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Electronic Colloquium on Computational Complexity

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TR13-127 | 15th September 2013 02:41

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. In particular, we develop a randomized algorithm for the element distinctness problem whose time $T$ and space $S$ satisfy $T\in \tilde O(n^{3/2}/S^{1/2})$, 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 multi-way branching programs, and hence standard RAM and word-RAM models, to compute the number of distinct elements, $F_0$, over sliding windows. The same lower bound holds for computing the low-order bit of $F_0$ and computing any frequency moment $F_k$ for $k\ne 1$. This shows that frequency moments $F_k\ne 1$ and even the decision problem $F_0\bmod 2$ are strictly harder than element distinctness. We provide even stronger separations on average for inputs from $[n]$.
We complement this lower bound with a $T\in \tilde O(n^2/S)$ comparison-based deterministic RAM algorithm for exactly computing $F_k$ over sliding windows, nearly matching both our general lower bound for the sliding-window version and the comparison-based lower bounds for a single instance of the problem. We further exhibit a quantum algorithm for $F_0$ over sliding windows with $T\in \tilde O(n^{3/2}/S^{1/2})$.
Finally, we consider the computations of order statistics over sliding windows.

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