Abstract. We study the k-party 'number on the forehead' communication
The spectral norm of a Boolean function f : {0, 1} n → {−1, 1} is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as r(f ) log(n/r(f )) where r(f ) = max{r 0 , r 1 }, and r 0 and r 1 are the smallest integers less than n/2 such that f (x) or f (x) · PARITY(x) is constant for all x with x i ∈ [r 0 , n − r 1 ]. We mention some applications to the decision tree and communication complexity of symmetric functions.
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Kushilevitz [1989] initiated the study of information-theoretic privacy within the context of communication complexity. Unfortunately, it has been shown that most interesting functions are not privately computable [Kushilevitz 1989, Brandt and Sandholm 2008]. The unattainability of perfect privacy for many functions motivated the study of approximate privacy . Feigenbaum et al. [2010a, 2010b] define notions of worst-case as well as average-case approximate privacy and present several interesting upper bounds as well as some open problems for further study. In this article, we obtain asymptotically tight bounds on the trade-offs between both the worst-case and average-case approximate privacy of protocols and their communication cost for Vickrey auctions. Further, we relate the notion of average-case approximate privacy to other measures based on information cost of protocols. This enables us to prove exponential lower bounds on the subjective approximate privacy of protocols for computing the Intersection function, independent of its communication cost. This proves a conjecture of Feigenbaum et al. [2010a].
The notion of communication complexity was introduced by Yao in his sem-inal paper [Yao79]. In [BFS86], Babai Frankl and Simon developed a rich structure of communication complexity classes to understand the relationships between various models of communication complexity. This made it apparent that communication complexity was a self-contained mini-world within complexity theory. In this thesis, we study the place of regular languages within this mini-world. In particular, we are interested in the non-deterministic communication complexity of regular languages. We show that a regular language has either O(1) or Ω(log n) non-determi-nistic complexity. We obtain several linear lower bound results which cover a wide range of regular languages having linear non-deterministic complexity. These lower bound results also imply a result in semigroup theory: we obtain sufficient conditions for not being in the positive variety P ol(Com). To obtain our results, we use algebraic techniques. In the study of regular languages, the algebraic point of view pioneered by Eilenberg ([Eil74]) has led to many interesting results. Viewing a semigroup as a computational device that recognizes languages has proven to be prolific from both semigroup theory and formal languages perspectives. In this thesis, we provide further instances of such mutualism. i Résumé La notion de complexité de communication a d'abordétéabordété introduite par Yao [Yao79]. Les travaux fondateurs de Babai et al. [BFS86] ont dévoilé une riche structures de classes de complexité de communication qui permettent de mieux comprendre la puissance de divers modles de complexité de communication. Ces résultats ont fait de la complexité de communication une sorte de maquette petite chelle du monde de la complexit. Dans ce mémoire, nousétudionsnousétudions la place des langages réguliers dans cette maquette. Plus précisément, nous chercherons déterminer la complexité de communication non-déterministe de ces langages. Nous montrons qu'un langage régulier a une complexité de communication soit O(log n), soit Ω(log n). NousétablissonsNousétablissons de plus des bornes inférieures linéaires sur la complexité non-déterministe d'une vaste classe de langages. Celles-ci fournissentégalementfournissentégalement des conditions suffisantes pour qu'un langage donné n'appartienne pas la variété positive P ol(Com). Nos résultats se basent sur des techniques algébriques. Dans l'´ etude des langages réguliers, le point de vue algébrique, développé initialement par Eilenberg [Eil74] s'est révélé comme un outil central. En effet, on peut voir un semigroupe fini comme une machine capable de reconnaˆıtrereconnaˆıtre des langages et cette perspective a permis des avancées tant en théorie des semigroupes qu'en théorie des langages formels. Dans ce mémoire, nousétablissonsnousétablissons de nouveaux exemples de ce mutualisme. ii
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The notion of communication complexity was introduced by Yao in his seminal paper [Yao79]. In [BFS86], Babai Frankl and Simon developed a rich structure of communication complexity classes to understand the relationships between various models of communication complexity. This made it apparent that communication complexity was a self-contained mini-world within complexity theory. In this thesis, we study the place of regular languages within this mini-world. In particular, we are interested in the nondeterministic communication complexity of regular languages.We show that a regular language has either O(1) or Ω(log n) non-deterministic complexity. We obtain several linear lower bound results which cover a wide range of regular languages having linear non-deterministic complexity.These lower bound results also imply a result in semigroup theory: we obtain sufficient conditions for not being in the positive variety P ol(Com).To obtain our results, we use algebraic techniques. In the study of regular languages, the algebraic point of view pioneered by Eilenberg ([Eil74]) has led to many interesting results. Viewing a semigroup as a computational device that recognizes languages has proven to be prolific from both semigroup theory and formal languages perspectives. In this thesis, we provide further instances of such mutualism. ii
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