Towards Optimal Approximate Streaming Pattern Matching by Matching Multiple Patterns in Multiple Streams

Golan, Shay; Kopelowitz, Tsvi; Porat, Ely

doi:10.4230/lipics.icalp.2018.65

Cited by 9 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Porat and Porat [37] also introduced the first streaming algorithm for the exact k-mismatch problem using Õ(k 2 ) time per character and Õ(k 3 ) space. Subsequent improvements [15,24] culminated in an algorithm by Clifford et al (SODA 2019) [16] which solves the streaming exact k-mismatch problem in Õ( √ k) time per character using Õ(k) space (this space consumption is optimal regardless of the running time).…”

Section: The New Time Bound Ismentioning

confidence: 99%

“…To this end, we could use a known streaming algorithm for pattern matching. However, because there are O(z) possible u i 's and O(p/z) possible v i 's, we actually need a streaming pattern matching algorithm that can handle multiple patterns and multiple text streams-luckily, this variant, known as multistream dictionary matching, has already been addressed in a recent paper by Golan et al [24]. The space bound is Õε (z + p/z), which becomes Õε ( √ p) = Õε ( √ k) by setting z = √ p, and the per-character running time is O ε (1).…”

Section: Further Consequences: An Overviewmentioning

confidence: 99%

“…After the arrival of a character to the ith stream, the algorithm has to report the longest pattern from D that matches a suffix of the ith text stream, or state that none of the patterns from D is a suffix of the ith text stream. We use the algorithm of Golan et al [24] for the multi-stream dictionary problem: Lemma 11.1 (immediate from [24, Theorem 2]). There exists an algorithm for the multi-stream dictionary matching problem on a dictionary D, with d patterns of length at most m each, which for t text streams costs O(d log m + t log m log d) words of space and O(log m + log d log log d) time per character.…”

Section: Streaming Algorithmsmentioning

confidence: 99%

See 2 more Smart Citations

Approximating Text-to-Pattern Hamming Distances

Chan¹,

Golan²,

Kociumaka³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

We revisit a fundamental problem in string matching: given a pattern of length m and a text of length n, both over an alphabet of size σ, compute the Hamming distance (i.e., the number of mismatches) between the pattern and the text at every location. Several randomized (1 + ε)-approximation algorithms have been proposed in the literature (e.g., by Karloff (Inf. Proc. Lett., 1993), Indyk (FOCS 1998), and Kopelowitz and Porat (SOSA 2018)), with running time of the form O(ε −O(1) n log n log m), all using fast Fourier transform (FFT). We describe a simple randomized (1 + ε)-approximation algorithm that is faster and does not need FFT. Combining our approach with additional ideas leads to numerous new results (all Monte-Carlo randomized) in different settings:

show abstract

Section: The New Time Bound Ismentioning

confidence: 99%

Section: Further Consequences: An Overviewmentioning

confidence: 99%

Section: Streaming Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

Approximating Text-to-Pattern Hamming Distances

Chan¹,

Golan²,

Kociumaka³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In that model, the text arrives in a stream, one character at a time, and the goal is to compute, or estimate, after the arrival of each text character, the Hamming distance between P and the current suffix of T . The state-of-the-art exact algorithm [15] uses Õ(k) space and costs Õ( √ k) time per character, which improves upon [14,23,36,38]. A recent approximate streaming algorithm [12] uses Õ(min(ε −2 √ k, ε −1.5 √ n)) space and costs Õ(ε −3 ) time per character, which improves upon [16,39].…”

Section: Related Workmentioning

confidence: 99%

Improved Circular $k$-Mismatch Sketches

Golan,

Kociumaka,

Kopelowitz

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

The shift distance sh(S 1 , S 2 ) between two strings S 1 and S 2 of the same length is defined as the minimum Hamming distance between S 1 and any rotation (cyclic shift) of S 2 . We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings S 1 and S 2 of length n are given to two identical players (encoders), who independently compute sketches (summaries) sk(S 1 ) and sk(S 2 ), respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) sh(S 1 , S 2 ) with high probability.This paper primarily focuses on the more general k-mismatch version of the problem, where the decoder is allowed to declare a failure if sh(S 1 , S 2 ) > k, where k is a parameter known to all parties. Andoni et al. (STOC'13) introduced exact circular k-mismatch sketches of size Õ(k + D(n)), where D(n) is the number of divisors of n. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches.We circumvent this lower bound by designing a (non-linear) exact circular k-mismatch sketch of size Õ(k); this size matches communication-complexity lower bounds. We also design (1 ± ε)approximate circular k-mismatch sketch of size Õ(min(ε −2 √ k, ε −1.5 √ n)), which improves upon an Õ(ε −2 √ n)-size sketch of Crouch and McGregor (APPROX'11).

show abstract

“…It started with a seminal paper of Porat and Porat in FOCS 2009 [52], who showed streaming algorithms for exact pattern matching and for the k-mismatches problem. The result of Porat and Porat was followed by a series of works on streaming pattern matching [7,13,34,14,57,33,30,16,32,53], search of repetitions in streams [20,18,19,28,48,47,29], and recognising formal languages in streams [45,24,25,22,23,27,4,21,26].…”

Section: Introductionmentioning

confidence: 99%

Small space and streaming pattern matching with k edits

Kociumaka,

Porat,

Starikovskaya

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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 ≥ 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 Õ(k 5 ) space and Õ(k 8 ) amortised time per character of the text, providing answers correct with high probability. (Hereafter, Õ(•) hides a poly(log n) factor.) This answers a decade-old question: since the discovery of a 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 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 readonly access. In this model, we give a deterministic algorithm that achieves slightly better complexity.Our central technical contribution is a new space-efficient deterministic encoding of two strings, called the greedy encoding, which encodes a set of all alignments of cost ≤ k with a certain property (we call such alignments greedy). On strings of length at most n, the encoding occupies Õ(k 2 ) space. We use the encoding to compress substrings of the text that are close to the pattern. In order to do so, we compute the encoding for substrings of the text and of the pattern, which requires read-only access to the latter.In order to develop the fully streaming algorithm, we further introduce a new edit distance sketch parametrised by integers n ≥ k. For any string of length at most n, the sketch is of size Õ(k 2 ) and it can be computed with an Õ(k 2 )-space streaming algorithm. Given the sketches of two strings, in Õ(k 3 ) time we can compute their edit distance or certify that it is larger than k. This result improves upon Õ(k 8 )-size sketches of Belazzougui and Zhu [FOCS 2016] and very recent Õ(k 3 )-size sketches of Jin, Nelson, and Wu [STACS 2021].

show abstract

Towards Optimal Approximate Streaming Pattern Matching by Matching Multiple Patterns in Multiple Streams

Cited by 9 publications

References 0 publications

Approximating Text-to-Pattern Hamming Distances

Approximating Text-to-Pattern Hamming Distances

Improved Circular $k$-Mismatch Sketches

Small space and streaming pattern matching with k edits

Contact Info

Product

Resources

About