Faster Approximate Pattern Matching: A Unified Approach

Charalampopoulos, Panagiotis; Kociumaka, Tomasz; Wellnitz, Philip

doi:10.1109/focs46700.2020.00095

Cited by 29 publications

(64 citation statements)

References 27 publications

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“…) is reported when the algorithm processes T [p], we need a buffer storing the active k-edit occurrences of P i−1 . We implement it using a recent combinatorial characterization of k-edit occurrences [12], which classifies strings based on the following notion of approximate periodicity: Definition 3.2 (k-periodic string). A string X is k-periodic if there exists a primitive string Q with |Q| ≤ |X|/128k such that the edit distance between X and a prefix of…”

Section: (Semi-)streaming Algorithm For Pattern Matching With K Editsmentioning

confidence: 99%

“…The current best result was achieved by a series of work [55,17] with the running time (for some range of parameters) O(n + k 4 n/m). Very recently, Charalampopoulos et al [12] studied the problem for both distances in the grammar-compressed setting. Their result, in particular, implies an O(k 4 )space and O(nk 4 )-time algorithm for the read-only model, where random access to characters in P and T is allowed and one accounts only for the "extra" space, the space required beyond the space needed to store the pattern and the text.…”

Section: Introductionmentioning

confidence: 99%

“…The authors would like to thank Panagiotis Charalampopoulos for an early write-up of the proof of this lemma, which was originally meant to be included into[12].…”

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confidence: 99%

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Small space and streaming pattern matching with k edits

Kociumaka,

Porat,

Starikovskaya

2021

Preprint

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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].

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Section: (Semi-)streaming Algorithm For Pattern Matching With K Editsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Small space and streaming pattern matching with k edits

Kociumaka,

Porat,

Starikovskaya

2021

Preprint

Self Cite

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“…Kociumaka and Seddighin [28] provided an algorithm for maintaining the exact LIS of a dynamic string with polynomial sublinear time per update. In addition to the above, the problems of finding the longest palindrome substring in a text [3,22], finding all the runs in a text [5], approximated pattern matching [17], and computing the edit distance between two strings [16] have all been recently investigated in dynamic settings.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Suffix Array with Sub-linear update time and Poly-logarithmic Lookup Time

Amir¹,

Boneh²

2021

Preprint

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The suffix array SAS[1 . . . n] of an n-length string S is a lexicographically sorted array of the suffixes of S. The suffix array is one of the most well known and widely used data structures in string algorithms. We present a data structure for maintaining a representation of the suffix array of a dynamic string which undergoes symbol substitutions, deletions, and insertions.For every string manipulation, our data structure can be updated in Õ(n

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“…A natural direction after having determined the complexity of a particular problem on strings is to consider the more general version in which we need to answer queries on substrings of the input string. This has been done for alignment [34,35], pattern matching [24,28], approximate pattern matching [17], dictionary matching [14,15], compression [24], periodicity [27,28], counting palindromes [33], longest common substring [3], computing minimal and maximal suffixes [5,26], and computing the lexicographically k-th suffix [4].…”

Section: Introductionmentioning

confidence: 99%

An Almost Optimal Edit Distance Oracle

Charalampopoulos,

Gawrychowski,

Mozes

et al. 2021

Preprint

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We consider the problem of preprocessing two strings S and T , of lengths m and n, respectively, in order to be able to efficiently answer the following queries: Given positions i, j in S and positions a, b in T , return the optimal alignment of S[i . . j] and T [a . . b]. Let N = mn. We present an oracle with preprocessing time N 1+o(1) and space N 1+o(1) that answers queries in log 2+o(1) N time. In other words, we show that we can query the alignment of every two substrings in almost the same time it takes to compute just the alignment of S and T . Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS).The best previously known oracle with construction time and size O(N ) has slow Ω( √ N ) query time [Sakai, TCS 2019], and the one with size N 1+o(1) and query time log 2+o(1) N (using a planar graph distance oracle) has slow Ω(N 3/2) construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a √ N factor.

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Faster Approximate Pattern Matching: A Unified Approach

Cited by 29 publications

References 27 publications

Small space and streaming pattern matching with k edits

Small space and streaming pattern matching with k edits

Dynamic Suffix Array with Sub-linear update time and Poly-logarithmic Lookup Time

An Almost Optimal Edit Distance Oracle

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