Daniel M. Kane

An optimal algorithm for the distinct elements problem

We give the first optimal algorithm for estimating the number of distinct elements in a data stream, closing a long line of theoretical research on this problem begun by Flajolet and Martin in their seminal paper in FOCS 1983. This problem has applications to query optimization, Internet routing, network topology, and data mining. For a stream of indices in {1,...,n}, our algorithm computes a (1 ± ε)-approximation using an optimal O(1/ε-2 + log(n)) bits of space with 2/3 success probability, where 0<ε<1 is given. This probability can be amplified by independent repetition. Furthermore, our algorithm processes each stream update in O(1) worst-case time, and can report an estimate at any point midstream in O(1) worst-case time, thus settling both the space and time complexities simultaneously. We also give an algorithm to estimate the Hamming norm of a stream, a generalization of the number of distinct elements, which is useful in data cleaning, packet tracing, and database auditing. Our algorithm uses nearly optimal space, and has optimal O(1) update and reporting times.

Sparser Johnson-Lindenstrauss Transforms

We give two different and simple constructions for dimensionality reduction in <i>ℓ</i>₂ via linear mappings that are sparse: only an <i>O</i>(<i>ϵ</i>)-fraction of entries in each column of our embedding matrices are non-zero to achieve distortion 1 + <i>ϵ</i> with high probability, while still achieving the asymptotically optimal number of rows. These are the first constructions to provide subconstant sparsity for all values of parameters, improving upon previous works of Achlioptas [2003] and Dasgupta et al. [2010]. Such distributions can be used to speed up applications where <i>ℓ</i>₂ dimensionality reduction is used.

On the complexity of two-player win-lose games

The efficient computation of Nash equilibria is one of the most formidable challenges in computational complexity today. The problem remains open for two-player games. We show that the complexity of two-player Nash equilibria is unchanged when all outcomes are restricted to be 0 or 1. That is, win-or-lose games are as complex as the general case for two-player games.

Bounded Independence Fools Degree-2 Threshold Functions

For an

Fast moment estimation in data streams in optimal space

n

Robust Estimators in High Dimensions without the Computational Intractability

-variate degree–

The Gaussian Surface Area and Noise Sensitivity of Degree-d Polynomial Threshold Functions

2

Counting arbitrary subgraphs in data streams

real polynomial

Testing identity of structured distributions

p

NEW RESULTS ON THE LEAST COMMON MULTIPLE OF CONSECUTIVE INTEGERS

, we prove that