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Documents authored by Kozachinskiy, Alexander

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RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.50
Towards Simpler Sorting Networks and Monotone Circuits for Majority

Authors: Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, and Vladimir Podolskii

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
In this paper, we study the problem of computing the majority function by low-depth monotone circuits and a related problem of constructing low-depth sorting networks. We consider both the classical setting with elementary operations of arity 2 and the generalized setting with operations of arity k, where k is a parameter. For both problems and both settings, there are various constructions known, the minimal known depth being logarithmic. However, there is currently no known efficient deterministic construction that simultaneously achieves sub-log-squared depth, simplicity, and has a potential to be used in practice. In this paper we make progress towards resolution of this problem. For computing majority by standard monotone circuits (gates of arity 2) we provide an explicit monotone circuit of depth O(log₂^{5/3} n). The construction is a combination of several known and not too complicated ideas. Essentially, for this result we gradually derandomize the construction of Valiant (1984). As one of the intermediate steps in our result we need an efficient construction of a sorting network with gates of arity k for arbitrary fixed k. For this we provide a new sorting network architecture inspired by representation of inputs as a high-dimensional cube. As a result we obtain a simple construction that improves previous upper bound of 4 log_k² n to 2 log_k² n. We prove the similar bound for the depth of the circuit computing majority of n bits consisting of gates computing majority of k bits. Note, that for both problems there is an explicit construction of depth O(log_k n) known, but the construction is complicated and the constant hidden in O-notation is huge.
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Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, and Vladimir Podolskii. Towards Simpler Sorting Networks and Monotone Circuits for Majority. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dobrokhotovamaikova_et_al:LIPIcs.APPROX/RANDOM.2024.50,
  author =	{Dobrokhotova-Maikova, Natalia and Kozachinskiy, Alexander and Podolskii, Vladimir},
  title =	{{Towards Simpler Sorting Networks and Monotone Circuits for Majority}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{50:1--50:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.50},
  URN =		{urn:nbn:de:0030-drops-210436},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.50},
  annote =	{Keywords: Sorting networks, constant depth, lower bounds, threshold circuits}
}
Document
DOI: 10.4230/LIPIcs.CSL.2024.34
Energy Games over Totally Ordered Groups

Authors: Alexander Kozachinskiy

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract
Kopczyński (ICALP 2006) conjectured that prefix-independent half-positional winning conditions are closed under finite unions. We refute this conjecture over finite arenas. For that, we introduce a new class of prefix-independent bi-positional winning conditions called energy conditions over totally ordered groups. We give an example of two such conditions whose union is not half-positional. We also conjecture that every prefix-independent bi-positional winning condition coincides with some energy condition over a totally ordered group on periodic sequences.
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Alexander Kozachinskiy. Energy Games over Totally Ordered Groups. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kozachinskiy:LIPIcs.CSL.2024.34,
  author =	{Kozachinskiy, Alexander},
  title =	{{Energy Games over Totally Ordered Groups}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{34:1--34:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.34},
  URN =		{urn:nbn:de:0030-drops-196778},
  doi =		{10.4230/LIPIcs.CSL.2024.34},
  annote =	{Keywords: Games on graphs, half-positionality, ordered groups}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.43
Constant-Depth Sorting Networks

Authors: Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, and Vladimir Podolskii

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
In this paper, we address sorting networks that are constructed from comparators of arity k > 2. I.e., in our setting the arity of the comparators - or, in other words, the number of inputs that can be sorted at the unit cost - is a parameter. We study its relationship with two other parameters - n, the number of inputs, and d, the depth. This model received considerable attention. Partly, its motivation is to better understand the structure of sorting networks. In particular, sorting networks with large arity are related to recursive constructions of ordinary sorting networks. Additionally, studies of this model have natural correspondence with a recent line of work on constructing circuits for majority functions from majority gates of lower fan-in. Motivated by these questions, we initiate the studies of lower bounds for constant-depth sorting networks. More precisely, we consider sorting networks of constant depth d and estimate the minimal k for which there is such a network with comparators of arity k. We prove tight lower bounds for d ≤ 4. More precisely, for depths d = 1,2 we observe that k = n. For d = 3 we show that k = ⌈n/2⌉. As our main result, we show that for d = 4 the minimal arity becomes sublinear: k = Θ(n^{2/3}). This contrasts with the case of majority circuits, in which k = O(n^{2/3}) is achievable already for depth d = 3. To prove these results, we develop a new combinatorial technique based on the notion of access to cells of a sorting network.
Cite as

Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, and Vladimir Podolskii. Constant-Depth Sorting Networks. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dobrokhotovamaikova_et_al:LIPIcs.ITCS.2023.43,
  author =	{Dobrokhotova-Maikova, Natalia and Kozachinskiy, Alexander and Podolskii, Vladimir},
  title =	{{Constant-Depth Sorting Networks}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{43:1--43:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.43},
  URN =		{urn:nbn:de:0030-drops-175468},
  doi =		{10.4230/LIPIcs.ITCS.2023.43},
  annote =	{Keywords: Sorting networks, constant depth, lower bounds, threshold circuits}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.43
One-To-Two-Player Lifting for Mildly Growing Memory

Authors: Alexander Kozachinskiy

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
We investigate a phenomenon of "one-to-two-player lifting" in infinite-duration two-player games on graphs with zero-sum objectives. More specifically, let 𝒞 be a class of strategies. It turns out that in many cases, to show that all two-player games on graphs with a given payoff function are determined in 𝒞, it is sufficient to do so for one-player games. That is, in many cases the determinacy in 𝒞 can be "lifted" from one-player games to two-player games. Namely, Gimbert and Zielonka (CONCUR 2005) have shown this for the class of positional strategies. Recently, Bouyer et al. (CONCUR 2020) have extended this to the classes of arena-independent finite-memory strategies. Informally, these are finite-memory strategies that use the same way of storing memory in all game graphs. In this paper, we put the lifting technique into the context of memory complexity. The memory complexity of a payoff function measures, how many states of memory we need to play optimally in game graphs with up to n nodes, depending on n. We address the following question. Assume that we know the memory complexity of our payoff function in one-player games. Then what can be said about its memory complexity in two-player games? In particular, when is it finite? In this paper, we answer this questions for strategies with "chromatic" memory. These are strategies that only accumulate sequences of colors of edges in their memory. We obtain the following results. - Assume that the chromatic memory complexity in one-player games is sublinear in n on some infinite subsequence. Then the chromatic memory complexity in two-player games is finite. - We provide an example in which (a) the chromatic memory complexity in one-player games is linear in n; (b) the memory complexity in two-player games is infinite. Thus, we obtain the exact barrier for the one-to-two-player lifting theorems in the setting of chromatic finite-memory strategies. Previous results only cover payoff functions with constant chromatic memory complexity.
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Alexander Kozachinskiy. One-To-Two-Player Lifting for Mildly Growing Memory. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 43:1-43:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kozachinskiy:LIPIcs.STACS.2022.43,
  author =	{Kozachinskiy, Alexander},
  title =	{{One-To-Two-Player Lifting for Mildly Growing Memory}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{43:1--43:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.43},
  URN =		{urn:nbn:de:0030-drops-158535},
  doi =		{10.4230/LIPIcs.STACS.2022.43},
  annote =	{Keywords: Games on graphs, one-to-two-player lifting, strategy complexity, positional determinacy, finite-memory determinacy}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2021.10
Continuous Positional Payoffs

Authors: Alexander Kozachinskiy

Published in: LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Abstract
What payoffs are positionally determined for deterministic two-player antagonistic games on finite directed graphs? In this paper we study this question for payoffs that are continuous. The main reason why continuous positionally determined payoffs are interesting is that they include the multi-discounted payoffs. We show that for continuous payoffs positional determinacy is equivalent to a simple property called prefix-monotonicity. We provide three proofs of it, using three major techniques of establishing positional determinacy - inductive technique, fixed point technique and strategy improvement technique. A combination of these approaches provides us with better understanding of the structure of continuous positionally determined payoffs as well as with some algorithmic results.
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Alexander Kozachinskiy. Continuous Positional Payoffs. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kozachinskiy:LIPIcs.CONCUR.2021.10,
  author =	{Kozachinskiy, Alexander},
  title =	{{Continuous Positional Payoffs}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-203-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{203},
  editor =	{Haddad, Serge and Varacca, Daniele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.10},
  URN =		{urn:nbn:de:0030-drops-143874},
  doi =		{10.4230/LIPIcs.CONCUR.2021.10},
  annote =	{Keywords: Games on graphs, positional strategies, continuous payoffs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.24
Multiparty Karchmer - Wigderson Games and Threshold Circuits

