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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among n Boolean vectors in d dimensions. The OV Conjecture (OVC) posits that OV requires n^{2-o(1)} time to solve, for all d=omega(log n). Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in P, such as Edit Distance, Frechet Distance, Longest Common Subsequence, and approximating the diameter of a graph.
We prove that OVC is true in several computational models of interest:
- For all sufficiently large n and d, OV for n vectors in {0,1}^d has branching program complexity Theta~(n * min(n,2^d)). In particular, the lower and upper bounds match up to polylog factors.
- OV has Boolean formula complexity Theta~(n * min(n,2^d)), over all complete bases of O(1) fan-in.
- OV requires Theta~(n * min(n,2^d)) wires, in formulas comprised of gates computing arbitrary symmetric functions of unbounded fan-in.
Our lower bounds basically match the best known (quadratic) lower bounds for any explicit function in those models. Analogous lower bounds hold for many related problems shown to be hard under OVC, such as Batch Partial Match, Batch Subset Queries, and Batch Hamming Nearest Neighbors, all of which have very succinct reductions to OV.
The proofs use a certain kind of input restriction that is different from typical random restrictions where variables are assigned independently. We give a sense in which independent random restrictions cannot be used to show hardness, in that OVC is false in the "average case" even for AC^0 formulas:
For all p in (0,1) there is a delta_p > 0 such that for every n and d, OV instances with input bits independently set to 1 with probability p (and 0 otherwise) can be solved with AC^0 formulas of O(n^{2-delta_p}) size, on all but a o_n(1) fraction of instances. Moreover, lim_{p - > 1}delta_p = 1.

Daniel M. Kane and Richard Ryan Williams. The Orthogonal Vectors Conjecture for Branching Programs and Formulas. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kane_et_al:LIPIcs.ITCS.2019.48, author = {Kane, Daniel M. and Williams, Richard Ryan}, title = {{The Orthogonal Vectors Conjecture for Branching Programs and Formulas}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {48:1--48:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.48}, URN = {urn:nbn:de:0030-drops-101418}, doi = {10.4230/LIPIcs.ITCS.2019.48}, annote = {Keywords: fine-grained complexity, orthogonal vectors, branching programs, symmetric functions, Boolean formulas} }

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

Let H be an arbitrary family of hyper-planes in d-dimensions. We show that the point-location problem for H can be solved by a linear decision tree that only uses a special type of queries called generalized comparison queries. These queries correspond to hyperplanes that can be written as a linear combination of two hyperplanes from H; in particular, if all hyperplanes in H are k-sparse then generalized comparisons are 2k-sparse. The depth of the obtained linear decision tree is polynomial in d and logarithmic in |H|, which is comparable to previous results in the literature that use general linear queries.
This extends the study of comparison trees from a previous work by the authors [Kane {et al.}, FOCS 2017]. The main benefit is that using generalized comparison queries allows to overcome limitations that apply for the more restricted type of comparison queries.
Our analysis combines a seminal result of Forster regarding sets in isotropic position [Forster, JCSS 2002], the margin-based inference dimension analysis for comparison queries from [Kane {et al.}, FOCS 2017], and compactness arguments.

Daniel M. Kane, Shachar Lovett, and Shay Moran. Generalized Comparison Trees for Point-Location Problems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 82:1-82:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kane_et_al:LIPIcs.ICALP.2018.82, author = {Kane, Daniel M. and Lovett, Shachar and Moran, Shay}, title = {{Generalized Comparison Trees for Point-Location Problems}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {82:1--82:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.82}, URN = {urn:nbn:de:0030-drops-90862}, doi = {10.4230/LIPIcs.ICALP.2018.82}, annote = {Keywords: linear decision trees, comparison queries, point location problems} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We investigate the problem of testing the equivalence between two discrete histograms. A k-histogram over [n] is a probability distribution that is piecewise constant over some set of k intervals over [n]. Histograms have been extensively studied in computer science and statistics. Given a set of samples from two k-histogram distributions p, q over [n], we want to distinguish (with high probability) between the cases that p = q and ||p ? q||_1 >= epsilon. The main contribution of this paper is a new algorithm for this testing problem and a nearly matching information-theoretic lower bound. Specifically, the sample complexity of our algorithm matches our lower bound up to a logarithmic factor, improving on previous work by polynomial factors in the relevant parameters. Our algorithmic approach applies in a more general setting and yields improved sample upper bounds for testing closeness of other structured distributions as well.

