12 Search Results for "Kosolobov, Dmitry"

Document
DOI: 10.4230/LIPIcs.SEA.2024.9
Top- k Frequent Patterns in Streams and Parameterized-Space LZ Compression

Authors: Patrick Dinklage, Johnnes Fischer, and Nicola Prezza

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
We present novel online approximations of the Lempel-Ziv 77 (LZ77) and Lempel-Ziv 78 (LZ78) compression schemes [Lempel & Ziv, 1977/1978] with parameterizable space usage based on estimating which k patterns occur the most frequently in the streamed input for parameter k. This new approach overcomes the issue of finding only local repetitions, which is a natural limitation of algorithms that compress using a sliding window or by partitioning the input into blocks. For this, we introduce the top-k trie, a summary for maintaining online the top-k frequent consecutive patterns in a stream of characters based on a combination of the Lempel-Ziv 78 compression scheme and the Misra-Gries algorithm for frequent item estimation in streams. Using straightforward encoding, our implementations yield compression ratios (output over input size) competitive with established general-purpose LZ-based compression utilities such as gzip or xz.
Cite as

Patrick Dinklage, Johnnes Fischer, and Nicola Prezza. Top- k Frequent Patterns in Streams and Parameterized-Space LZ Compression. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dinklage_et_al:LIPIcs.SEA.2024.9,
  author =	{Dinklage, Patrick and Fischer, Johnnes and Prezza, Nicola},
  title =	{{Top- k Frequent Patterns in Streams and Parameterized-Space LZ Compression}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{9:1--9:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.9},
  URN =		{urn:nbn:de:0030-drops-203748},
  doi =		{10.4230/LIPIcs.SEA.2024.9},
  annote =	{Keywords: compression, streaming, heavy hitters, algorithm engineering}
}
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.2024.19
Simplified Tight Bounds for Monotone Minimal Perfect Hashing

Authors: Dmitry Kosolobov

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

Abstract
Given an increasing sequence of integers x₁,…,x_n from a universe {0,…,u-1}, the monotone minimal perfect hash function (MMPHF) for this sequence is a data structure that answers the following rank queries: rank(x) = i if x = x_i, for i ∈ {1,…,n}, and rank(x) is arbitrary otherwise. Assadi, Farach-Colton, and Kuszmaul recently presented at SODA'23 a proof of the lower bound Ω(n min{log log log u, log n}) for the bits of space required by MMPHF, provided u ≥ n 2^{2^{√{log log n}}}, which is tight since there is a data structure for MMPHF that attains this space bound (and answers the queries in O(log u) time). In this paper, we close the remaining gap by proving that, for u ≥ (1+ε)n, where ε > 0 is any constant, the tight lower bound is Ω(n min{log log log u/n, log n}), which is also attainable; we observe that, for all reasonable cases when n < u < (1+ε)n, known facts imply tight bounds, which virtually settles the problem. Along the way we substantially simplify the proof of Assadi et al. replacing a part of their heavy combinatorial machinery by trivial observations. However, an important part of the proof still remains complicated. This part of our paper repeats arguments of Assadi et al. and is not novel. Nevertheless, we include it, for completeness, offering a somewhat different perspective on these arguments.
Cite as

Dmitry Kosolobov. Simplified Tight Bounds for Monotone Minimal Perfect Hashing. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kosolobov:LIPIcs.CPM.2024.19,
  author =	{Kosolobov, Dmitry},
  title =	{{Simplified Tight Bounds for Monotone Minimal Perfect Hashing}},
  booktitle =	{35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)},
  pages =	{19:1--19:13},
  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.2024.19},
  URN =		{urn:nbn:de:0030-drops-201296},
  doi =		{10.4230/LIPIcs.CPM.2024.19},
  annote =	{Keywords: monotone minimal perfect hashing, lower bound, MMPHF, hash}
}
Document
DOI: 10.4230/LIPIcs.CPM.2024.20
Construction of Sparse Suffix Trees and LCE Indexes in Optimal Time and Space

Authors: Dmitry Kosolobov and Nikita Sivukhin

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.CPM.2021.6
Internal Shortest Absent Word Queries

Authors: Golnaz Badkobeh, Panagiotis Charalampopoulos, and Solon P. Pissis

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

Abstract
Given a string T of length n over an alphabet Σ ⊂ {1,2,…,n^{𝒪(1)}} of size σ, we are to preprocess T so that given a range [i,j], we can return a representation of a shortest string over Σ that is absent in the fragment T[i]⋯ T[j] of T. For any positive integer k ∈ [1,log log_σ n], we present an 𝒪((n/k)⋅ log log_σ n)-size data structure, which can be constructed in 𝒪(nlog_σ n) time, and answers queries in time 𝒪(log log_σ k).
Cite as

