n) and multiplicative error O(1) for the reporting variant for a special class of patterns.
The error bounds hold with high probability. All of these algorithms return a witness, that is, if there exists a substring of S with distance at most k to P, then the algorithm returns a substring of S with distance at most (1+γ)k+α to P.
Further, we complement these results by a lower bound, showing that any algorithm for the existence variant which also returns a witness must have an additive error of Ω(ε^{-1}log n) with constant probability.

Teresa Anna Steiner. Differentially Private Approximate Pattern Matching. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 94:1-94:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{steiner:LIPIcs.ITCS.2024.94, author = {Steiner, Teresa Anna}, title = {{Differentially Private Approximate Pattern Matching}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {94:1--94:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.94}, URN = {urn:nbn:de:0030-drops-196223}, doi = {10.4230/LIPIcs.ITCS.2024.94}, annote = {Keywords: Differential privacy, pattern matching} }

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**Published in:** LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

The fundamental question considered in algorithms on strings is that of indexing, that is, preprocessing a given string for specific queries. By now we have a number of efficient solutions for this problem when the queries ask for an exact occurrence of a given pattern P. However, practical applications motivate the necessity of considering more complex queries, for example concerning near occurrences of two patterns. Recently, Bille et al. [CPM 2021] introduced a variant of such queries, called gapped consecutive occurrences, in which a query consists of two patterns P₁ and P₂ and a range [a,b], and one must find all consecutive occurrences (q₁,q₂) of P₁ and P₂ such that q₂-q₁ ∈ [a,b]. By their results, we cannot hope for a very efficient indexing structure for such queries, even if a = 0 is fixed (although at the same time they provided a non-trivial upper bound). Motivated by this, we focus on a text given as a straight-line program (SLP) and design an index taking space polynomial in the size of the grammar that answers such queries in time optimal up to polylog factors.

Paweł Gawrychowski, Garance Gourdel, Tatiana Starikovskaya, and Teresa Anna Steiner. Compressed Indexing for Consecutive Occurrences. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 12:1-12:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gawrychowski_et_al:LIPIcs.CPM.2023.12, author = {Gawrychowski, Pawe{\l} and Gourdel, Garance and Starikovskaya, Tatiana and Steiner, Teresa Anna}, title = {{Compressed Indexing for Consecutive Occurrences}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-276-1}, ISSN = {1868-8969}, year = {2023}, volume = {259}, editor = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.12}, URN = {urn:nbn:de:0030-drops-179666}, doi = {10.4230/LIPIcs.CPM.2023.12}, annote = {Keywords: Compressed indexing, two patterns, consecutive occurrences} }

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**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

Given two strings S and P, the Episode Matching problem is to find the shortest substring of S that contains P as a subsequence. The best known upper bound for this problem is Õ(nm) by Das et al. (1997), where n,m are the lengths of S and P, respectively. Although the problem is well studied and has many applications in data mining, this bound has never been improved. In this paper we show why this is the case by proving that no O((nm)^{1-ε}) algorithm (even for binary strings) exists, unless the Strong Exponential Time Hypothesis (SETH) is false.
We then consider the indexing version of the problem, where S is preprocessed into a data structure for answering episode matching queries P. We show that for any τ, there is a data structure using O(n+(n/(τ)) ^k) space that answers episode matching queries for any P of length k in O(k⋅ τ ⋅ log log n) time. We complement this upper bound with an almost matching lower bound, showing that any data structure that answers episode matching queries for patterns of length k in time O(n^δ), must use Ω(n^{k-kδ-o(1)}) space, unless the Strong k-Set Disjointness Conjecture is false. Finally, for the special case of k = 2, we present a faster construction of the data structure using fast min-plus multiplication of bounded integer matrices.

Philip Bille, Inge Li Gørtz, Shay Mozes, Teresa Anna Steiner, and Oren Weimann. The Fine-Grained Complexity of Episode Matching. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 4:1-4:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bille_et_al:LIPIcs.CPM.2022.4, author = {Bille, Philip and G{\o}rtz, Inge Li and Mozes, Shay and Steiner, Teresa Anna and Weimann, Oren}, title = {{The Fine-Grained Complexity of Episode Matching}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {4:1--4:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.4}, URN = {urn:nbn:de:0030-drops-161312}, doi = {10.4230/LIPIcs.CPM.2022.4}, annote = {Keywords: Pattern matching, fine-grained complexity, longest common subsequence} }

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**Published in:** LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)

