Anil Ada

The Hardness of Being Private

In 1989 Kushilevitz initiated the study of iinformation-theoretic privacy within the context of communication complexity. Unfortunately, it has been shown that most interesting functions are not privately computable. The unattainability of perfect privacy for many functions motivated the study of approximate privacy. Feigenbaum et al. define notions of worst-case as well as average-case approximate privacy, and present several interesting upper bounds, and some open problems for further study. In this paper, we obtain asymptotically tight bounds on the tradeoffs 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.

Spectral Norm of Symmetric Functions

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 \(f(x) \cdot \textnormal{\textsc{parity}}(x)\) 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.

On the Non-deterministic Communication Complexity of Regular Languages

In this paper we study the non-deterministic communication complexity of regular languages. We show that a regular language has either constant or at least logarithmic non-deterministic communication complexity. We prove several linear lower bounds which we know cover a wide range of regular languages with linear complexity. Furthermore we find evidence that previous techniques (Tesson and Therien 2005) for proving linear lower bounds, for instance in deterministic and probabilistic models, do not work in the non-deterministic setting.

ON THE NON-DETERMINISTIC COMMUNICATION COMPLEXITY OF REGULAR LANGUAGES

In this paper we study the non-deterministic communication complexity of regular languages. We show that a regular language has either constant or at least logarithmic non-deterministic communication complexity. We prove several linear lower bounds which we know cover a wide range of regular languages with linear complexity. Furthermore we find evidence that previous techniques (Tesson and Therien 2005) for proving linear lower bounds, for instance in deterministic and probabilistic models, do not work in the non-deterministic setting.