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RANDOM

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

A tournament is a complete directed graph. A source in a tournament is a vertex that has no in-neighbours (every other vertex is reachable from it via a path of length 1), and a king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king. In particular, a maximum out-degree vertex is a king. The tasks of finding a king and a maximum out-degree vertex in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of finding a king, of finding a maximum out-degree vertex, and of finding a source (if it exists) in a tournament, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments:
- We show that the communication task of finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. As a result, known bounds on the communication complexity of CIS [Yannakakis, JCSS'91, Göös, Pitassi, Watson, SICOMP'18] imply a bound of Θ̃(log² n) for finding a source (if it exists, or outputting that there is no source) in a tournament.
- The deterministic and randomized communication complexities of finding a king are Θ(n). The quantum communication complexity of finding a king is Θ̃(√n).
- The deterministic, randomized, and quantum communication complexities of finding a maximum out-degree vertex are Θ(n log n), Θ̃(n) and Θ̃(√n), respectively. Our upper bounds above hold for all partitions of edges, and the lower bounds for a specific partition of the edges.
One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness. An interesting point to note here is that while the deterministic query complexity of finding a king has been open for over two decades [Shen, Sheng, Wu, SICOMP'03], we are able to essentially resolve the complexity of this problem in a model (communication complexity) that is usually harder to analyze than query complexity.

Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Communication Complexity of Finding a King in a Tournament. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 64:1-64:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mande_et_al:LIPIcs.APPROX/RANDOM.2024.64, author = {Mande, Nikhil S. and Paraashar, Manaswi and Sanyal, Swagato and Saurabh, Nitin}, title = {{On the Communication Complexity of Finding a King in a Tournament}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {64:1--64:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.64}, URN = {urn:nbn:de:0030-drops-210571}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.64}, annote = {Keywords: Communication complexity, tournaments, query complexity} }

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**Published in:** LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Given an Abelian group 𝒢, a Boolean-valued function f: 𝒢 → {-1,+1}, is said to be s-sparse, if it has at most s-many non-zero Fourier coefficients over the domain 𝒢. In a seminal paper, Gopalan et al. [Gopalan et al., 2011] proved "Granularity" for Fourier coefficients of Boolean valued functions over ℤ₂ⁿ, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over ℤ₂ⁿ which are approximately Fourier-sparse. In this work, we obtain structural results for approximately Fourier-sparse Boolean valued functions over Abelian groups 𝒢 of the form, 𝒢: = ℤ_{p_1}^{n_1} × ⋯ × ℤ_{p_t}^{n_t}, for distinct primes p_i. We also obtain a lower bound of the form 1/(m²s)^⌈φ(m)/2⌉, on the absolute value of the smallest non-zero Fourier coefficient of an s-sparse function, where m = p_1 ⋯ p_t, and φ(m) = (p_1-1) ⋯ (p_t-1). We carefully apply probabilistic techniques from [Gopalan et al., 2011], to obtain our structural results, and use some non-trivial results from algebraic number theory to get the lower bound.
We construct a family of at most s-sparse Boolean functions over ℤ_pⁿ, where p > 2, for arbitrarily large enough s, where the minimum non-zero Fourier coefficient is o(1/s). The "Granularity" result of Gopalan et al. implies that the absolute values of non-zero Fourier coefficients of any s-sparse Boolean valued function over ℤ₂ⁿ are Ω(1/s). So, our result shows that one cannot expect such a lower bound for general Abelian groups.
Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient sparsity testing algorithm for Boolean function, which tests whether the given function is s-sparse, or ε-far from any sparse Boolean function, and it requires poly((ms)^φ(m),1/ε)-many queries. Further, we generalize the notion of degree of a Boolean function over an Abelian group 𝒢. We use it to prove an Ω(√s) lower bound on the query complexity of any adaptive sparsity testing algorithm.