The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient pattern matching queries. Typical queries include existential queries (decide if the pattern occurs in S), reporting queries (return all positions where the pattern occurs), and counting queries (return the number of occurrences of the pattern). In this paper we consider a variant of string indexing, where the goal is to compactly represent the string such that given two patterns P₁ and P₂ and a gap range [α, β] we can quickly find the consecutive occurrences of P₁ and P₂ with distance in [α, β], i.e., pairs of subsequent occurrences with distance within the range. We present data structures that use Õ(n) space and query time Õ(|P₁|+|P₂|+n^{2/3}) for existence and counting and Õ(|P₁|+|P₂|+n^{2/3}occ^{1/3}) for reporting. We complement this with a conditional lower bound based on the set intersection problem showing that any solution using Õ(n) space must use Ω̃(|P₁| + |P₂| + √n) query time. To obtain our results we develop new techniques and ideas of independent interest including a new suffix tree decomposition and hardness of a variant of the set intersection problem.

Philip Bille, Inge Li Gørtz, Max Rishøj Pedersen, and Teresa Anna Steiner. Gapped Indexing for Consecutive Occurrences. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 10:1-10:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bille_et_al:LIPIcs.CPM.2021.10, author = {Bille, Philip and G{\o}rtz, Inge Li and Pedersen, Max Rish{\o}j and Steiner, Teresa Anna}, title = {{Gapped Indexing for Consecutive Occurrences}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {10:1--10:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-186-3}, ISSN = {1868-8969}, year = {2021}, volume = {191}, editor = {Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.10}, URN = {urn:nbn:de:0030-drops-139615}, doi = {10.4230/LIPIcs.CPM.2021.10}, annote = {Keywords: String indexing, two patterns, consecutive occurrences, conditional lower bound} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string P, report all occurrences of P within S. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-k close consecutive occurrences problem (Sitcco). Here, a consecutive occurrence is a pair (i,j), i < j, such that P occurs at positions i and j in S and there is no occurrence of P between i and j, and their distance is defined as j-i. Given a pattern P and a parameter k, the goal is to report the top-k consecutive occurrences of P in S of minimal distance. The challenge is to compactly represent S while supporting queries in time close to the length of P and k. We give two time-space trade-offs for the problem. Let n be the length of S, m the length of P, and ε ∈ (0,1]. Our first result achieves O(nlog n) space and optimal query time of O(m+k), and our second result achieves linear space and query time O(m+k^{1+ε}). Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.

Philip Bille, Inge Li Gørtz, Max Rishøj Pedersen, Eva Rotenberg, and Teresa Anna Steiner. String Indexing for Top-k Close Consecutive Occurrences. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 14:1-14:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bille_et_al:LIPIcs.FSTTCS.2020.14, author = {Bille, Philip and G{\o}rtz, Inge Li and Pedersen, Max Rish{\o}j and Rotenberg, Eva and Steiner, Teresa Anna}, title = {{String Indexing for Top-k Close Consecutive Occurrences}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {14:1--14:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.14}, URN = {urn:nbn:de:0030-drops-132558}, doi = {10.4230/LIPIcs.FSTTCS.2020.14}, annote = {Keywords: String indexing, pattern matching, consecutive occurrences} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Given a string S of length n, the classic string indexing problem is to preprocess S into a compact data structure that supports efficient subsequent pattern queries. In this paper we consider the basic variant where the pattern is given in compressed form and the goal is to achieve query time that is fast in terms of the compressed size of the pattern. This captures the common client-server scenario, where a client submits a query and communicates it in compressed form to a server. Instead of the server decompressing the query before processing it, we consider how to efficiently process the compressed query directly. Our main result is a novel linear space data structure that achieves near-optimal query time for patterns compressed with the classic Lempel-Ziv 1977 (LZ77) compression scheme. Along the way we develop several data structural techniques of independent interest, including a novel data structure that compactly encodes all LZ77 compressed suffixes of a string in linear space and a general decomposition of tries that reduces the search time from logarithmic in the size of the trie to logarithmic in the length of the pattern.

Philip Bille, Inge Li Gørtz, and Teresa Anna Steiner. String Indexing with Compressed Patterns. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 10:1-10:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bille_et_al:LIPIcs.STACS.2020.10, author = {Bille, Philip and G{\o}rtz, Inge Li and Steiner, Teresa Anna}, title = {{String Indexing with Compressed Patterns}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {10:1--10:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.10}, URN = {urn:nbn:de:0030-drops-118716}, doi = {10.4230/LIPIcs.STACS.2020.10}, annote = {Keywords: string indexing, compression, pattern matching} }

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