Golnaz Badkobeh, Panagiotis Charalampopoulos, and Solon P. Pissis. Internal Shortest Absent Word Queries. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{badkobeh_et_al:LIPIcs.CPM.2021.6,
  author =	{Badkobeh, Golnaz and Charalampopoulos, Panagiotis and Pissis, Solon P.},
  title =	{{Internal Shortest Absent Word Queries}},
  booktitle =	{32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)},
  pages =	{6:1--6:18},
  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.2021.6},
  URN =		{urn:nbn:de:0030-drops-139570},
  doi =		{10.4230/LIPIcs.CPM.2021.6},
  annote =	{Keywords: string algorithms, internal queries, shortest absent word, bit parallelism}
}
Document
DOI: 10.4230/LIPIcs.CPM.2021.8
Weighted Ancestors in Suffix Trees Revisited

Authors: Djamal Belazzougui, Dmitry Kosolobov, Simon J. Puglisi, and Rajeev Raman

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

Abstract
The weighted ancestor problem is a well-known generalization of the predecessor problem to trees. It is known to require Ω(log log n) time for queries provided 𝒪(n polylog n) space is available and weights are from [0..n], where n is the number of tree nodes. However, when applied to suffix trees, the problem, surprisingly, admits an 𝒪(n)-space solution with constant query time, as was shown by Gawrychowski, Lewenstein, and Nicholson (Proc. ESA 2014). This variant of the problem can be reformulated as follows: given the suffix tree of a string s, we need a data structure that can locate in the tree any substring s[p..q] of s in 𝒪(1) time (as if one descended from the root reading s[p..q] along the way). Unfortunately, the data structure of Gawrychowski et al. has no efficient construction algorithm, limiting its wider usage as an algorithmic tool. In this paper we resolve this issue, describing a data structure for weighted ancestors in suffix trees with constant query time and a linear construction algorithm. Our solution is based on a novel approach using so-called irreducible LCP values.
Cite as

Djamal Belazzougui, Dmitry Kosolobov, Simon J. Puglisi, and Rajeev Raman. Weighted Ancestors in Suffix Trees Revisited. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{belazzougui_et_al:LIPIcs.CPM.2021.8,
  author =	{Belazzougui, Djamal and Kosolobov, Dmitry and Puglisi, Simon J. and Raman, Rajeev},
  title =	{{Weighted Ancestors in Suffix Trees Revisited}},
  booktitle =	{32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)},
  pages =	{8:1--8:15},
  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.2021.8},
  URN =		{urn:nbn:de:0030-drops-139594},
  doi =		{10.4230/LIPIcs.CPM.2021.8},
  annote =	{Keywords: suffix tree, weighted ancestors, irreducible LCP, deterministic substring hashing}
}
Document
DOI: 10.4230/LIPIcs.CPM.2019.13
Compressed Multiple Pattern Matching

Authors: Dmitry Kosolobov and Nikita Sivukhin

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.WABI.2018.15
Minimum Segmentation for Pan-genomic Founder Reconstruction in Linear Time

Authors: Tuukka Norri, Bastien Cazaux, Dmitry Kosolobov, and Veli Mäkinen

Published in: LIPIcs, Volume 113, 18th International Workshop on Algorithms in Bioinformatics (WABI 2018)

Abstract
Given a threshold L and a set R = {R_1, ..., R_m} of m strings (haplotype sequences), each having length n, the minimum segmentation problem for founder reconstruction is to partition [1,n] into set P of disjoint segments such that each segment [a,b] in P has length at least L and the number d(a,b)=|{R_i[a,b] : 1 <= i <= m}| of distinct substrings at segment [a,b] is minimized over [a,b] in P. The distinct substrings in the segments represent founder blocks that can be concatenated to form max{d(a,b) : [a,b] in P} founder sequences representing the original R such that crossovers happen only at segment boundaries. We give an optimal O(mn) time algorithm to solve the problem, improving over earlier O(mn^2). This improvement enables to exploit the algorithm on a pan-genomic setting of input strings being aligned haplotype sequences of complete human chromosomes, with a goal of finding a representative set of references that can be indexed for read alignment and variant calling. We implemented the new algorithm and give some experimental evidence on the practicality of the approach on this pan-genomic setting.
Cite as

Tuukka Norri, Bastien Cazaux, Dmitry Kosolobov, and Veli Mäkinen. Minimum Segmentation for Pan-genomic Founder Reconstruction in Linear Time. In 18th International Workshop on Algorithms in Bioinformatics (WABI 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 113, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{norri_et_al:LIPIcs.WABI.2018.15,
  author =	{Norri, Tuukka and Cazaux, Bastien and Kosolobov, Dmitry and M\"{a}kinen, Veli},
  title =	{{Minimum Segmentation for Pan-genomic Founder Reconstruction in Linear Time}},
  booktitle =	{18th International Workshop on Algorithms in Bioinformatics (WABI 2018)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-082-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{113},
  editor =	{Parida, Laxmi and Ukkonen, Esko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2018.15},
  URN =		{urn:nbn:de:0030-drops-93175},
  doi =		{10.4230/LIPIcs.WABI.2018.15},
  annote =	{Keywords: Pan-genome indexing, founder reconstruction, dynamic programming, positional Burrows-Wheeler transform, range minimum query}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.46
Relations Between Greedy and Bit-Optimal LZ77 Encodings