Sourav Chakraborty, Swarnalipa Datta, Pranjal Dutta, Arijit Ghosh, and Swagato Sanyal. On Fourier Analysis of Sparse Boolean Functions over Certain Abelian Groups. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2024.40, author = {Chakraborty, Sourav and Datta, Swarnalipa and Dutta, Pranjal and Ghosh, Arijit and Sanyal, Swagato}, title = {{On Fourier Analysis of Sparse Boolean Functions over Certain Abelian Groups}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {40:1--40:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.40}, URN = {urn:nbn:de:0030-drops-205963}, doi = {10.4230/LIPIcs.MFCS.2024.40}, annote = {Keywords: Fourier coefficients, sparse, Abelian, granularity} }

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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Let 𝖱_ε denote randomized query complexity for error probability ε, and R: = 𝖱_{1/3}. In this work we investigate whether a perfect composition theorem 𝖱(f∘gⁿ) = Ω(𝖱(f)⋅ 𝖱(g)) holds for a relation f ⊆ {0,1}ⁿ × 𝒮 and a total inner function g:{0,1}^m → {0,1}.
Composition theorems of the form 𝖱(f∘gⁿ) = Ω(𝖱(f)⋅ 𝖬(g)) are known for various measures 𝖬. Such measures include the sabotage complexity RS defined by Ben-David and Kothari (ICALP 2015), the max-conflict complexity defined by Gavinsky, Lee, Santha and Sanyal (ICALP 2019), and the linearized complexity measure defined by Ben-David, Blais, Göös and Maystre (FOCS 2022). The above measures are asymptotically non-decreasing in the above order. However, for total Boolean functions no asymptotic separation is known between any two of them.
Let 𝖣^{prod} denote the maximum distributional query complexity with respect to any product (over variables) distribution . In this work we show that for any total Boolean function g, the sabotage complexity RS(g) = Ω̃(𝖣^{prod}(g)). This gives the composition theorem 𝖱(f∘gⁿ) = Ω̃(𝖱(f)⋅ 𝖣^{prod}(g)). In light of the minimax theorem which states that 𝖱(g) is the maximum distributional complexity of g over any distribution, our result makes progress towards answering the composition question.
We prove our result by means of a complexity measure 𝖱_ε^{prod} that we define for total Boolean functions. Informally, 𝖱_ε^{prod}(g) is the minimum complexity of any randomized decision tree with unlabelled leaves with the property that, for every product distribution μ over the inputs, the average bias of its leaves is at least ((1-ε)-ε)/2 = 1/2-ε. It follows by standard arguments that 𝖱_{1/3}^{prod}(g) = Ω(𝖣^{prod}(g)). We show that 𝖱_{1/3}^{prod} is equivalent to the sabotage complexity up to a logarithmic factor.
Ben-David and Kothari asked whether RS(g) = Θ(𝖱(g)) for total functions g. We generalize their question and ask if for any error ε, 𝖱_ε^{prod}(g) = Θ̃(𝖱_ε(g)). We observe that the work by Ben-David, Blais, Göös and Maystre (FOCS 2022) implies that for a perfect composition theorem 𝖱_{1/3}(f∘gⁿ) = Ω(𝖱_{1/3}(f)⋅𝖱_{1/3}(g)) to hold for any relation f and total function g, a necessary condition is that 𝖱_{1/3}(g) = O(1/(ε)⋅ 𝖱_{1/2-ε}(g)) holds for any total function g. We show that 𝖱_ε^{prod}(g) admits a similar error-reduction 𝖱_{1/3}^{prod}(g) = Õ(1/(ε)⋅𝖱_{1/2-ε}^{prod}(g)). Note that from the definition of 𝖱_ε^{prod} it is not immediately clear that 𝖱_ε^{prod} admits any error-reduction at all.
We ask if our bound RS(g) = Ω̃(𝖣^{prod}(g)) is tight. We answer this question in the negative, by showing that for the NAND tree function, sabotage complexity is polynomially larger than 𝖣^{prod}. Our proof yields an alternative and different derivation of the tight lower bound on the bounded error randomized query complexity of the NAND tree function (originally proved by Santha in 1985), which may be of independent interest. Our result shows that sometimes, 𝖱_{1/3}^{prod} and sabotage complexity may be useful in producing an asymptotically larger lower bound on 𝖱(f∘gⁿ) than Ω̃(𝖱(f)⋅ 𝖣^{prod}(g)). In addition, this gives an explicit polynomial separation between 𝖱 and 𝖣^{prod} which, to our knowledge, was not known prior to our work.

