4 Search Results for "Duraj, Lech"

Document
DOI: 10.4230/LIPIcs.ESA.2021.76
Faster Deterministic Modular Subset Sum

Authors: Krzysztof Potępa

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
We consider the Modular Subset Sum problem: given a multiset X of integers from ℤ_m and a target integer t, decide if there exists a subset of X with a sum equal to t (mod m). Recent independent works by Cardinal and Iacono (SOSA'21), and Axiotis et al. (SOSA'21) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in 𝒪(m log m) time, while Axiotis et al. gave a deterministic algorithm that runs in 𝒪(m polylog m) time. Both results work by reduction to a text problem, which is solved using a dynamic strings data structure. In this work, we develop a simple data structure, designed specifically to handle the text problem that arises in the algorithms for Modular Subset Sum. Our data structure, which we call the shift-tree, is a simple variant of a segment tree. We provide both a hashing-based and a deterministic variant of the shift-trees. We then apply our data structure to the Modular Subset Sum problem and obtain two algorithms. The first algorithm is Monte-Carlo randomized and matches the 𝒪(m log m) runtime of the Las-Vegas algorithm by Cardinal and Iacono. The second algorithm is fully deterministic and runs in 𝒪(m log m ⋅ α(m)) time, where α is the inverse Ackermann function.
Cite as

Krzysztof Potępa. Faster Deterministic Modular Subset Sum. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{potepa:LIPIcs.ESA.2021.76,
  author =	{Pot\k{e}pa, Krzysztof},
  title =	{{Faster Deterministic Modular Subset Sum}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{76:1--76:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.76},
  URN =		{urn:nbn:de:0030-drops-146574},
  doi =		{10.4230/LIPIcs.ESA.2021.76},
  annote =	{Keywords: Modular Subset Sum, String Problem, Segment Tree, Data Structure}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.41
A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem

Authors: Lech Duraj

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

Abstract
The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an ?(n²)-time algorithm and a SETH-based conditional lower bound of ?(n^{2-ε}). For LCS, there is also the Masek-Paterson ?(n²/log n)-time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a ?(n²/log^a n)-time algorithm for a = 1/6-o(1). The algorithm is not based on memorizing small chunks of data (often used for logarithmic speedups, including the "Four Russians Trick" in LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences.
Cite as

Lech Duraj. A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{duraj:LIPIcs.STACS.2020.41,
  author =	{Duraj, Lech},
  title =	{{A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{41:1--41:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.41},
  URN =		{urn:nbn:de:0030-drops-119020},
  doi =		{10.4230/LIPIcs.STACS.2020.41},
  annote =	{Keywords: longest common increasing subsequence, log-shaving, matching pairs}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.46
A Note on Two-Colorability of Nonuniform Hypergraphs

Authors: Lech Duraj, Grzegorz Gutowski, and Jakub Kozik

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
For a hypergraph H, let q(H) denote the expected number of monochromatic edges when the color of each vertex in H is sampled uniformly at random from the set of size 2. Let s_{min}(H) denote the minimum size of an edge in H. Erdös asked in 1963 whether there exists an unbounded function g(k) such that any hypergraph H with s_{min}(H) >=slant k and q(H) <=slant g(k) is two colorable. Beck in 1978 answered this question in the affirmative for a function g(k) = Theta(log^* k). We improve this result by showing that, for an absolute constant delta>0, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph H with s_{min}(H) >=slant k and q(H) <=slant delta * log k.
Cite as

Lech Duraj, Grzegorz Gutowski, and Jakub Kozik. A Note on Two-Colorability of Nonuniform Hypergraphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{duraj_et_al:LIPIcs.ICALP.2018.46,
  author =	{Duraj, Lech and Gutowski, Grzegorz and Kozik, Jakub},
  title =	{{A Note on Two-Colorability of Nonuniform Hypergraphs}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{46:1--46:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.46},
  URN =		{urn:nbn:de:0030-drops-90505},
  doi =		{10.4230/LIPIcs.ICALP.2018.46},
  annote =	{Keywords: Property B, Nonuniform Hypergraphs, Hypergraph Coloring, Random Greedy Coloring}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2017.15
Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

Authors: Lech Duraj, Marvin Künnemann, and Adam Polak

Published in: LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

Abstract
We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called k-LCIS: Given k integer sequences X_1,...,X_k of length at most n, the task is to determine the length of the longest common subsequence of X_1,...,X_k that is also strictly increasing. Especially for the case of k=2 (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case. Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as LCS. We further strengthen this lower bound to rule out O((nL)^{1-epsilon}) time algorithms for LCIS, where L denotes the solution size, and to rule out O(n^{k-epsilon}) time algorithms for k-LCIS. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.
Cite as

Lech Duraj, Marvin Künnemann, and Adam Polak. Tight Conditional Lower Bounds for Longest Common Increasing Subsequence. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{duraj_et_al:LIPIcs.IPEC.2017.15,
  author =	{Duraj, Lech and K\"{u}nnemann, Marvin and Polak, Adam},
  title =	{{Tight Conditional Lower Bounds for Longest Common Increasing Subsequence}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{15:1--15:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Lokshtanov, Daniel and Nishimura, Naomi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.15},
  URN =		{urn:nbn:de:0030-drops-85706},
  doi =		{10.4230/LIPIcs.IPEC.2017.15},
  annote =	{Keywords: fine-grained complexity, combinatorial pattern matching, sequence alignments, parameterized complexity, SETH}
}
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