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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In this paper we give an algorithm for streaming k-edit approximate pattern matching which uses space Õ(k²) and time Õ(k²) per arriving symbol. This improves substantially on the recent algorithm of Kociumaka, Porat and Starikovskaya [Kociumaka et al., 2022] which uses space Õ(k⁵) and time Õ(k⁸) per arriving symbol. In the k-edit approximate pattern matching problem we get a pattern P and text T and we want to identify all substrings of the text T that are at edit distance at most k from P. In the streaming version of this problem both the pattern and the text arrive in a streaming fashion symbol by symbol and after each symbol of the text we need to report whether there is a current suffix of the text with edit distance at most k from P. We measure the total space needed by the algorithm and time needed per arriving symbol.

Sudatta Bhattacharya and Michal Koucký. Streaming k-Edit Approximate Pattern Matching via String Decomposition. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bhattacharya_et_al:LIPIcs.ICALP.2023.22, author = {Bhattacharya, Sudatta and Kouck\'{y}, Michal}, title = {{Streaming k-Edit Approximate Pattern Matching via String Decomposition}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.22}, URN = {urn:nbn:de:0030-drops-180741}, doi = {10.4230/LIPIcs.ICALP.2023.22}, annote = {Keywords: Approximate pattern matching, edit distance, streaming algorithms} }

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

In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li [Li and Li, 2004] and several data structure problems such as non-adaptive function inversion of Hellman [M. Hellman, 1980] and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.

Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová. Data Structures Lower Bounds and Popular Conjectures. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.ESA.2021.39, author = {Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal and Kr\'{a}l, Karel and Sl{\'\i}vov\'{a}, Veronika}, title = {{Data Structures Lower Bounds and Popular Conjectures}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {39:1--39:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.39}, URN = {urn:nbn:de:0030-drops-146207}, doi = {10.4230/LIPIcs.ESA.2021.39}, annote = {Keywords: Data structures, Circuits, Lower bounds, Network Coding Conjecture} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We build boolean circuits of size 𝒪(nm²) and depth 𝒪(log(n) + m log(m)) for sorting n integers each of m-bits. We build also circuits that sort n integers each of m-bits according to their first k bits that are of size 𝒪(nmk (1 + log^*(n) - log^*(m))) and depth 𝒪(log³(n)). This improves on the results of Asharov et al. [Asharov et al., 2021] and resolves some of their open questions.

Michal Koucký and Karel Král. Sorting Short Integers. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 88:1-88:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{koucky_et_al:LIPIcs.ICALP.2021.88, author = {Kouck\'{y}, Michal and Kr\'{a}l, Karel}, title = {{Sorting Short Integers}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {88:1--88:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.88}, URN = {urn:nbn:de:0030-drops-141577}, doi = {10.4230/LIPIcs.ICALP.2021.88}, annote = {Keywords: sorting, small integers, boolean circuits} }

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Invited Talk

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

The edit distance (or Levenshtein distance) between two strings x, y is the minimum number of character insertions, deletions, and substitutions needed to convert x into y. It has numerous applications in various fields from text processing to bioinformatics so algorithms for edit distance computation attract lot of attention. In this talk I will survey recent progress on computational aspects of edit distance in several contexts: computing edit distance approximately, sketching and computing it in streaming model, exchanging strings in communication complexity model, and building error correcting codes for edit distance. I will point out many problems that are still open in those areas.

Michal Koucký. Computing Edit Distance (Invited Talk). In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{koucky:LIPIcs.CPM.2021.2, author = {Kouck\'{y}, Michal}, title = {{Computing Edit Distance}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {2:1--2:1}, 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.2021.2}, URN = {urn:nbn:de:0030-drops-139534}, doi = {10.4230/LIPIcs.CPM.2021.2}, annote = {Keywords: edit distance, streaming algorithms, approximation algorithms, sketching} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

In this paper we study the computational complexity of functions that have efficient card-based protocols. A study of card-based protocols was initiated by den Boer [den Boer, 1990] as a means for secure two-party computation. Our contribution is two-fold: We classify a large class of protocols with respect to the computational complexity of functions they compute, and we propose other encodings of inputs which require fewer cards than the usual 2-card representation.

Pavel Dvořák and Michal Koucký. Barrington Plays Cards: The Complexity of Card-Based Protocols. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2021.26, author = {Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal}, title = {{Barrington Plays Cards: The Complexity of Card-Based Protocols}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {26:1--26:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus 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.2021.26}, URN = {urn:nbn:de:0030-drops-136715}, doi = {10.4230/LIPIcs.STACS.2021.26}, annote = {Keywords: Efficient card-based protocol, Branching program, Turing machine} }

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

Given a Boolean function f:{-1,1}ⁿ→ {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f̂(S)². The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [E. Friedgut and G. Kalai, 1996] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that ℍ(f̂²)≤ C⋅ Inf(f), where ℍ(f̂²) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f?
In this paper we present three new contributions towards the FEI conjecture:
ii) Our first contribution shows that ℍ(f̂²) ≤ 2⋅ aUC^⊕(f), where aUC^⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [S. Chakraborty et al., 2016]. We further improve this bound for unambiguous DNFs.
iii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [R. O'Donnell et al., 2011; R. O'Donnell, 2014] which asks if ℍ_{∞}(f̂²) ≤ C⋅ Inf(f), where ℍ_{∞}(f̂²) is the min-entropy of the Fourier distribution. We show ℍ_{∞}(f̂²) ≤ 2⋅?_{min}^⊕(f), where ?_{min}^⊕(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have ℍ_{∞}(f̂²) ≤ 2log (‖f̂‖_{1,ε}/(1-ε)), where ‖f̂‖_{1,ε} is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k).
iv) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2^ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.

Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf. Improved Bounds on Fourier Entropy and Min-Entropy. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arunachalam_et_al:LIPIcs.STACS.2020.45, author = {Arunachalam, Srinivasan and Chakraborty, Sourav and Kouck\'{y}, Michal and Saurabh, Nitin and de Wolf, Ronald}, title = {{Improved Bounds on Fourier Entropy and Min-Entropy}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {45:1--45:19}, 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.2020.45}, URN = {urn:nbn:de:0030-drops-119062}, doi = {10.4230/LIPIcs.STACS.2020.45}, annote = {Keywords: Fourier analysis of Boolean functions, FEI conjecture, query complexity, polynomial approximation, approximate degree, certificate complexity} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

We consider the approximate pattern matching problem under edit distance. In this problem we are given a pattern P of length m and a text T of length n over some alphabet Sigma, and a positive integer k. The goal is to find all the positions j in T such that there is a substring of T ending at j which has edit distance at most k from the pattern P. Recall, the edit distance between two strings is the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. For a position t in {1,...,n}, let k_t be the smallest edit distance between P and any substring of T ending at t. In this paper we give a constant factor approximation to the sequence k_1,k_2,...,k_n. We consider both offline and online settings.
In the offline setting, where both P and T are available, we present an algorithm that for all t in {1,...,n}, computes the value of k_t approximately within a constant factor. The worst case running time of our algorithm is O~(n m^(3/4)).
In the online setting, we are given P and then T arrives one symbol at a time. We design an algorithm that upon arrival of the t-th symbol of T computes k_t approximately within O(1)-multiplicative factor and m^(8/9)-additive error. Our algorithm takes O~(m^(1-(7/54))) amortized time per symbol arrival and takes O~(m^(1-(1/54))) additional space apart from storing the pattern P. Both of our algorithms are randomized and produce correct answer with high probability. To the best of our knowledge this is the first algorithm that takes worst-case sublinear (in the length of the pattern) time and sublinear extra space for the online approximate pattern matching problem. To get our result we build on the technique of Chakraborty, Das, Goldenberg, Koucký and Saks [FOCS'18] for computing a constant factor approximation of edit distance in sub-quadratic time.

Diptarka Chakraborty, Debarati Das, and Michal Koucký. Approximate Online Pattern Matching in Sublinear Time. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2019.10, author = {Chakraborty, Diptarka and Das, Debarati and Kouck\'{y}, Michal}, title = {{Approximate Online Pattern Matching in Sublinear Time}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.10}, URN = {urn:nbn:de:0030-drops-115726}, doi = {10.4230/LIPIcs.FSTTCS.2019.10}, annote = {Keywords: Approximate Pattern Matching, Online Pattern Matching, Edit Distance, Sublinear Algorithm, Streaming Algorithm} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A quasi-Gray code of dimension n and length l over an alphabet Sigma is a sequence of distinct words w_1,w_2,...,w_l from Sigma^n such that any two consecutive words differ in at most c coordinates, for some fixed constant c>0. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in order to transform any word w_i into its successor w_{i+1}.
We present construction of quasi-Gray codes of dimension n and length 3^n over the ternary alphabet {0,1,2} with worst-case read complexity O(log n) and write complexity 2. This generalizes to arbitrary odd-size alphabets. For the binary alphabet, we present quasi-Gray codes of dimension n and length at least 2^n - 20n with worst-case read complexity 6+log n and write complexity 2. This complements a recent result by Raskin [Raskin '17] who shows that any quasi-Gray code over binary alphabet of length 2^n has read complexity Omega(n).
Our results significantly improve on previously known constructions and for the odd-size alphabets we break the Omega(n) worst-case barrier for space-optimal (non-redundant) quasi-Gray codes with constant number of writes. We obtain our results via a novel application of algebraic tools together with the principles of catalytic computation [Buhrman et al. '14, Ben-Or and Cleve '92, Barrington '89, Coppersmith and Grossman '75].

Diptarka Chakraborty, Debarati Das, Michal Koucký, and Nitin Saurabh. Space-Optimal Quasi-Gray Codes with Logarithmic Read Complexity. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chakraborty_et_al:LIPIcs.ESA.2018.12, author = {Chakraborty, Diptarka and Das, Debarati and Kouck\'{y}, Michal and Saurabh, Nitin}, title = {{Space-Optimal Quasi-Gray Codes with Logarithmic Read Complexity}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.12}, URN = {urn:nbn:de:0030-drops-94750}, doi = {10.4230/LIPIcs.ESA.2018.12}, annote = {Keywords: Gray code, Space-optimal counter, Decision assignment tree, Cell probe model} }

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

In this paper we propose models of combinatorial algorithms for the Boolean
Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models.
First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976),
and we prove that the time required by any algorithm
for the BMM is at least Omega(n^3 / 2^{O( sqrt{ log n })}). Subsequently, we propose a more general model capable of simulating the
"Four Russian Algorithm". We prove a lower bound of Omega(n^{7/3} / 2^{O(sqrt{ log n })}) for the BMM under this model.
We use a special class of graphs, called (r,t)-graphs, originally discovered by Rusza and Szemeredi (1978),
along with randomization, to construct matrices that are hard instances for our combinatorial models.

