Knowledge implies time/space efficiency

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 35-36).The probabilistically checkable proof (PCP) system enables proofs to be verified in time polylogarithmic in the length of a classical proof. Computationally sound (CS) proofs improve upon PCPs by additionally shortening the length of the transmitted proof to be polylogarithmic in the length of the classical proof. In this thesis we explore the ultimate limits of non-interactive proof systems with respect to time/space efficiency and the new criterion of composability. We deduce the existence of our proposed proof system by way of a natural new assumption about proofs of knowledge. In fact, a main contribution of our result is showing that knowledge can be "traded" for time and space efficiency in noninteractive proof systems.by Paul Valiant.S.M

Testing symmetric properties of distributions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 65-66).We introduce the notion of a Canonical Tester for a class of properties on distributions, that is, a tester strong and general enough that "a distribution property in the class is testable if and only if the Canonical Tester tests it". We construct a Canonical Tester for the class of symmetric properties of one or two distributions, satisfying a certain weak continuity condition. Analyzing the performance of the Canonical Tester on specific properties resolves several open problems, establishing lower bounds that match known upper bounds: we show that distinguishing between entropy p on distributions over [n] requires nc/P-O(1) samples, and distinguishing whether a pair of distributions has statistical distance 0 requires n1-o(1) samples. Our techniques also resolve a conjecture about a property that our Canonical Tester does not apply to: distinguishing identical distributions from those with statistical distance > 0 requires Q(n2/3) samples.by Paul Valiant.Ph.D

The Shifting Sands Algorithm

We resolve the problem of small-space approximate selection in random-order streams. Specifically, we present an algorithm that reads the n elements of a set in random order and returns an element whose rank differs from the true median by at most n 1/3+o(1) while storing a constant number of elements and counters at any one time. This is optimal: it was previously shown that achieving better accuracy required poly(n) memory. However, it was conjectured that the lower bound was not tight and that a previous algorithm achieving an n 1/2+o(1) approximation was optimal. We therefore consider the new result a surprising resolution to a natural and basic question.