Authors: Dmitry Kosolobov

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
This paper investigates the size in bits of the LZ77 encoding, which is the most popular and efficient variant of the Lempel-Ziv encodings used in data compression. We prove that, for a wide natural class of variable-length encoders for LZ77 phrases, the size of the greedily constructed LZ77 encoding on constant alphabets is within a factor O(log n / log log log n) of the optimal LZ77 encoding, where n is the length of the processed string. We describe a series of examples showing that, surprisingly, this bound is tight, thus improving both the previously known upper and lower bounds. Further, we obtain a more detailed bound O(min{z, log n / log log z}), which uses the number z of phrases in the greedy LZ77 encoding as a parameter, and construct a series of examples showing that this bound is tight even for binary alphabet. We then investigate the problem on non-constant alphabets: we show that the known O(log n) bound is tight even for alphabets of logarithmic size, and provide tight bounds for some other important cases.
Cite as

Dmitry Kosolobov. Relations Between Greedy and Bit-Optimal LZ77 Encodings. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kosolobov:LIPIcs.STACS.2018.46,
  author =	{Kosolobov, Dmitry},
  title =	{{Relations Between Greedy and Bit-Optimal LZ77 Encodings}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.46},
  URN =		{urn:nbn:de:0030-drops-84894},
  doi =		{10.4230/LIPIcs.STACS.2018.46},
  annote =	{Keywords: Lempel-Ziv, LZ77 encoding, greedy LZ77, bit optimal LZ77}
}
Document
DOI: 10.4230/LIPIcs.ESA.2017.53
LZ-End Parsing in Linear Time

Authors: Dominik Kempa and Dmitry Kosolobov

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract
We present a deterministic algorithm that constructs in linear time and space the LZ-End parsing (a variation of LZ77) of a given string over an integer polynomially bounded alphabet.
Cite as

Dominik Kempa and Dmitry Kosolobov. LZ-End Parsing in Linear Time. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kempa_et_al:LIPIcs.ESA.2017.53,
  author =	{Kempa, Dominik and Kosolobov, Dmitry},
  title =	{{LZ-End Parsing in Linear Time}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{53:1--53:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.53},
  URN =		{urn:nbn:de:0030-drops-78471},
  doi =		{10.4230/LIPIcs.ESA.2017.53},
  annote =	{Keywords: LZ-End, LZ77, construction algorithm, linear time}
}
Document
DOI: 10.4230/LIPIcs.CPM.2017.23
Palindromic Length in Linear Time

Authors: Kirill Borozdin, Dmitry Kosolobov, Mikhail Rubinchik, and Arseny M. Shur

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

Abstract
Palindromic length of a string is the minimum number of palindromes whose concatenation is equal to this string. The problem of finding the palindromic length drew some attention, and a few O(n log n) time online algorithms were recently designed for it. In this paper we present the first linear time online algorithm for this problem.
Cite as

Kirill Borozdin, Dmitry Kosolobov, Mikhail Rubinchik, and Arseny M. Shur. Palindromic Length in Linear Time. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 23:1-23:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{borozdin_et_al:LIPIcs.CPM.2017.23,
  author =	{Borozdin, Kirill and Kosolobov, Dmitry and Rubinchik, Mikhail and Shur, Arseny M.},
  title =	{{Palindromic Length in Linear Time}},
  booktitle =	{28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)},
  pages =	{23:1--23:12},
  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.2017.23},
  URN =		{urn:nbn:de:0030-drops-73389},
  doi =		{10.4230/LIPIcs.CPM.2017.23},
  annote =	{Keywords: palindrome, palindromic length, palindromic factorization, online}
}
Document
DOI: 10.4230/LIPIcs.STACS.2015.582
Lempel-Ziv Factorization May Be Harder Than Computing All Runs

Authors: Dmitry Kosolobov

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
The complexity of computing the Lempel-Ziv decomposition and the set of all runs (= maximal repetitions) is studied in the decision tree model of computation over ordered alphabet. It is known that both these problems can be solved by RAM algorithms in O(n\log\sigma) time, where n is the length of the input string and \sigma is the number of distinct letters in it. We prove an \Omega(n\log\sigma) lower bound on the number of comparisons required to construct the Lempel-Ziv decomposition and thereby conclude that a popular technique of computation of runs using the Lempel-Ziv decomposition cannot achieve an o(n\log\sigma) time bound. In contrast with this, we exhibit an O(n) decision tree algorithm finding all runs in a string. Therefore, in the decision tree model the runs problem is easier than the Lempel-Ziv decomposition. Thus we support the conjecture that there is a linear RAM algorithm finding all runs.
Cite as

Dmitry Kosolobov. Lempel-Ziv Factorization May Be Harder Than Computing All Runs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 582-593, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kosolobov:LIPIcs.STACS.2015.582,
  author =	{Kosolobov, Dmitry},
  title =	{{Lempel-Ziv Factorization May Be Harder Than Computing All Runs}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{582--593},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.582},
  URN =		{urn:nbn:de:0030-drops-49438},
  doi =		{10.4230/LIPIcs.STACS.2015.582},
  annote =	{Keywords: Lempel-Ziv factorization, runs, repetitions, decision tree, lower bounds}
}
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