Debarati Das, Michal Koucký, and Michael Saks. Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{das_et_al:LIPIcs.STACS.2018.23, author = {Das, Debarati and Kouck\'{y}, Michal and Saks, Michael}, title = {{Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {23:1--23:14}, 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.2018.23}, URN = {urn:nbn:de:0030-drops-85050}, doi = {10.4230/LIPIcs.STACS.2018.23}, annote = {Keywords: Lower bounds, Combinatorial algorithm, Boolean matrix multiplication} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson (2002), and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the "NC^1 reasoning"). As a corollary, we prove the assumption made by Jerabek (2011) that a construction of certain bipartite expander graphs can be formalized in VNC^1. This in turn implies that every proof in Gentzen's sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlak (2002).

Sam Buss, Valentine Kabanets, Antonina Kolokolova, and Michal Koucky. Expander Construction in VNC1. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 31:1-31:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buss_et_al:LIPIcs.ITCS.2017.31, author = {Buss, Sam and Kabanets, Valentine and Kolokolova, Antonina and Koucky, Michal}, title = {{Expander Construction in VNC1}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {31:1--31:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.31}, URN = {urn:nbn:de:0030-drops-81871}, doi = {10.4230/LIPIcs.ITCS.2017.31}, annote = {Keywords: expander graphs, bounded arithmetic, alternating log time, sequent calculus, monotone propositional logic} }

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**Published in:** Dagstuhl Reports, Volume 7, Issue 3 (2017)

This report documents the program and the outcomes of Dagstuhl Seminar 17121 "Computational Complexity of Discrete Problems". The first section gives an overview of the topics covered and the organization of the meeting. Section 2 lists the talks given in alphabetical order. The last section contains the abstracts of the talks.

Anna Gál, Michal Koucký, Oded Regev, and Till Tantau. Computational Complexity of Discrete Problems (Dagstuhl Seminar 17121). In Dagstuhl Reports, Volume 7, Issue 3, pp. 45-69, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{gal_et_al:DagRep.7.3.45, author = {G\'{a}l, Anna and Kouck\'{y}, Michal and Regev, Oded and Tantau, Till}, title = {{Computational Complexity of Discrete Problems (Dagstuhl Seminar 17121)}}, pages = {45--69}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {7}, number = {3}, editor = {G\'{a}l, Anna and Kouck\'{y}, Michal and Regev, Oded and Tantau, Till}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.3.45}, URN = {urn:nbn:de:0030-drops-73611}, doi = {10.4230/DagRep.7.3.45}, annote = {Keywords: Computational Complexity} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds.
To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.

Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Lower Bounds for Elimination via Weak Regularity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2017.21, author = {Chattopadhyay, Arkadev and Dvor\'{a}k, Pavel and Kouck\'{y}, Michal and Loff, Bruno and Mukhopadhyay, Sagnik}, title = {{Lower Bounds for Elimination via Weak Regularity}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert 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.2017.21}, URN = {urn:nbn:de:0030-drops-70128}, doi = {10.4230/LIPIcs.STACS.2017.21}, annote = {Keywords: communication complexity, elimination, discrepancy, regularity, greater-than} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation.
Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.

Harry Buhrman, Michal Koucký, Bruno Loff, and Florian Speelman. Catalytic Space: Non-determinism and Hierarchy. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{buhrman_et_al:LIPIcs.STACS.2016.24, author = {Buhrman, Harry and Kouck\'{y}, Michal and Loff, Bruno and Speelman, Florian}, title = {{Catalytic Space: Non-determinism and Hierarchy}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.24}, URN = {urn:nbn:de:0030-drops-57258}, doi = {10.4230/LIPIcs.STACS.2016.24}, annote = {Keywords: catalytic computation, Immerman–Szelepcs\'{e}nyi theorem, space hierarchy} }

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**Published in:** Dagstuhl Reports, Volume 4, Issue 3 (2014)

This report documents the program and the outcomes of Dagstuhl Seminar 14121 "Computational Complexity of Discrete Problems". The first section gives an overview of the topics covered and the organization of the meeting. Section 2 lists the talks given in chronological order. The last section contains the abstracts of the talks.

Anna Gal, Michal Koucky, Oded Regev, and Rüdiger Reischuk. Computational Complexity of Discrete Problems (Dagstuhl Seminar 14121). In Dagstuhl Reports, Volume 4, Issue 3, pp. 62-84, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Article{gal_et_al:DagRep.4.3.62, author = {Gal, Anna and Koucky, Michal and Regev, Oded and Reischuk, R\"{u}diger}, title = {{Computational Complexity of Discrete Problems (Dagstuhl Seminar 14121)}}, pages = {62--84}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2014}, volume = {4}, number = {3}, editor = {Gal, Anna and Koucky, Michal and Regev, Oded and Reischuk, R\"{u}diger}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.3.62}, URN = {urn:nbn:de:0030-drops-45921}, doi = {10.4230/DagRep.4.3.62}, annote = {Keywords: discrete problems, computational complexity, Turing machines, Boolean circuits, arithmetic circuits, quantum computing, communication complexity, pseudorandomness, derandomization, approximation, data streams} }

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**Published in:** Dagstuhl Reports, Volume 1, Issue 3 (2011)

This report documents the program and the outcomes of Dagstuhl Seminar
11121 ``Computational Complexity of Discrete Problems''.
The first section gives an overview of the topics covered and the organization
of the meeting. Section~2 lists the talks given in chronological order.
The last section contains the abstracts of the talks.