Swagato Sanyal. Randomized Query Composition and Product Distributions. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 56:1-56:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{sanyal:LIPIcs.STACS.2024.56, author = {Sanyal, Swagato}, title = {{Randomized Query Composition and Product Distributions}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {56:1--56:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.56}, URN = {urn:nbn:de:0030-drops-197668}, doi = {10.4230/LIPIcs.STACS.2024.56}, annote = {Keywords: Randomized query complexity, Decision Tree, Boolean function complexity, Analysis of Boolean functions} }

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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. While decision tree complexity is long known to be polynomially related with many other measures, the optimal exponents of many of these relations are not known. It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic.
Proving quadratic relations between these measures would resolve several open questions in decision tree complexity. For example, it will imply a tight relation between decision tree complexity and square of randomized decision tree complexity and a tight relation between zero-error randomized decision tree complexity and square of fractional block sensitivity, resolving an open question raised by Aaronson [Aaronson, 2008]. In this work, we investigate the tightness of the existing cubic upper bounds.
We improve the cubic upper bounds for many interesting classes of Boolean functions. We show that for graph properties and for functions with a constant number of alternations, the cubic upper bounds can be improved to quadratic. We define a class of Boolean functions, which we call the zebra functions, that comprises Boolean functions where each monotone path from 0ⁿ to 1ⁿ has an equal number of alternations. This class contains the symmetric and monotone functions as its subclasses. We show that for any zebra function, decision tree complexity is at most the square of block sensitivity, and certificate complexity is at most the square of degree.
Finally, we show using a lifting theorem of communication complexity by Göös, Pitassi and Watson [Göös et al., 2017] that the task of proving an improved upper bound on the decision tree complexity for all functions is in a sense equivalent to the potentially easier task of proving a similar upper bound on communication complexity for each bi-partition of the input variables, for all functions. In particular, this implies that to bound the decision tree complexity it suffices to bound smaller measures like parity decision tree complexity, subcube decision tree complexity and decision tree rank, that are defined in terms of models that can be efficiently simulated by communication protocols.

Rahul Chugh, Supartha Podder, and Swagato Sanyal. Decision Tree Complexity Versus Block Sensitivity and Degree. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 27:1-27:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chugh_et_al:LIPIcs.FSTTCS.2023.27, author = {Chugh, Rahul and Podder, Supartha and Sanyal, Swagato}, title = {{Decision Tree Complexity Versus Block Sensitivity and Degree}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {27:1--27:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.27}, URN = {urn:nbn:de:0030-drops-194001}, doi = {10.4230/LIPIcs.FSTTCS.2023.27}, annote = {Keywords: Query complexity, Graph Property, Boolean functions} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

For any Boolean functions f and g, the question whether R(f∘g) = Θ̃(R(f) ⋅ R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg̃(f∘g) = Θ̃(deg̃(f)⋅deg̃(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily.
It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg̃ compose.
A recent landmark result (Ben-David and Blais, 2020) showed that R(f∘g) = Ω(noisyR(f)⋅ R(g)). This implies that composition holds whenever noisyR(f) = Θ̃(R(f)). We show two results:
1. When R(f) = Θ(n), then noisyR(f) = Θ(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full.
2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg̃(f∘g) = Ω(M(f) ⋅ deg̃(g)) (for some non-trivial complexity measure M(⋅)) was known to the best of our knowledge. We prove that deg̃(f∘g) = Ω̃(√{bs(f)} ⋅ deg̃(g)), where bs(f) is the block sensitivity of f. This implies that deg̃ composes when deg̃(f) is asymptotically equal to √{bs(f)}.
It is already known that both R and deg̃ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.

Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Composition of Randomized Query Complexity and Approximate Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 63:1-63:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chakraborty_et_al:LIPIcs.APPROX/RANDOM.2023.63, author = {Chakraborty, Sourav and Kayal, Chandrima and Mittal, Rajat and Paraashar, Manaswi and Sanyal, Swagato and Saurabh, Nitin}, title = {{On the Composition of Randomized Query Complexity and Approximate Degree}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {63:1--63:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.63}, URN = {urn:nbn:de:0030-drops-188883}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.63}, annote = {Keywords: Approximate degree, Boolean functions, Composition Theorem, Partial functions, Randomized Query Complexity} }

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

We show that the deterministic decision tree complexity of a (partial) function or relation f lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation f∘g as long as the gadget g satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of f, which could be exponentially smaller than its deterministic counterpart when either f is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to f. Reducing the gadget size to a constant is an important open problem at the frontier of current research.
Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., Res(⊕), of the unsatisfiability of closely related constant-width CNF formulas.

Arkadev Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. Lifting to Parity Decision Trees via Stifling. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2023.33, author = {Chattopadhyay, Arkadev and Mande, Nikhil S. and Sanyal, Swagato and Sherif, Suhail}, title = {{Lifting to Parity Decision Trees via Stifling}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {33:1--33:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.33}, URN = {urn:nbn:de:0030-drops-175362}, doi = {10.4230/LIPIcs.ITCS.2023.33}, annote = {Keywords: Decision trees, parity decision trees, lifting theorems} }

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

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. More generally, we show the following when the gadget is Inner Product on 2b input bits for all b ≥ 2, denoted IP.
- If f is a total Boolean function that depends on all of its n input bits, then the bounded-error one-way quantum communication complexity of f∘IP equals Ω(n(b-1)).
- If f is a partial Boolean function, then the deterministic one-way communication complexity of f∘IP is at least Ω(b ⋅ 𝖣_{dt}^ → (f)), where 𝖣_{dt}^ → (f) denotes non-adaptive decision tree complexity of f. To prove our quantum lower bound, we first show a lower bound on the VC-dimension of f∘IP. We then appeal to a result of Klauck [STOC'00], which immediately yields our quantum lower bound. Our deterministic lower bound relies on a combinatorial result independently proven by Ahlswede and Khachatrian [Adv. Appl. Math.'98], and Frankl and Tokushige [Comb.'99].
It is known due to a result of Montanaro and Osborne [arXiv'09] that the deterministic one-way communication complexity of f∘XOR equals the non-adaptive parity decision tree complexity of f. In contrast, we show the following when the inner gadget is the AND function on 2 input bits.
- There exists a function for which even the quantum non-adaptive AND decision tree complexity of f is exponentially large in the deterministic one-way communication complexity of f∘AND.
- However, for symmetric functions f, the non-adaptive AND decision tree complexity of f is at most quadratic in the (even two-way) communication complexity of f∘AND. In view of the first bullet, a lower bound on non-adaptive AND decision tree complexity of f does not lift to a lower bound on one-way communication complexity of f∘AND. The proof of the first bullet above uses the well-studied Odd-Max-Bit function. For the second bullet, we first observe a connection between the one-way communication complexity of f and the Möbius sparsity of f, and then give a lower bound on the Möbius sparsity of symmetric functions. An upper bound on the non-adaptive AND decision tree complexity of symmetric functions follows implicitly from prior work on combinatorial group testing; for the sake of completeness, we include a proof of this result.
It is well known that the rank of the communication matrix of a function F is an upper bound on its deterministic one-way communication complexity. This bound is known to be tight for some F. However, in our final result we show that this is not the case when F = f∘AND. More precisely we show that for all f, the deterministic one-way communication complexity of F = f∘AND is at most (rank(M_{F}))(1 - Ω(1)), where M_{F} denotes the communication matrix of F.

Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. One-Way Communication Complexity and Non-Adaptive Decision Trees. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mande_et_al:LIPIcs.STACS.2022.49, author = {Mande, Nikhil S. and Sanyal, Swagato and Sherif, Suhail}, title = {{One-Way Communication Complexity and Non-Adaptive Decision Trees}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {49:1--49:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.49}, URN = {urn:nbn:de:0030-drops-158598}, doi = {10.4230/LIPIcs.STACS.2022.49}, annote = {Keywords: Decision trees, communication complexity, composed Boolean functions} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Chang’s lemma (Duke Mathematical Journal, 2002) is a classical result in mathematics, with applications spanning across additive combinatorics, combinatorial number theory, analysis of Boolean functions, communication complexity and algorithm design. For a Boolean function f that takes values in {-1, 1} let r(f) denote its Fourier rank (i.e., the dimension of the span of its Fourier support). For each positive threshold t, Chang’s lemma provides a lower bound on δ(f) := Pr[f(x) = -1] in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least 1/t. In this work we examine the tightness of Chang’s lemma with respect to the following three natural settings of the threshold:
- the Fourier sparsity of f, denoted k(f),
- the Fourier max-supp-entropy of f, denoted k'(f), defined to be the maximum value of the reciprocal of the absolute value of a non-zero Fourier coefficient,
- the Fourier max-rank-entropy of f, denoted k''(f), defined to be the minimum t such that characters whose coefficients are at least 1/t in magnitude span a r(f)-dimensional space. In this work we prove new lower bounds on δ(f) in terms of the above measures. One of our lower bounds, δ(f) = Ω(r(f)²/(k(f) log² k(f))), subsumes and refines the previously best known upper bound r(f) = O(√{k(f)}log k(f)) on r(f) in terms of k(f) by Sanyal (Theory of Computing, 2019). We improve upon this bound and show r(f) = O(√{k(f)δ(f)}log k(f)). Another lower bound, δ(f) = Ω(r(f)/(k''(f) log k(f))), is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of absolute values of level-1 Fourier coefficients in terms of 𝔽₂-degree. We further show that Chang’s lemma for the above-mentioned choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved.
Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions witnessing their tightness. All the functions we construct are modifications of the Addressing function, where we replace certain input variables by suitable functions. Our final contribution is to construct Boolean functions f for which our lower bounds asymptotically match δ(f), and for any choice of the threshold t, the lower bound obtained from Chang’s lemma is asymptotically smaller than δ(f).
Our results imply more refined deterministic one-way communication complexity upper bounds for XOR functions. Given the wide-ranging application of Chang’s lemma to areas like additive combinatorics, learning theory and communication complexity, we strongly feel that our refinements of Chang’s lemma will find many more applications.

Sourav Chakraborty, Nikhil S. Mande, Rajat Mittal, Tulasimohan Molli, Manaswi Paraashar, and Swagato Sanyal. Tight Chang’s-Lemma-Type Bounds for Boolean Functions. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2021.10, author = {Chakraborty, Sourav and Mande, Nikhil S. and Mittal, Rajat and Molli, Tulasimohan and Paraashar, Manaswi and Sanyal, Swagato}, title = {{Tight Chang’s-Lemma-Type Bounds for Boolean Functions}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {10:1--10:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.10}, URN = {urn:nbn:de:0030-drops-155215}, doi = {10.4230/LIPIcs.FSTTCS.2021.10}, annote = {Keywords: Analysis of Boolean functions, Chang’s lemma, Parity decision trees, Fourier dimension} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Our contributions are as follows. Let f : 𝔽₂ⁿ → {-1, 1} be a Boolean function with Fourier support 𝒮 and Fourier sparsity k.
- We prove via the probabilistic method that there exists a parity decision tree of depth O(√k) that computes f. This matches the best known upper bound on the parity decision tree complexity of Boolean functions (Tsang, Wong, Xie, and Zhang, FOCS 2013). Moreover, while previous constructions (Tsang et al., FOCS 2013, Shpilka, Tal, and Volk, Comput. Complex. 2017) build the trees by carefully choosing the parities to be queried in each step, our proof shows that a naive sampling of the parities suffices.
- We generalize the above result by showing that if the Fourier spectra of Boolean functions satisfy a natural "folding property", then the above proof can be adapted to establish existence of a tree of complexity polynomially smaller than O(√ k). More concretely, the folding property we consider is that for most distinct γ, δ in 𝒮, there are at least a polynomial (in k) number of pairs (α, β) of parities in 𝒮 such that α+β = γ+δ. We make a conjecture in this regard which, if true, implies that the communication complexity of an XOR function is bounded above by the fourth root of the rank of its communication matrix, improving upon the previously known upper bound of square root of rank (Tsang et al., FOCS 2013, Lovett, J. ACM. 2016).
- Motivated by the above, we present some structural results about the Fourier spectra of Boolean functions. It can be shown by elementary techniques that for any Boolean function f and all (α, β) in binom(𝒮,2), there exists another pair (γ, δ) in binom(𝒮,2) such that α + β = γ + δ. One can view this as a "trivial" folding property that all Boolean functions satisfy. Prior to our work, it was conceivable that for all (α, β) ∈ binom(𝒮,2), there exists exactly one other pair (γ, δ) ∈ binom(𝒮,2) with α + β = γ + δ. We show, among other results, that there must exist several γ ∈ 𝔽₂ⁿ such that there are at least three pairs of parities (α₁, α₂) ∈ binom(𝒮,2) with α₁+α₂ = γ. This, in particular, rules out the possibility stated earlier.

