Almost-Smooth Histograms and Sliding-Window Graph Algorithms

We study algorithms for the sliding-window model, an important variant of the data-stream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We extend the smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes all subadditive functions. Specifically, we show that if a subadditive function can be $(1+\epsilon)$-approximated in the insertion-only streaming model, then it can be $(2+\epsilon)$-approximated also in the sliding-window model with space complexity larger by factor $O(\epsilon^{-1}\log w)$, where $w$ is the window size. We demonstrate how our framework yields new approximation algorithms with relatively little effort for a variety of problems that do not admit the smooth-histogram technique. For example, in the frequency-vector model, a symmetric norm is subadditive and thus we obtain a sliding-window $(2+\epsilon)$-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window $(\sqrt{2}+\epsilon)$-approximation algorithm for Schatten $4$-norm. We then consider graph streams and show that many graph problems are subadditive, including maximum submodular matching, minimum vertex-cover, and maximum $k$-cover, thereby deriving sliding-window $O(1)$-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every $d\in (1,2]$ an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly $d$

Parallel Streaming Frequency-Based Aggregates

We present efficient parallel streaming algorithms for fundamental frequency-based aggregates in both the sliding window and the infinite window settings. In the sliding window setting, we give a parallel algorithm for maintaining a space-bounded block counter (SBBC). Using SBBC, we derive algorithms for basic counting, frequency estimation, and heavy hitters that perform no more work than their best sequential counterparts. In the infinite window setting, we present algorithms for frequency estimation, heavy hitters, and count-min sketch. For both the infinite window and sliding window settings, our parallel algorithms process a minibatch of items using linear work and polylog parallel depth. We also prove a lower bound showing that the work of the parallel algorithm is optimal in the case of heavy hitters and frequency estimation. To our knowledge, these are the first parallel algorithms for these problems that are provably work efficient and have low depth

Parallel approach to sliding window sums

Sliding window sums are widely used in bioinformatics applications, including sequence assembly, k-mer generation, hashing and compression. New vector algorithms which utilize the advanced vector extension (AVX) instructions available on modern processors, or the parallel compute units on GPUs and FPGAs, would provide a significant performance boost for the bioinformatics applications. We develop a generic vectorized sliding sum algorithm with speedup for window size w and number of processors P is O(P/w) for a generic sliding sum. For a sum with commutative operator the speedup is improved to O(P/log(w)). When applied to the genomic application of minimizer based k-mer table generation using AVX instructions, we obtain a speedup of over 5X.Comment: 10 pages, 5 figure

Sliding Window Property Testing for Regular Languages

We study the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window n and a stream of symbols. At each time instant, we must decide whether the suffix of length n of the current stream ("the active window") belongs to a given regular language. Recent works [Moses Ganardi et al., 2018; Moses Ganardi et al., 2016] showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic or linear in the window size n and provided natural language theoretic characterizations of the space complexity classes. Subsequently, [Moses Ganardi et al., 2018] extended this result to randomized algorithms to show that any such algorithm admits either constant, double logarithmic, logarithmic or linear space complexity. In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject it if it is far from the language. We show that for every regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space