Martin Grohe, Michal Koucky, Rüdiger Reischik, and Dieter van Melkebeek. Computational Complexity of Discrete Problems (Dagstuhl Seminar 11121). In Dagstuhl Reports, Volume 1, Issue 3, pp. 42-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@Article{grohe_et_al:DagRep.1.3.42, author = {Grohe, Martin and Koucky, Michal and Reischik, R\"{u}diger and van Melkebeek, Dieter}, title = {{Computational Complexity of Discrete Problems (Dagstuhl Seminar 11121)}}, pages = {42--66}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2011}, volume = {1}, number = {3}, editor = {Grohe, Martin and Koucky, Michal and Reischik, R\"{u}diger and van Melkebeek, Dieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.1.3.42}, URN = {urn:nbn:de:0030-drops-31935}, doi = {10.4230/DagRep.1.3.42}, annote = {Keywords: Discrete problems, computational complexity, Turing machines, Boolean circuits, quantum computing, communication and query complexity, extractors, pseudorandomness, derandomization, approximation, coding cryptography, algorithmic learning} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 7411, Algebraic Methods in Computational Complexity (2008)

We study the two party problem of randomly selecting a string among
all the strings of length n. We want the protocol to have the
property that the output distribution has high entropy, even
when one of the two parties is dishonest and deviates from the
protocol. We develop protocols that achieve high, close to n,
entropy.
In the literature the randomness guarantee is usually expressed as
being close to the uniform distribution or in terms of resiliency.
The notion of entropy is not directly comparable to that of
resiliency, but we establish a connection between the two that
allows us to compare our protocols with the existing ones.
We construct an
explicit protocol that yields entropy n - O(1) and has 4log^* n
rounds, improving over the protocol of Goldwasser
et al. that also achieves this entropy but needs O(n)
rounds. Both these protocols need O(n^2) bits of communication.
Next we reduce the communication in our protocols. We show the existence,
non-explicitly, of a protocol that has 6-rounds, 2n + 8log n bits
of communication and yields entropy n- O(log n) and min-entropy
n/2 - O(log n). Our protocol achieves the same entropy bound as
the recent, also non-explicit, protocol of Gradwohl
et al., however achieves much higher min-entropy: n/2 -
O(log n) versus O(log n).
Finally we exhibit very simple explicit protocols. We connect the
security parameter of these geometric protocols with the well
studied Kakeya problem motivated by harmonic analysis and analytical
number theory. We are only able to prove that these protocols have
entropy 3n/4 but still n/2 - O(log n) min-entropy. Therefore
they do not perform as well with respect to the explicit
constructions of Gradwohl et al. entropy-wise, but still
have much better min-entropy. We conjecture that these simple
protocols achieve n -o(n) entropy. Our geometric
construction and its relation to the Kakeya problem follows a new and
different approach to the random selection problem than any of the
previously known protocols.

Nikolai K. Vereshchagin, Harry Buhrman, Matthias Cristandl, Michal Koucky, Zvi Lotker, and Boaz Patt-Shamir. High Entropy Random Selection Protocols. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 7411, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{vereshchagin_et_al:DagSemProc.07411.5, author = {Vereshchagin, Nikolai K. and Buhrman, Harry and Cristandl, Matthias and Koucky, Michal and Lotker, Zvi and Patt-Shamir, Boaz}, title = {{High Entropy Random Selection Protocols}}, booktitle = {Algebraic Methods in Computational Complexity}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {7411}, editor = {Manindra Agrawal and Harry Buhrman and Lance Fortnow and Thomas Thierauf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07411.5}, URN = {urn:nbn:de:0030-drops-13091}, doi = {10.4230/DagSemProc.07411.5}, annote = {Keywords: Shannon entropy, Random string ds} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

We propose a new model of restricted branching programs
which we call {em incremental branching programs}.
We show that {em syntactic}
incremental branching programs capture previously studied
structured models of computation for the problem GEN, namely marking
machines [Cook74].
and Poon's extension [Poon93] of jumping automata
on graphs [CookRackoff80].
We then prove
exponential size lower bounds for our syntactic incremental model,
and for some other restricted branching program models as well.
We further show that nondeterministic syntactic incremental
branching programs are
provably stronger than their deterministic counterpart when solving a
natural NL-complete GEN subproblem.
It remains open if syntactic incremental branching programs are as powerful
as unrestricted branching programs for GEN problems.
Joint work with Anna GÃƒÂ¡l and Michal KouckÃƒÂ½

Anna Gál, Pierre McKenzie, and Michal Koucký. Incremental branching programs. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{gal_et_al:DagSemProc.06111.10, author = {G\'{a}l, Anna and McKenzie, Pierre and Kouck\'{y}, Michal}, title = {{Incremental branching programs}}, booktitle = {Complexity of Boolean Functions}, pages = {1--20}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.10}, URN = {urn:nbn:de:0030-drops-6230}, doi = {10.4230/DagSemProc.06111.10}, annote = {Keywords: Complexity theory, branching programs, logarithmic space, marking machines} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In this paper we give an algorithm for streaming k-edit approximate pattern matching which uses space Õ(k²) and time Õ(k²) per arriving symbol. This improves substantially on the recent algorithm of Kociumaka, Porat and Starikovskaya [Kociumaka et al., 2022] which uses space Õ(k⁵) and time Õ(k⁸) per arriving symbol. In the k-edit approximate pattern matching problem we get a pattern P and text T and we want to identify all substrings of the text T that are at edit distance at most k from P. In the streaming version of this problem both the pattern and the text arrive in a streaming fashion symbol by symbol and after each symbol of the text we need to report whether there is a current suffix of the text with edit distance at most k from P. We measure the total space needed by the algorithm and time needed per arriving symbol.

