Small space and streaming pattern matching with k edits

In this work, we revisit the fundamental and well-studied problem of approximate pattern matching under edit distance. Given an integer $k$, a pattern $P$ of length $m$, and a text $T$ of length $n \ge m$, the task is to find substrings of $T$ that are within edit distance $k$ from $P$. Our main result is a streaming algorithm that solves the problem in $\tilde{O}(k^5)$ space and $\tilde{O}(k^8)$ amortised time per character of the text, providing answers correct with high probability. (Hereafter, $\tilde{O}(\cdot)$ hides a $\mathrm{poly}(\log n)$ factor.) This answers a decade-old question: since the discovery of a $\mathrm{poly}(k\log n)$-space streaming algorithm for pattern matching under Hamming distance by Porat and Porat [FOCS 2009], the existence of an analogous result for edit distance remained open. Up to this work, no $\mathrm{poly}(k\log n)$-space algorithm was known even in the simpler semi-streaming model, where $T$ comes as a stream but $P$ is available for read-only access. In this model, we give a deterministic algorithm that achieves slightly better complexity. In order to develop the fully streaming algorithm, we introduce a new edit distance sketch parametrised by integers $n\ge k$. For any string of length at most $n$, the sketch is of size $\tilde{O}(k^2)$ and it can be computed with an $\tilde{O}(k^2)$-space streaming algorithm. Given the sketches of two strings, in $\tilde{O}(k^3)$ time we can compute their edit distance or certify that it is larger than $k$. This result improves upon $\tilde{O}(k^8)$-size sketches of Belazzougui and Zhu [FOCS 2016] and very recent $\tilde{O}(k^3)$-size sketches of Jin, Nelson, and Wu [STACS 2021]

Multilinguals and Wikipedia Editing

This article analyzes one month of edits to Wikipedia in order to examine the role of users editing multiple language editions (referred to as multilingual users). Such multilingual users may serve an important function in diffusing information across different language editions of the encyclopedia, and prior work has suggested this could reduce the level of self-focus bias in each edition. This study finds multilingual users are much more active than their single-edition (monolingual) counterparts. They are found in all language editions, but smaller-sized editions with fewer users have a higher percentage of multilingual users than larger-sized editions. About a quarter of multilingual users always edit the same articles in multiple languages, while just over 40% of multilingual users edit different articles in different languages. When non-English users do edit a second language edition, that edition is most frequently English. Nonetheless, several regional and linguistic cross-editing patterns are also present

Approximating Properties of Data Streams

In this dissertation, we present algorithms that approximate properties in the data stream model, where elements of an underlying data set arrive sequentially, but algorithms must use space sublinear in the size of the underlying data set. We first study the problem of finding all k-periods of a length-n string S, presented as a data stream. S is said to have k-period p if its prefix of length n − p differs from its suffix of length n − p in at most k locations. We give algorithms to compute the k-periods of a string S using poly(k, log n) bits of space and we complement these results with comparable lower bounds. We then study the problem of identifying a longest substring of strings S and T of length n that forms a d-near-alignment under the edit distance, in the simultaneous streaming model. In this model, symbols of strings S and T are streamed at the same time and form a d-near-alignment if the distance between them in some given metric is at most d. We give several algorithms, including an exact one-pass algorithm that uses O(d2 + d log n) bits of space. We then consider the distinct elements and `p-heavy hitters problems in the sliding window model, where only the most recent n elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and `p-heavy hitters that is nearly optimal in both n and c. Finally, we consider the problem of estimating the maximum weighted matching of a graph whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum cardinality matching problem that only doubles the approximation factor of a streaming algorithm developed for the maximum cardinality matching problem. As an application, we obtain an estimator for the weight of a maximum weighted matching in bounded-arboricity graphs and in particular, a (48 + )-approximation estimator for the weight of a maximum weighted matching in planar graphs

FPGA Acceleration of Pre-Alignment Filters for Short Read Mapping With HLS

Pre-alignment filters are useful for reducing the computational requirements of genomic sequence mappers. Most of them are based on estimating or computing the edit distance between sequences and their candidate locations in a reference genome using a subset of the dynamic programming table used to compute Levenshtein distance. Some of their FPGA implementations of use classic HDL toolchains, thus limiting their portability. Currently, most FPGA accelerators offered by heterogeneous cloud providers support C/C++ HLS. In this work, we implement and optimize several state-of-the-art pre-alignment filters using C/C++ based-HLS to expand their portability to a wide range of systems supporting the OpenCL runtime. Moreover, we perform a complete analysis of the performance and accuracy of the filters and analyze the implications of the results. The maximum throughput obtained by an exact filter is 95.1 MPairs/s including memory transfers using 100 bp sequences, which is the highest ever reported for a comparable system and more than two times faster than previous HDL-based results. The best energy efficiency obtained from the accelerator (not considering host CPU) is 2.1 MPairs/J, more than one order of magnitude higher than other accelerator-based comparable approaches from the state of the art.10.13039/501100008530-European Union Regional Development Fund (ERDF) within the framework of the ERDF Operational Program of Catalonia 2014-2020 with a grant of 50% of the total cost eligible under the Designing RISC-V based Accelerators for next generation computers project (DRAC) (Grant Number: [001-P-001723]) 10.13039/501100002809-Catalan Government (Grant Number: 2017-SGR-313 and 2017-SGR-1624) 10.13039/501100004837-Spanish Ministry of Science, Innovation and Universities (Grant Number: PID2020-113614RB-C21 and RTI2018-095209-B-C22)Peer ReviewedPostprint (published version