Recursive Sketching For Frequency Moments

In a ground-breaking paper, Indyk and Woodruff (STOC 05) showed how to compute $F_k$ (for $k>2$) in space complexity O(\mbox{\em poly-log}(n,m)\cdot n^{1-\frac2k}), which is optimal up to (large) poly-logarithmic factors in $n$ and $m$, where $m$ is the length of the stream and $n$ is the upper bound on the number of distinct elements in a stream. The best known lower bound for large moments is $\Omega(\log(n)n^{1-\frac2k})$. A follow-up work of Bhuvanagiri, Ganguly, Kesh and Saha (SODA 2006) reduced the poly-logarithmic factors of Indyk and Woodruff to $O(\log^2(m)\cdot (\log n+ \log m)\cdot n^{1-{2\over k}})$. Further reduction of poly-log factors has been an elusive goal since 2006, when Indyk and Woodruff method seemed to hit a natural "barrier." Using our simple recursive sketch, we provide a different yet simple approach to obtain a $O(\log(m)\log(nm)\cdot (\log\log n)^4\cdot n^{1-{2\over k}})$ algorithm for constant $\epsilon$ (our bound is, in fact, somewhat stronger, where the $(\log\log n)$ term can be replaced by any constant number of $\log$ iterations instead of just two or three, thus approaching $log^*n$. Our bound also works for non-constant $\epsilon$ (for details see the body of the paper). Further, our algorithm requires only $4$-wise independence, in contrast to existing methods that use pseudo-random generators for computing large frequency moments

Approximate Hamming distance in a stream

We consider the problem of computing a $(1+\epsilon)$-approximation of the Hamming distance between a pattern of length $n$ and successive substrings of a stream. We first look at the one-way randomised communication complexity of this problem, giving Alice the first half of the stream and Bob the second half. We show the following: (1) If Alice and Bob both share the pattern then there is an $O(\epsilon^{-4} \log^2 n)$ bit randomised one-way communication protocol. (2) If only Alice has the pattern then there is an $O(\epsilon^{-2}\sqrt{n}\log n)$ bit randomised one-way communication protocol. We then go on to develop small space streaming algorithms for $(1+\epsilon)$-approximate Hamming distance which give worst case running time guarantees per arriving symbol. (1) For binary input alphabets there is an $O(\epsilon^{-3} \sqrt{n} \log^{2} n)$ space and $O(\epsilon^{-2} \log{n})$ time streaming $(1+\epsilon)$-approximate Hamming distance algorithm. (2) For general input alphabets there is an $O(\epsilon^{-5} \sqrt{n} \log^{4} n)$ space and $O(\epsilon^{-4} \log^3 {n})$ time streaming $(1+\epsilon)$-approximate Hamming distance algorithm.Comment: Submitted to ICALP' 201