Sudatta Bhattacharya and Michal Koucký. Streaming k-Edit Approximate Pattern Matching via String Decomposition. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bhattacharya_et_al:LIPIcs.ICALP.2023.22, author = {Bhattacharya, Sudatta and Kouck\'{y}, Michal}, title = {{Streaming k-Edit Approximate Pattern Matching via String Decomposition}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.22}, URN = {urn:nbn:de:0030-drops-180741}, doi = {10.4230/LIPIcs.ICALP.2023.22}, annote = {Keywords: Approximate pattern matching, edit distance, streaming algorithms} }

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

In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li [Li and Li, 2004] and several data structure problems such as non-adaptive function inversion of Hellman [M. Hellman, 1980] and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.

Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová. Data Structures Lower Bounds and Popular Conjectures. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.ESA.2021.39, author = {Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal and Kr\'{a}l, Karel and Sl{\'\i}vov\'{a}, Veronika}, title = {{Data Structures Lower Bounds and Popular Conjectures}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {39:1--39:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.39}, URN = {urn:nbn:de:0030-drops-146207}, doi = {10.4230/LIPIcs.ESA.2021.39}, annote = {Keywords: Data structures, Circuits, Lower bounds, Network Coding Conjecture} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We build boolean circuits of size 𝒪(nm²) and depth 𝒪(log(n) + m log(m)) for sorting n integers each of m-bits. We build also circuits that sort n integers each of m-bits according to their first k bits that are of size 𝒪(nmk (1 + log^*(n) - log^*(m))) and depth 𝒪(log³(n)). This improves on the results of Asharov et al. [Asharov et al., 2021] and resolves some of their open questions.

Michal Koucký and Karel Král. Sorting Short Integers. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 88:1-88:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{koucky_et_al:LIPIcs.ICALP.2021.88, author = {Kouck\'{y}, Michal and Kr\'{a}l, Karel}, title = {{Sorting Short Integers}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {88:1--88:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.88}, URN = {urn:nbn:de:0030-drops-141577}, doi = {10.4230/LIPIcs.ICALP.2021.88}, annote = {Keywords: sorting, small integers, boolean circuits} }

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Invited Talk

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

The edit distance (or Levenshtein distance) between two strings x, y is the minimum number of character insertions, deletions, and substitutions needed to convert x into y. It has numerous applications in various fields from text processing to bioinformatics so algorithms for edit distance computation attract lot of attention. In this talk I will survey recent progress on computational aspects of edit distance in several contexts: computing edit distance approximately, sketching and computing it in streaming model, exchanging strings in communication complexity model, and building error correcting codes for edit distance. I will point out many problems that are still open in those areas.

Michal Koucký. Computing Edit Distance (Invited Talk). In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{koucky:LIPIcs.CPM.2021.2, author = {Kouck\'{y}, Michal}, title = {{Computing Edit Distance}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {2:1--2:1}, 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.2021.2}, URN = {urn:nbn:de:0030-drops-139534}, doi = {10.4230/LIPIcs.CPM.2021.2}, annote = {Keywords: edit distance, streaming algorithms, approximation algorithms, sketching} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

In this paper we study the computational complexity of functions that have efficient card-based protocols. A study of card-based protocols was initiated by den Boer [den Boer, 1990] as a means for secure two-party computation. Our contribution is two-fold: We classify a large class of protocols with respect to the computational complexity of functions they compute, and we propose other encodings of inputs which require fewer cards than the usual 2-card representation.

Pavel Dvořák and Michal Koucký. Barrington Plays Cards: The Complexity of Card-Based Protocols. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2021.26, author = {Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal}, title = {{Barrington Plays Cards: The Complexity of Card-Based Protocols}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {26:1--26:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus 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.2021.26}, URN = {urn:nbn:de:0030-drops-136715}, doi = {10.4230/LIPIcs.STACS.2021.26}, annote = {Keywords: Efficient card-based protocol, Branching program, Turing machine} }

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

Given a Boolean function f:{-1,1}ⁿ→ {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f̂(S)². The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [E. Friedgut and G. Kalai, 1996] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that ℍ(f̂²)≤ C⋅ Inf(f), where ℍ(f̂²) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f?
In this paper we present three new contributions towards the FEI conjecture:
ii) Our first contribution shows that ℍ(f̂²) ≤ 2⋅ aUC^⊕(f), where aUC^⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [S. Chakraborty et al., 2016]. We further improve this bound for unambiguous DNFs.
iii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [R. O'Donnell et al., 2011; R. O'Donnell, 2014] which asks if ℍ_{∞}(f̂²) ≤ C⋅ Inf(f), where ℍ_{∞}(f̂²) is the min-entropy of the Fourier distribution. We show ℍ_{∞}(f̂²) ≤ 2⋅?_{min}^⊕(f), where ?_{min}^⊕(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have ℍ_{∞}(f̂²) ≤ 2log (‖f̂‖_{1,ε}/(1-ε)), where ‖f̂‖_{1,ε} is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k).
iv) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2^ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.

Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf. Improved Bounds on Fourier Entropy and Min-Entropy. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arunachalam_et_al:LIPIcs.STACS.2020.45, author = {Arunachalam, Srinivasan and Chakraborty, Sourav and Kouck\'{y}, Michal and Saurabh, Nitin and de Wolf, Ronald}, title = {{Improved Bounds on Fourier Entropy and Min-Entropy}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {45:1--45:19}, 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.2020.45}, URN = {urn:nbn:de:0030-drops-119062}, doi = {10.4230/LIPIcs.STACS.2020.45}, annote = {Keywords: Fourier analysis of Boolean functions, FEI conjecture, query complexity, polynomial approximation, approximate degree, certificate complexity} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

We consider the approximate pattern matching problem under edit distance. In this problem we are given a pattern P of length m and a text T of length n over some alphabet Sigma, and a positive integer k. The goal is to find all the positions j in T such that there is a substring of T ending at j which has edit distance at most k from the pattern P. Recall, the edit distance between two strings is the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. For a position t in {1,...,n}, let k_t be the smallest edit distance between P and any substring of T ending at t. In this paper we give a constant factor approximation to the sequence k_1,k_2,...,k_n. We consider both offline and online settings.
In the offline setting, where both P and T are available, we present an algorithm that for all t in {1,...,n}, computes the value of k_t approximately within a constant factor. The worst case running time of our algorithm is O~(n m^(3/4)).
In the online setting, we are given P and then T arrives one symbol at a time. We design an algorithm that upon arrival of the t-th symbol of T computes k_t approximately within O(1)-multiplicative factor and m^(8/9)-additive error. Our algorithm takes O~(m^(1-(7/54))) amortized time per symbol arrival and takes O~(m^(1-(1/54))) additional space apart from storing the pattern P. Both of our algorithms are randomized and produce correct answer with high probability. To the best of our knowledge this is the first algorithm that takes worst-case sublinear (in the length of the pattern) time and sublinear extra space for the online approximate pattern matching problem. To get our result we build on the technique of Chakraborty, Das, Goldenberg, Koucký and Saks [FOCS'18] for computing a constant factor approximation of edit distance in sub-quadratic time.

Diptarka Chakraborty, Debarati Das, and Michal Koucký. Approximate Online Pattern Matching in Sublinear Time. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2019.10, author = {Chakraborty, Diptarka and Das, Debarati and Kouck\'{y}, Michal}, title = {{Approximate Online Pattern Matching in Sublinear Time}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.10}, URN = {urn:nbn:de:0030-drops-115726}, doi = {10.4230/LIPIcs.FSTTCS.2019.10}, annote = {Keywords: Approximate Pattern Matching, Online Pattern Matching, Edit Distance, Sublinear Algorithm, Streaming Algorithm} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A quasi-Gray code of dimension n and length l over an alphabet Sigma is a sequence of distinct words w_1,w_2,...,w_l from Sigma^n such that any two consecutive words differ in at most c coordinates, for some fixed constant c>0. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in order to transform any word w_i into its successor w_{i+1}.
We present construction of quasi-Gray codes of dimension n and length 3^n over the ternary alphabet {0,1,2} with worst-case read complexity O(log n) and write complexity 2. This generalizes to arbitrary odd-size alphabets. For the binary alphabet, we present quasi-Gray codes of dimension n and length at least 2^n - 20n with worst-case read complexity 6+log n and write complexity 2. This complements a recent result by Raskin [Raskin '17] who shows that any quasi-Gray code over binary alphabet of length 2^n has read complexity Omega(n).
Our results significantly improve on previously known constructions and for the odd-size alphabets we break the Omega(n) worst-case barrier for space-optimal (non-redundant) quasi-Gray codes with constant number of writes. We obtain our results via a novel application of algebraic tools together with the principles of catalytic computation [Buhrman et al. '14, Ben-Or and Cleve '92, Barrington '89, Coppersmith and Grossman '75].

Diptarka Chakraborty, Debarati Das, Michal Koucký, and Nitin Saurabh. Space-Optimal Quasi-Gray Codes with Logarithmic Read Complexity. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chakraborty_et_al:LIPIcs.ESA.2018.12, author = {Chakraborty, Diptarka and Das, Debarati and Kouck\'{y}, Michal and Saurabh, Nitin}, title = {{Space-Optimal Quasi-Gray Codes with Logarithmic Read Complexity}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.12}, URN = {urn:nbn:de:0030-drops-94750}, doi = {10.4230/LIPIcs.ESA.2018.12}, annote = {Keywords: Gray code, Space-optimal counter, Decision assignment tree, Cell probe model} }

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

In this paper we propose models of combinatorial algorithms for the Boolean
Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models.
First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976),
and we prove that the time required by any algorithm
for the BMM is at least Omega(n^3 / 2^{O( sqrt{ log n })}). Subsequently, we propose a more general model capable of simulating the
"Four Russian Algorithm". We prove a lower bound of Omega(n^{7/3} / 2^{O(sqrt{ log n })}) for the BMM under this model.
We use a special class of graphs, called (r,t)-graphs, originally discovered by Rusza and Szemeredi (1978),
along with randomization, to construct matrices that are hard instances for our combinatorial models.

