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We prove that if a linear error-correcting code C : f0;1g n ! f0;1g m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2 (n). We also present several extensions of this result. We show a reduction from the complexity of one-round, information-theoretic Private Information… (More)

We investigate variants of Lloyd's heuristic for clustering high-dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a <i>clusterability</i> criterion for data sets. We present variants of Lloyd's heuristic that… (More)

We present a fairly general method for nding deterministic constructions obeying what we call k-restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2… (More)

1 Introduction This paper is concerned with the question of whether it is possible to sustain computation in the presence of noise; and if so, at what cost in efficiency and reliability. Shannon in his classical coding theorem showed that data can successfully and efficiently be transmitted in a noisy environment [27]. The outstanding features of the… (More)

Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless in each direction. We study the following question: if in… (More)

In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cut-matching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a… (More)

— We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.

We provide positive and negative results concerning the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.

Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function $f: X \times Y \rightarrow \{0,1\}$ and a probability distribution $D$ over $X \times Y$, we define the sampling complexity of $(f,D)$ as the minimum number of bits Alice and Bob must communicate for Alice to pick $x \in X$… (More)