Authors: Alexander Kozachinskiy and Vladimir Podolskii

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
We suggest a generalization of Karchmer - Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables. This resolves a conjecture of Cohen et al. (CRYPTO 2013).
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Alexander Kozachinskiy and Vladimir Podolskii. Multiparty Karchmer - Wigderson Games and Threshold Circuits. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 24:1-24:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kozachinskiy_et_al:LIPIcs.CCC.2020.24,
  author =	{Kozachinskiy, Alexander and Podolskii, Vladimir},
  title =	{{Multiparty Karchmer - Wigderson Games and Threshold Circuits}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{24:1--24:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.24},
  URN =		{urn:nbn:de:0030-drops-125767},
  doi =		{10.4230/LIPIcs.CCC.2020.24},
  annote =	{Keywords: Karchmer-Wigderson Games, Threshold Circuits, threshold gates, majority function}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.4
From Expanders to Hitting Distributions and Simulation Theorems

Authors: Alexander Kozachinskiy

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
In this paper we explore hitting distributions, a notion that arose recently in the context of deterministic "query-to-communication" simulation theorems. We show that any expander in which any two distinct vertices have at most one common neighbor can be transformed into a gadget possessing good hitting distributions. We demonstrate that this result is applicable to affine plane expanders and to Lubotzky-Phillips-Sarnak construction of Ramanujan graphs . In particular, from affine plane expanders we extract a gadget achieving the best known trade-off between the arity of outer function and the size of gadget. More specifically, when this gadget has k bits on input, it admits a simulation theorem for all outer function of arity roughly 2^(k/2) or less (the same was also known for k-bit Inner Product). In addition we show that, unlike Inner Product, underlying hitting distributions in our new gadget are "polynomial-time listable" in the sense that their supports can be written down in time 2^O(k), i.e. in time polynomial in size of gadget's matrix. We also obtain two results showing that with current technique no better trade-off between the arity of outer function and the size of gadget can be achieved. Namely, we observe that no gadget can have hitting distributions with significantly better parameters than Inner Product or our new affine plane gadget. We also show that Thickness Lemma, a place which causes restrictions on the arity of outer functions in proofs of simulation theorems, is unimprovable.
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Alexander Kozachinskiy. From Expanders to Hitting Distributions and Simulation Theorems. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kozachinskiy:LIPIcs.MFCS.2018.4,
  author =	{Kozachinskiy, Alexander},
  title =	{{From Expanders to Hitting Distributions and Simulation Theorems}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{4:1--4:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.4},
  URN =		{urn:nbn:de:0030-drops-95863},
  doi =		{10.4230/LIPIcs.MFCS.2018.4},
  annote =	{Keywords: simulation theorems, hitting distributions, expanders}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.7
One-Sided Error Communication Complexity of Gap Hamming Distance

Authors: Egor Klenin and Alexander Kozachinskiy

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
Assume that Alice has a binary string x and Bob a binary string y, both strings are of length n. Their goal is to output 0, if x and y are at least L-close in Hamming distance, and output 1, if x and y are at least U-far in Hamming distance, where L < U are some integer parameters known to both parties. If the Hamming distance between x and y lies in the interval (L, U), they are allowed to output anything. This problem is called the Gap Hamming Distance. In this paper we study public-coin one-sided error communication complexity of this problem. The error with probability at most 1/2 is allowed only for pairs at Hamming distance at least U. In this paper we determine this complexity up to factors logarithmic in L. The protocol we construct for the upper bound is simultaneous.
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Egor Klenin and Alexander Kozachinskiy. One-Sided Error Communication Complexity of Gap Hamming Distance. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{klenin_et_al:LIPIcs.MFCS.2018.7,
  author =	{Klenin, Egor and Kozachinskiy, Alexander},
  title =	{{One-Sided Error Communication Complexity of Gap Hamming Distance}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.7},
  URN =		{urn:nbn:de:0030-drops-95893},
  doi =		{10.4230/LIPIcs.MFCS.2018.7},
  annote =	{Keywords: Communication Complexity, Gap Hamming Distance, one-sided error}
}
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