4 Search Results for "Wedel Vildhøj, Hjalte"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.39
Optimal Bounds for Distinct Quartics

Authors: Panagiotis Charalampopoulos, Paweł Gawrychowski, and Samah Ghazawi

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
A fundamental concept related to strings is that of repetitions. It has been extensively studied in many versions, from both purely combinatorial and algorithmic angles. One of the most basic questions is how many distinct squares, i.e., distinct strings of the form UU, a string of length n can contain as fragments. It turns out that this is always 𝒪(n), and the bound cannot be improved to sublinear in n [Fraenkel and Simpson, JCTA 1998]. Several similar questions about repetitions in strings have been considered, and by now we seem to have a good understanding of their repetitive structure. For higher-dimensional strings, the basic concept of periodicity has been successfully extended and applied to design efficient algorithms - it is inherently more complex than for regular strings. Extending the notion of repetitions and understanding the repetitive structure of higher-dimensional strings is however far from complete. Quartics were introduced by Apostolico and Brimkov [TCS 2000] as analogues of squares in two dimensions. Charalampopoulos, Radoszewski, Rytter, Waleń, and Zuba [ESA 2020] proved that the number of distinct quartics in an n×n 2D string is 𝒪(n²log²n) and that they can be computed in 𝒪(n²log²n) time. Gawrychowski, Ghazawi, and Landau [SPIRE 2021] constructed an infinite family of n×n 2D strings with Ω(n²log n) distinct quartics. This brings the challenge of determining asymptotically tight bounds. Here, we settle both the combinatorial and the algorithmic aspects of this question: the number of distinct quartics in an n×n 2D string is 𝒪(n²log n) and they can be computed in the worst-case optimal 𝒪(n²log n) time. As expected, our solution heavily exploits the periodic structure implied by occurrences of quartics. However, the two-dimensional nature of the problem introduces some technical challenges. Somewhat surprisingly, we overcome the final challenge for the combinatorial bound using a result of Marcus and Tardos [JCTA 2004] for permutation avoidance on matrices.
Cite as

Panagiotis Charalampopoulos, Paweł Gawrychowski, and Samah Ghazawi. Optimal Bounds for Distinct Quartics. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{charalampopoulos_et_al:LIPIcs.ICALP.2024.39,
  author =	{Charalampopoulos, Panagiotis and Gawrychowski, Pawe{\l} and Ghazawi, Samah},
  title =	{{Optimal Bounds for Distinct Quartics}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{39:1--39:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.39},
  URN =		{urn:nbn:de:0030-drops-201823},
  doi =		{10.4230/LIPIcs.ICALP.2024.39},
  annote =	{Keywords: 2D strings, quartics, repetitions, periodicity}
}
Document
DOI: 10.4230/LIPIcs.CPM.2017.16
Time-Space Trade-Offs for Lempel-Ziv Compressed Indexing

Authors: Philip Bille, Mikko Berggren Ettienne, Inge Li Gørtz, and Hjalte Wedel Vildhøj

Published in: LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

Abstract
Given a string S, the compressed indexing problem is to preprocess S into a compressed representation that supports fast substring queries. The goal is to use little space relative to the compressed size of S while supporting fast queries. We present a compressed index based on the Lempel-Ziv 1977 compression scheme. Let n, and z denote the size of the input string, and the compressed LZ77 string, respectively. We obtain the following time-space trade-offs. Given a pattern string P of length m, we can solve the problem in (i) O(m + occ lglg n) time using O(z lg(n/z) lglg z) space, or (ii) O(m(1 + lg^e z / lg(n/z)) + occ(lglg n + lg^e z)) time using O(z lg(n/z)) space, for any 0 < e < 1 In particular, (i) improves the leading term in the query time of the previous best solution from O(m lg m) to O(m) at the cost of increasing the space by a factor lglg z. Alternatively, (ii) matches the previous best space bound, but has a leading term in the query time of O(m(1+lg^e z / lg(n/z))). However, for any polynomial compression ratio, i.e., z = O(n^{1-d}), for constant d > 0, this becomes O(m). Our index also supports extraction of any substring of length l in O(l + lg(n/z)) time. Technically, our results are obtained by novel extensions and combinations of existing data structures of independent interest, including a new batched variant of weak prefix search.
Cite as