Ilias Diakonikolas, Daniel M. Kane, and Vladimir Nikishkin. Near-Optimal Closeness Testing of Discrete Histogram Distributions. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{diakonikolas_et_al:LIPIcs.ICALP.2017.8, author = {Diakonikolas, Ilias and Kane, Daniel M. and Nikishkin, Vladimir}, title = {{Near-Optimal Closeness Testing of Discrete Histogram Distributions}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.8}, URN = {urn:nbn:de:0030-drops-74937}, doi = {10.4230/LIPIcs.ICALP.2017.8}, annote = {Keywords: distribution testing, histograms, closeness testing} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We construct and analyze a new pseudorandom generator for degree 2 polynomial threshold functions with respect to the Gaussian measure. In particular, we obtain one whose seed length is polylogarithmic in both the dimension and the desired error, a substantial improvement over existing constructions.
Our generator is obtained as an appropriate weighted average of pseudorandom generators against read once branching programs. The analysis requires a number of ideas including a hybrid argument and a structural result that allows us to treat our degree 2 threshold function as a function of a number of linear polynomials and one approximately linear polynomial.

Daniel M. Kane. A Polylogarithmic PRG for Degree 2 Threshold Functions in the Gaussian Setting. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 567-581, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kane:LIPIcs.CCC.2015.567, author = {Kane, Daniel M.}, title = {{A Polylogarithmic PRG for Degree 2 Threshold Functions in the Gaussian Setting}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {567--581}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.567}, URN = {urn:nbn:de:0030-drops-50534}, doi = {10.4230/LIPIcs.CCC.2015.567}, annote = {Keywords: polynomial threshold function, pseudorandom generator, Gaussian distribution} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

"Twenty questions" is a guessing game played by two players: Bob thinks of an integer between 1 and n, and Alice’s goal is to recover it using a minimal number of Yes/No questions. Shannon’s entropy has a natural interpretation in this context. It characterizes the average number of questions used by an optimal strategy in the distributional variant of the game: let μ be a distribution over [n], then the average number of questions used by an optimal strategy that recovers x∼ μ is between H(μ) and H(μ)+1.
We consider an extension of this game where at most k questions can be answered falsely. We extend the classical result by showing that an optimal strategy uses roughly H(μ) + k H_2(μ) questions, where H_2(μ) = ∑_x μ(x)log log 1/μ(x). This also generalizes a result by Rivest et al. (1980) for the uniform distribution.
Moreover, we design near optimal strategies that only use comparison queries of the form "x ≤ c?" for c ∈ [n]. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.

Yuval Dagan, Yuval Filmus, Daniel Kane, and Shay Moran. The Entropy of Lies: Playing Twenty Questions with a Liar. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dagan_et_al:LIPIcs.ITCS.2021.1, author = {Dagan, Yuval and Filmus, Yuval and Kane, Daniel and Moran, Shay}, title = {{The Entropy of Lies: Playing Twenty Questions with a Liar}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {1:1--1:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.1}, URN = {urn:nbn:de:0030-drops-135400}, doi = {10.4230/LIPIcs.ITCS.2021.1}, annote = {Keywords: entropy, twenty questions, algorithms, sorting} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We construct and analyze a pseudorandom generator for degree 2 boolean polynomial threshold functions. Random constructions achieve the optimal seed length of O(log n + log 1/epsilon), however the best known explicit construction of [Ilias Diakonikolas, 2010] uses a seed length of O(log n * epsilon^{-8}). In this work we give an explicit construction that uses a seed length of O(log n + (1/epsilon)^{o(1)}). Note that this improves the seed length substantially and that the dependence on the error epsilon is additive and only grows subpolynomially as opposed to the previously known multiplicative polynomial dependence.
Our generator uses dimensionality reduction on a Nisan-Wigderson based pseudorandom generator given by Lu, Kabanets [Kabanets and Lu, 2018].

Daniel Kane and Sankeerth Rao. A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 2:1-2:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kane_et_al:LIPIcs.CCC.2018.2, author = {Kane, Daniel and Rao, Sankeerth}, title = {{A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {2:1--2:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.2}, URN = {urn:nbn:de:0030-drops-88861}, doi = {10.4230/LIPIcs.CCC.2018.2}, annote = {Keywords: Pseudorandomness, Polynomial Threshold Functions} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9111, Computational Geometry (2009)

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of some convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution is the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron given the metric, and prove a pseudopolynomial bound on its running time.

Daniel Kane, Gregory Nathan Price, and Erik Demaine. A Pseudopolynomial Algorithm for Alexandrov's Theorem. In Computational Geometry. Dagstuhl Seminar Proceedings, Volume 9111, pp. 1-22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{kane_et_al:DagSemProc.09111.2, author = {Kane, Daniel and Price, Gregory Nathan and Demaine, Erik}, title = {{A Pseudopolynomial Algorithm for Alexandrov's Theorem}}, booktitle = {Computational Geometry}, pages = {1--22}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {9111}, editor = {Pankaj Kumar Agarwal and Helmut Alt and Monique Teillaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09111.2}, URN = {urn:nbn:de:0030-drops-20328}, doi = {10.4230/DagSemProc.09111.2}, annote = {Keywords: Folding, metrics, pseudopolynomial, algorithms} }

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