Debarati Das, Michal Koucký, and Michael Saks. Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{das_et_al:LIPIcs.STACS.2018.23, author = {Das, Debarati and Kouck\'{y}, Michal and Saks, Michael}, title = {{Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {23:1--23:14}, 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.2018.23}, URN = {urn:nbn:de:0030-drops-85050}, doi = {10.4230/LIPIcs.STACS.2018.23}, annote = {Keywords: Lower bounds, Combinatorial algorithm, Boolean matrix multiplication} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson (2002), and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the "NC^1 reasoning"). As a corollary, we prove the assumption made by Jerabek (2011) that a construction of certain bipartite expander graphs can be formalized in VNC^1. This in turn implies that every proof in Gentzen's sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlak (2002).

Sam Buss, Valentine Kabanets, Antonina Kolokolova, and Michal Koucky. Expander Construction in VNC1. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 31:1-31:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buss_et_al:LIPIcs.ITCS.2017.31, author = {Buss, Sam and Kabanets, Valentine and Kolokolova, Antonina and Koucky, Michal}, title = {{Expander Construction in VNC1}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {31:1--31:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.31}, URN = {urn:nbn:de:0030-drops-81871}, doi = {10.4230/LIPIcs.ITCS.2017.31}, annote = {Keywords: expander graphs, bounded arithmetic, alternating log time, sequent calculus, monotone propositional logic} }

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**Published in:** Dagstuhl Reports, Volume 7, Issue 3 (2017)

This report documents the program and the outcomes of Dagstuhl Seminar 17121 "Computational Complexity of Discrete Problems". The first section gives an overview of the topics covered and the organization of the meeting. Section 2 lists the talks given in alphabetical order. The last section contains the abstracts of the talks.

Anna Gál, Michal Koucký, Oded Regev, and Till Tantau. Computational Complexity of Discrete Problems (Dagstuhl Seminar 17121). In Dagstuhl Reports, Volume 7, Issue 3, pp. 45-69, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{gal_et_al:DagRep.7.3.45, author = {G\'{a}l, Anna and Kouck\'{y}, Michal and Regev, Oded and Tantau, Till}, title = {{Computational Complexity of Discrete Problems (Dagstuhl Seminar 17121)}}, pages = {45--69}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {7}, number = {3}, editor = {G\'{a}l, Anna and Kouck\'{y}, Michal and Regev, Oded and Tantau, Till}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.3.45}, URN = {urn:nbn:de:0030-drops-73611}, doi = {10.4230/DagRep.7.3.45}, annote = {Keywords: Computational Complexity} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds.
To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.

Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Lower Bounds for Elimination via Weak Regularity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2017.21, author = {Chattopadhyay, Arkadev and Dvor\'{a}k, Pavel and Kouck\'{y}, Michal and Loff, Bruno and Mukhopadhyay, Sagnik}, title = {{Lower Bounds for Elimination via Weak Regularity}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert 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.2017.21}, URN = {urn:nbn:de:0030-drops-70128}, doi = {10.4230/LIPIcs.STACS.2017.21}, annote = {Keywords: communication complexity, elimination, discrepancy, regularity, greater-than} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation.
Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.

Harry Buhrman, Michal Koucký, Bruno Loff, and Florian Speelman. Catalytic Space: Non-determinism and Hierarchy. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{buhrman_et_al:LIPIcs.STACS.2016.24, author = {Buhrman, Harry and Kouck\'{y}, Michal and Loff, Bruno and Speelman, Florian}, title = {{Catalytic Space: Non-determinism and Hierarchy}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.24}, URN = {urn:nbn:de:0030-drops-57258}, doi = {10.4230/LIPIcs.STACS.2016.24}, annote = {Keywords: catalytic computation, Immerman–Szelepcs\'{e}nyi theorem, space hierarchy} }

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**Published in:** Dagstuhl Reports, Volume 4, Issue 3 (2014)

This report documents the program and the outcomes of Dagstuhl Seminar 14121 "Computational Complexity of Discrete Problems". The first section gives an overview of the topics covered and the organization of the meeting. Section 2 lists the talks given in chronological order. The last section contains the abstracts of the talks.

Anna Gal, Michal Koucky, Oded Regev, and Rüdiger Reischuk. Computational Complexity of Discrete Problems (Dagstuhl Seminar 14121). In Dagstuhl Reports, Volume 4, Issue 3, pp. 62-84, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Article{gal_et_al:DagRep.4.3.62, author = {Gal, Anna and Koucky, Michal and Regev, Oded and Reischuk, R\"{u}diger}, title = {{Computational Complexity of Discrete Problems (Dagstuhl Seminar 14121)}}, pages = {62--84}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2014}, volume = {4}, number = {3}, editor = {Gal, Anna and Koucky, Michal and Regev, Oded and Reischuk, R\"{u}diger}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.3.62}, URN = {urn:nbn:de:0030-drops-45921}, doi = {10.4230/DagRep.4.3.62}, annote = {Keywords: discrete problems, computational complexity, Turing machines, Boolean circuits, arithmetic circuits, quantum computing, communication complexity, pseudorandomness, derandomization, approximation, data streams} }

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**Published in:** Dagstuhl Reports, Volume 1, Issue 3 (2011)

This report documents the program and the outcomes of Dagstuhl Seminar
11121 ``Computational Complexity of Discrete Problems''.
The first section gives an overview of the topics covered and the organization
of the meeting. Section~2 lists the talks given in chronological order.
The last section contains the abstracts of the talks.

