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

The quantum complexity of approximating the frequency moments

The $k$'th frequency moment of a sequence of integers is defined as $F_k = \sum_j n_j^k$, where $n_j$ is the number of times that $j$ occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate $F_k$ up to relative error $\epsilon$. We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements of the input one at a time, and seek to minimise the amount of storage space, or passes over the data, used to approximate $F_k$. We describe quantum algorithms for $F_0$, $F_2$ and $F_\infty$ in this model which substantially outperform the best possible classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio

Towards Optimal Moment Estimation in Streaming and Distributed Models

One of the oldest problems in the data stream model is to approximate the p-th moment ||X||_p^p = sum_{i=1}^n X_i^p of an underlying non-negative vector X in R^n, which is presented as a sequence of poly(n) updates to its coordinates. Of particular interest is when p in (0,2]. Although a tight space bound of Theta(epsilon^-2 log n) bits is known for this problem when both positive and negative updates are allowed, surprisingly there is still a gap in the space complexity of this problem when all updates are positive. Specifically, the upper bound is O(epsilon^-2 log n) bits, while the lower bound is only Omega(epsilon^-2 + log n) bits. Recently, an upper bound of O~(epsilon^-2 + log n) bits was obtained under the assumption that the updates arrive in a random order. We show that for p in (0, 1], the random order assumption is not needed. Namely, we give an upper bound for worst-case streams of O~(epsilon^-2 + log n) bits for estimating |X |_p^p. Our techniques also give new upper bounds for estimating the empirical entropy in a stream. On the other hand, we show that for p in (1,2], in the natural coordinator and blackboard distributed communication topologies, there is an O~(epsilon^-2) bit max-communication upper bound based on a randomized rounding scheme. Our protocols also give rise to protocols for heavy hitters and approximate matrix product. We generalize our results to arbitrary communication topologies G, obtaining an O~(epsilon^2 log d) max-communication upper bound, where d is the diameter of G. Interestingly, our upper bound rules out natural communication complexity-based approaches for proving an Omega(epsilon^-2 log n) bit lower bound for p in (1,2] for streaming algorithms. In particular, any such lower bound must come from a topology with large diameter

Element Distinctness, Frequency Moments, and Sliding Windows

We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than previous lower bounds for comparison-based algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and space efficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows, where the task is to take an input of length 2n-1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a time-space tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to compute the number of distinct elements over sliding windows. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k, k neq 1. This shows that those frequency moments and the decision problem F_0 mod 2 are strictly harder than element distinctness. We complement this lower bound with a T in O(n^2/S) comparison-based deterministic RAM algorithm for exactly computing F_k over sliding windows, nearly matching both our lower bound for the sliding-window version and the comparison-based lower bounds for the single-window version. We further exhibit a quantum algorithm for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437