Dynamic Longest Common Substring in Polylogarithmic Time

Charalampopoulos, Panagiotis; Gawrychowski, Paweł; Pokorski, Karol

doi:10.4230/lipics.icalp.2020.27

Cited by 14 publications

(20 citation statements)

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“…Instead, we will design a different data structure that satisfies these two requirements, and can solve the Two String Families LCP problem on the maintained subset in Õ( √ rd) quantum time. This time complexity is worse than the poly log(n) time achieved by the classical data structure of [CGP20], but suffices for our application.…”

Section: Longest Common Substringmentioning

confidence: 88%

“…Instead, we will design a different data structure that satisfies the history-independence and worst-case update time requirements, and can solve the Two String Families LCP problem on the maintained instance in Õ( √ rd) quantum time. This time complexity is much worse than the poly log(n) time achieved by the classical data structure of [CGP20], but is sufficient for our purpose. As mentioned in Section 1.2.1, one challenge is the lack of a comparison-based data structure for 2D range query that also satisfies the two requirements above.…”

Section: Overviewmentioning

confidence: 93%

“…Hence, we should try to dynamically maintain the solution using data structures, instead of solving it from scratch every time. In fact, such a data structure with poly log(n) time per operation was already given by Charalampopoulos, Gawrychowski, and Pokorski [CGP20], and was used to obtain a classical data structure for maintaining Longest Common Substring under character substitutions. However, this data structure cannot be applied to the quantum walk algorithm, since it violates two requirements that are crucial for the correctness of quantum walk algorithms: (1) It should have worst-case time complexity (instead of being amortized), and (2) it should be history-independent (see the discussion in Section 3.2.1 for more details).…”

Section: Longest Common Substringmentioning

confidence: 99%

“…Instead, we dynamically maintain the solution using some data structure, which efficiently handles each update step during the quantum walk where we insert one string pair (P, Q) into (and remove one from) the current Two String Families LCP instance. As mentioned in Section 1.2.1, the classical data structure for this task given by Charalampopoulos, Gawrychowski, and Pokorski [CGP20] is not applicable here, since it violates both requirements mentioned above: it maintains a heavy-light decomposition of the compact tries of the input strings, and rebuilds them from time to time to ensure amortized poly log(n) time complexity. It is not clear how to implement this strategy in a history-independent way and with worst-case time complexity per operation.…”

Section: Overviewmentioning

confidence: 99%

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Near-Optimal Quantum Algorithms for String Problems

Akmal¹,

Jin²

2021

Preprint

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We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that:• Longest Common Substring can be solved by a quantum algorithm in Õ(n 2/3 ) time, improving upon the recent Õ(n 5/6 )-time algorithm by Le Gall and Seddighin (2020). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, 2019) and generalized difference covers.• Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n 1/2+o(1) time, improving upon the recent Õ(n 3/4 )-time algorithm by Wang and Ying (2020). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding.• Longest Square Substring can be solved by a quantum algorithm in Õ( √ n) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2020) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies.Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix.

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Section: Longest Common Substringmentioning

confidence: 88%

Section: Overviewmentioning

confidence: 93%

Section: Longest Common Substringmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

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Near-Optimal Quantum Algorithms for String Problems

Akmal¹,

Jin²

2021

Preprint

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“…Other results on the LCS problem include the linear-time computation of an LCS of several strings over an integer alphabet [46], trade-offs between the time and the working space for computing an LCS of two strings [13,53,60], and the dynamic maintenance of an LCS [2,3,27]. Very recently, a strongly sublinear-time quantum algorithm and a lower bound for the quantum setting were shown [41].…”

Section: Other Related Workmentioning

confidence: 99%

Faster Algorithms for Longest Common Substring

Charalampopoulos,

Kociumaka,

Pissis

et al. 2021

Preprint

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In the classic longest common substring (LCS) problem, we are given two strings S and T , each of length at most n, over an alphabet of size σ, and we are asked to find a longest string occurring as a fragment of both S and T . Weiner, in his seminal paper that introduced the suffix tree, presented an O(n log σ)-time algorithm for this problem [SWAT 1973]. For polynomially-bounded integer alphabets, the linear-time construction of suffix trees by Farach yielded an O(n)-time algorithm for the LCS problem [FOCS 1997]. However, for small alphabets, this is not necessarily optimal for the LCS problem in the word RAM model of computation, in which the strings can be stored in O(n log σ/ log n) space and read in O(n log σ/ log n) time. We show that, in this model, we canWe then lift our ideas to the problem of computing a k-mismatch LCS, which has received considerable attention in recent years. In this problem, the aim is to compute a longest substring of S that occurs in T with at most k mismatches. Flouri et al. showed how to compute a 1-mismatch LCS in O(n log n) time [IPL 2015]. Thankachan et al. extended this result to computing a k-mismatch LCS in O(n log k n) time for k = O(1) [J. Comput. Biol. 2016]. We show an O(n log k−1/2 n)-time algorithm, for any constant k > 0 and irrespective of the alphabet size, using O(n) space as the previous approaches. We thus notably break through the well-known n log k n barrier, which stems from a recursive heavy-path decomposition technique that was first introduced in the seminal paper of Cole et al. [STOC 2004] for string indexing with k errors.

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Minimal Unique Palindromic Substrings After Single-Character Substitution

Funakoshi

Mieno

2021

String Processing and Information Retrieval

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Dynamic Longest Common Substring in Polylogarithmic Time

Cited by 14 publications

References 0 publications

Near-Optimal Quantum Algorithms for String Problems

Near-Optimal Quantum Algorithms for String Problems

Faster Algorithms for Longest Common Substring

Minimal Unique Palindromic Substrings After Single-Character Substitution

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