Martin Grohe, Michal Koucky, Rüdiger Reischik, and Dieter van Melkebeek. Computational Complexity of Discrete Problems (Dagstuhl Seminar 11121). In Dagstuhl Reports, Volume 1, Issue 3, pp. 42-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@Article{grohe_et_al:DagRep.1.3.42, author = {Grohe, Martin and Koucky, Michal and Reischik, R\"{u}diger and van Melkebeek, Dieter}, title = {{Computational Complexity of Discrete Problems (Dagstuhl Seminar 11121)}}, pages = {42--66}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2011}, volume = {1}, number = {3}, editor = {Grohe, Martin and Koucky, Michal and Reischik, R\"{u}diger and van Melkebeek, Dieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.1.3.42}, URN = {urn:nbn:de:0030-drops-31935}, doi = {10.4230/DagRep.1.3.42}, annote = {Keywords: Discrete problems, computational complexity, Turing machines, Boolean circuits, quantum computing, communication and query complexity, extractors, pseudorandomness, derandomization, approximation, coding cryptography, algorithmic learning} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 7411, Algebraic Methods in Computational Complexity (2008)

We study the two party problem of randomly selecting a string among
all the strings of length n. We want the protocol to have the
property that the output distribution has high entropy, even
when one of the two parties is dishonest and deviates from the
protocol. We develop protocols that achieve high, close to n,
entropy.
In the literature the randomness guarantee is usually expressed as
being close to the uniform distribution or in terms of resiliency.
The notion of entropy is not directly comparable to that of
resiliency, but we establish a connection between the two that
allows us to compare our protocols with the existing ones.
We construct an
explicit protocol that yields entropy n - O(1) and has 4log^* n
rounds, improving over the protocol of Goldwasser
et al. that also achieves this entropy but needs O(n)
rounds. Both these protocols need O(n^2) bits of communication.
Next we reduce the communication in our protocols. We show the existence,
non-explicitly, of a protocol that has 6-rounds, 2n + 8log n bits
of communication and yields entropy n- O(log n) and min-entropy
n/2 - O(log n). Our protocol achieves the same entropy bound as
the recent, also non-explicit, protocol of Gradwohl
et al., however achieves much higher min-entropy: n/2 -
O(log n) versus O(log n).
Finally we exhibit very simple explicit protocols. We connect the
security parameter of these geometric protocols with the well
studied Kakeya problem motivated by harmonic analysis and analytical
number theory. We are only able to prove that these protocols have
entropy 3n/4 but still n/2 - O(log n) min-entropy. Therefore
they do not perform as well with respect to the explicit
constructions of Gradwohl et al. entropy-wise, but still
have much better min-entropy. We conjecture that these simple
protocols achieve n -o(n) entropy. Our geometric
construction and its relation to the Kakeya problem follows a new and
different approach to the random selection problem than any of the
previously known protocols.

Nikolai K. Vereshchagin, Harry Buhrman, Matthias Cristandl, Michal Koucky, Zvi Lotker, and Boaz Patt-Shamir. High Entropy Random Selection Protocols. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 7411, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{vereshchagin_et_al:DagSemProc.07411.5, author = {Vereshchagin, Nikolai K. and Buhrman, Harry and Cristandl, Matthias and Koucky, Michal and Lotker, Zvi and Patt-Shamir, Boaz}, title = {{High Entropy Random Selection Protocols}}, booktitle = {Algebraic Methods in Computational Complexity}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {7411}, editor = {Manindra Agrawal and Harry Buhrman and Lance Fortnow and Thomas Thierauf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07411.5}, URN = {urn:nbn:de:0030-drops-13091}, doi = {10.4230/DagSemProc.07411.5}, annote = {Keywords: Shannon entropy, Random string ds} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

We propose a new model of restricted branching programs
which we call {em incremental branching programs}.
We show that {em syntactic}
incremental branching programs capture previously studied
structured models of computation for the problem GEN, namely marking
machines [Cook74].
and Poon's extension [Poon93] of jumping automata
on graphs [CookRackoff80].
We then prove
exponential size lower bounds for our syntactic incremental model,
and for some other restricted branching program models as well.
We further show that nondeterministic syntactic incremental
branching programs are
provably stronger than their deterministic counterpart when solving a
natural NL-complete GEN subproblem.
It remains open if syntactic incremental branching programs are as powerful
as unrestricted branching programs for GEN problems.
Joint work with Anna GÃƒÂ¡l and Michal KouckÃƒÂ½

Anna Gál, Pierre McKenzie, and Michal Koucký. Incremental branching programs. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{gal_et_al:DagSemProc.06111.10, author = {G\'{a}l, Anna and McKenzie, Pierre and Kouck\'{y}, Michal}, title = {{Incremental branching programs}}, booktitle = {Complexity of Boolean Functions}, pages = {1--20}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.10}, URN = {urn:nbn:de:0030-drops-6230}, doi = {10.4230/DagSemProc.06111.10}, annote = {Keywords: Complexity theory, branching programs, logarithmic space, marking machines} }

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