Ameya Velingker

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We prove that a random linear code over Fq, with probability arbitrarily close to 1, is list decodable at radius 1− 1/q− with list size L = O(1/ ) and rate R = Ωq( 2/(log(1/ ))). Up to the polylogarithmic factor in 1/ and constant factors depending on q, this matches the lower bound L = Ωq(1/ ) for the list size and upper bound R = Oq( ) for the rate.(More)
We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2<sup>O(d)</sup>(n log n + m)), where n is the input size, m is the output point set size,(More)
Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to(More)
We prove a lower estimate on the increase in entropy when two copies of a conditional random variable X|Y , with X supported on Zq = {0, 1, . . . , q − 1} for prime q, are summed modulo q. Specifically, given two i.i.d. copies (X1, Y1) and (X2, Y2) of a pair of random variables (X,Y ), with X taking values in Zq, we show H(X1 +X2 | Y1, Y2)−H(X|Y ) ≥ α(q)(More)
Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and(More)
The central goal of this thesis is to better understand, and explicitly construct, expanding towers G1,G2, . . ., which are expander families with the additional constraint that Gn+1 is a lift of Gn . A lift G of H is a graph that locally looks like H , but may be globally di erent; lifts have been proposed as a more structured setting for elementary(More)
We study the communication rate of coding schemes for interactive communication that transform any two-party interactive protocol into a protocol that is robust to noise. Recently, Haeupler [Hae14] showed that if an > 0 fraction of transmissions are corrupted, adversarially or randomly, then it is possible to achieve a communication rate of 1 − Õ( √ ).(More)
We consider the problem of estimating the value of MAX-CUT in a graph in the streaming model of computation. We show that there exists a constant ∗ > 0 such that any randomized streaming algorithm that computes a (1 + ∗)-approximation to MAX-CUT requires Ω(n) space on an n vertex graph. By contrast, there are algorithms that produce a (1 + )-approximation(More)
We introduce the notion of one-way communication schemes with partial noiseless feedback. In this setting, Alice wishes to communicate a message to Bob by using a communication scheme that involves sending a sequence of bits over a channel while receiving feedback bits from Bob for δ fraction of the transmissions. An adversary is allowed to corrupt up to a(More)
Locally testable codes (LTCs) of constant minimum (absolute) distance that allow the tester to make a nearly linear number of queries have become the focus of attention recently, due to their connections to central questions in approximability theory. In particular, the binary Reed-Muller code of block length N and absolute distance d is known to be(More)