Sliding Windows with Limited Storage
Paul Beame, Raphael Clifford, Widad Machmouchi

TL;DR
This paper investigates the computational complexity of sliding window problems, establishing lower bounds and algorithms for frequency moments, element distinctness, and order statistics, revealing fundamental time-space tradeoffs.
Contribution
It provides new lower bounds and algorithms for sliding window computations, highlighting the inherent complexity and separations among different statistical functions.
Findings
Omega(n^2) lower bound for computing F_0 in sliding windows
Deterministic O(n^2) time-space algorithm for F_k
Logarithmic increase in time for order statistics like max/min
Abstract
We consider time-space tradeoffs for exactly computing frequency moments and order statistics over sliding windows. Given an input of length 2n-1, the task is to output the function of each window of length n, giving n outputs in total. Computations over sliding windows are related to direct sum problems except that inputs to instances almost completely overlap. We show an average case and randomized time-space tradeoff lower bound of TS in Omega(n^2) for multi-way branching programs, and hence standard RAM and word-RAM models, to compute the number of distinct elements, F_0, in sliding windows over alphabet [n]. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k for k not equal to 1. We complement this lower bound with a TS in \tilde O(n^2) deterministic RAM algorithm for exactly computing F_k in sliding windows. We show…
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Taxonomy
TopicsAlgorithms and Data Compression · semigroups and automata theory · Machine Learning and Algorithms