Philip Bille, Mikko Berggren Ettienne, Inge Li Gørtz, and Hjalte Wedel Vildhøj. Time-Space Trade-Offs for Lempel-Ziv Compressed Indexing. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bille_et_al:LIPIcs.CPM.2017.16,
  author =	{Bille, Philip and Ettienne, Mikko Berggren and G{\o}rtz, Inge Li and Vildh{\o}j, Hjalte Wedel},
  title =	{{Time-Space Trade-Offs for Lempel-Ziv Compressed Indexing}},
  booktitle =	{28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-039-2},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{78},
  editor =	{K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.16},
  URN =		{urn:nbn:de:0030-drops-73254},
  doi =		{10.4230/LIPIcs.CPM.2017.16},
  annote =	{Keywords: compressed indexing, pattern matching, LZ77, prefix search}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2016.18
Dynamic Relative Compression, Dynamic Partial Sums, and Substring Concatenation

Authors: Philip Bille, Patrick Hagge Cording, Inge Li Gørtz, Frederik Rye Skjoldjensen, Hjalte Wedel Vildhøj, and Søren Vind

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract
Given a static reference string R and a source string S, a relative compression of S with respect to R is an encoding of S as a sequence of references to substrings of R. Relative compression schemes are a classic model of compression and have recently proved very successful for compressing highly-repetitive massive data sets such as genomes and web-data. We initiate the study of relative compression in a dynamic setting where the compressed source string S is subject to edit operations. The goal is to maintain the compressed representation compactly, while supporting edits and allowing efficient random access to the (uncompressed) source string. We present new data structures that achieve optimal time for updates and queries while using space linear in the size of the optimal relative compression, for nearly all combinations of parameters. We also present solutions for restricted and extended sets of updates. To achieve these results, we revisit the dynamic partial sums problem and the substring concatenation problem. We present new optimal or near optimal bounds for these problems. Plugging in our new results we also immediately obtain new bounds for the string indexing for patterns with wildcards problem and the dynamic text and static pattern matching problem.
Cite as

Philip Bille, Patrick Hagge Cording, Inge Li Gørtz, Frederik Rye Skjoldjensen, Hjalte Wedel Vildhøj, and Søren Vind. Dynamic Relative Compression, Dynamic Partial Sums, and Substring Concatenation. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bille_et_al:LIPIcs.ISAAC.2016.18,
  author =	{Bille, Philip and Cording, Patrick Hagge and G{\o}rtz, Inge Li and Skjoldjensen, Frederik Rye and Vildh{\o}j, Hjalte Wedel and Vind, S{\o}ren},
  title =	{{Dynamic Relative Compression, Dynamic Partial Sums, and Substring Concatenation}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{18:1--18:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.18},
  URN =		{urn:nbn:de:0030-drops-67872},
  doi =		{10.4230/LIPIcs.ISAAC.2016.18},
  annote =	{Keywords: Relative compression, dynamic compression, dynamic partial sum, sub-string concatenation, external macro compression}
}
Document
DOI: 10.4230/LIPIcs.CPM.2016.20
Boxed Permutation Pattern Matching

Authors: Mika Amit, Philip Bille, Patrick Hagge Cording, Inge Li Gørtz, and Hjalte Wedel Vildhøj

Published in: LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

Abstract
Given permutations T and P of length n and m, respectively, the Permutation Pattern Matching problem asks to find all m-length subsequences of T that are order-isomorphic to P. This problem has a wide range of applications but is known to be NP-hard. In this paper, we study the special case, where the goal is to only find the boxed subsequences of T that are order-isomorphic to P. This problem was introduced by Bruner and Lackner who showed that it can be solved in O(n^3) time. Cho et al. [CPM 2015] gave an O(n^2m) time algorithm and improved it to O(n^2 log m). In this paper we present a solution that uses only O(n^2) time. In general, there are instances where the output size is Omega(n^2) and hence our bound is optimal. To achieve our results, we introduce several new ideas including a novel reduction to 2D offline dominance counting. Our algorithm is surprisingly simple and straightforward to implement.
Cite as

Mika Amit, Philip Bille, Patrick Hagge Cording, Inge Li Gørtz, and Hjalte Wedel Vildhøj. Boxed Permutation Pattern Matching. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 20:1-20:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{amit_et_al:LIPIcs.CPM.2016.20,
  author =	{Amit, Mika and Bille, Philip and Hagge Cording, Patrick and Li G{\o}rtz, Inge and Wedel Vildh{\o}j, Hjalte},
  title =	{{Boxed Permutation Pattern Matching}},
  booktitle =	{27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)},
  pages =	{20:1--20:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-012-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{54},
  editor =	{Grossi, Roberto and Lewenstein, Moshe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.20},
  URN =		{urn:nbn:de:0030-drops-60744},
  doi =		{10.4230/LIPIcs.CPM.2016.20},
  annote =	{Keywords: Permutation, Subsequence, Pattern Matching, Order Preserving, Boxed Mesh Pattern}
}
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