Nikhil S. Mande and Swagato Sanyal. On Parity Decision Trees for Fourier-Sparse Boolean Functions. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mande_et_al:LIPIcs.FSTTCS.2020.29, author = {Mande, Nikhil S. and Sanyal, Swagato}, title = {{On Parity Decision Trees for Fourier-Sparse Boolean Functions}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {29:1--29:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.29}, URN = {urn:nbn:de:0030-drops-132703}, doi = {10.4230/LIPIcs.FSTTCS.2020.29}, annote = {Keywords: Parity decision trees, log-rank conjecture, analysis of Boolean functions, communication complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)}) , where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g.
The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n).
The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g.

Dmitry Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gavinsky_et_al:LIPIcs.ICALP.2019.64, author = {Gavinsky, Dmitry and Lee, Troy and Santha, Miklos and Sanyal, Swagato}, title = {{A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {64:1--64:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.64}, URN = {urn:nbn:de:0030-drops-106407}, doi = {10.4230/LIPIcs.ICALP.2019.64}, annote = {Keywords: query complexity, lower bounds} }

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

We initiate a systematic study of linear sketching over F_2. For a given Boolean function treated as f : F_2^n -> F_2 a randomized F_2-sketch is a distribution M over d x n matrices with elements over F_2 such that Mx suffices for computing f(x) with high probability. Such sketches for d << n can be used to design small-space distributed and streaming algorithms.
Motivated by these applications we study a connection between F_2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F_2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F_2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F_2-sketching is optimal (up to constant factors) for uniformly distributed inputs.
Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F_2 can be constructed as F_2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length O(n) constructed through uniformly random updates.

Sampath Kannan, Elchanan Mossel, Swagato Sanyal, and Grigory Yaroslavtsev. Linear Sketching over F_2. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 8:1-8:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kannan_et_al:LIPIcs.CCC.2018.8, author = {Kannan, Sampath and Mossel, Elchanan and Sanyal, Swagato and Yaroslavtsev, Grigory}, title = {{Linear Sketching over F\underline2}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {8:1--8:37}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.8}, URN = {urn:nbn:de:0030-drops-88819}, doi = {10.4230/LIPIcs.CCC.2018.8}, annote = {Keywords: Linear sketch, Streaming algorithms, XOR-functions, Communication complexity} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

Let the randomized query complexity of a relation for error probability epsilon be denoted by R_epsilon(). We prove that for any relation f contained in {0,1}^n times R and Boolean function g:{0,1}^m -> {0,1}, R_{1/3}(f o g^n) = Omega(R_{4/9}(f).R_{1/2-1/n^4}(g)), where f o g^n is the relation obtained by composing f and g. We also show using an XOR lemma that R_{1/3}(f o (g^{xor}_{O(log n)})^n) = Omega(log n . R_{4/9}(f) . R_{1/3}(g))$, where g^{xor}_{O(log n)} is the function obtained by composing the XOR function on O(log n) bits and g.

Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{anshu_et_al:LIPIcs.FSTTCS.2017.10, author = {Anshu, Anurag and Gavinsky, Dmitry and Jain, Rahul and Kundu, Srijita and Lee, Troy and Mukhopadhyay, Priyanka and Santha, Miklos and Sanyal, Swagato}, title = {{A Composition Theorem for Randomized Query Complexity}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {10:1--10:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.10}, URN = {urn:nbn:de:0030-drops-83967}, doi = {10.4230/LIPIcs.FSTTCS.2017.10}, annote = {Keywords: Query algorithms and complexity, Decision trees, Composition theorem, XOR lemma, Hardness amplification} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

The pointer function of Goos, Pitassi and Watson and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and quantum query complexity, exact and approximate polynomial degree, etc. In particular, Ambainis et al. (STOC 2016) obtained the widest possible (quadratic) separations between deterministic and zero-error randomized query complexity, as well as between bounded-error and zero-error randomized query complexity by considering variants of this pointer function.
However, as Ambainis et al. pointed out in their work, the precise zero-error complexity of the original pointer function was
not known. We show a lower bound of ~Omega(n^{3/4}) on the zero-error randomized query complexity of the pointer function on Theta(n * log(n)) bits; since an ~O(n^{3/4}) upper bound was already shown by Mukhopadhyay and Sanyal (FSTTCS 2015), our lower bound is optimal up to polylog factors. We, in fact, consider a generalization of the original function and obtain lower bounds for it that are optimal up to polylog factors.

Jaikumar Radhakrishnan and Swagato Sanyal. The Zero-Error Randomized Query Complexity of the Pointer Function. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 16:1-16:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{radhakrishnan_et_al:LIPIcs.FSTTCS.2016.16, author = {Radhakrishnan, Jaikumar and Sanyal, Swagato}, title = {{The Zero-Error Randomized Query Complexity of the Pointer Function}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {16:1--16:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.16}, URN = {urn:nbn:de:0030-drops-68514}, doi = {10.4230/LIPIcs.FSTTCS.2016.16}, annote = {Keywords: Deterministic Decision Tree, Randomized Decision Tree, Query Complexity, Models of Computation.} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

We show that there exists a Boolean function F which gives the following separations among deterministic query complexity (D(F)), randomized zero error query complexity (R_0(F)) and randomized one-sided error query complexity (R_1(F)): R_1(F) = ~O(sqrt{D(F)) and R_0(F)=~O(D(F))^3/4. This refutes the conjecture made by Saks and Wigderson that for any Boolean function f, R_0(f)=Omega(D(f))^0.753.. . This also shows widest separation between R_1(f) and D(f) for any Boolean function. The function F was defined by Göös, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function F certify optimal (quadratic) separation between D(f) and R_0(f), and polynomial separation between R_0(f) and R_1(f). Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considered in the work of Ambainis et al are different variants of F, in this work we show that the original function F itself is sufficient to refute the Saks-Wigderson conjecture and obtain widest possible separation between the deterministic and one-sided error randomized query complexity.

Sagnik Mukhopadhyay and Swagato Sanyal. Towards Better Separation between Deterministic and Randomized Query Complexity. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 206-220, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{mukhopadhyay_et_al:LIPIcs.FSTTCS.2015.206, author = {Mukhopadhyay, Sagnik and Sanyal, Swagato}, title = {{Towards Better Separation between Deterministic and Randomized Query Complexity}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {206--220}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.206}, URN = {urn:nbn:de:0030-drops-56467}, doi = {10.4230/LIPIcs.FSTTCS.2015.206}, annote = {Keywords: Deterministic Decision Tree, Randomized Decision Tree, Query Complexity, Models of Computation.} }

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