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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)
Consider a process in which information is transmitted from a given root node on a noisy tree network T. We start with an unbiased random bit R at the root of the tree, and send it down the edges of T. On every edge the bit can be reversed with probability , and these errors occur independently. The goal is to reconstruct R from the values which arrive at(More)
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)
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)
Attempts to find new quantum algorithms that outperform classical computation have fo-cused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples(More)