Measuring sequence similarity and compressing texts are among the most fundamental tasks in string algorithms. In this work, we develop near-optimal quantum algorithms for the central problems in these two areas: computing the edit distance of two strings [Levenshtein, 1965] and building the Lempel-Ziv factorization of a string [Ziv & Lempel, 1977], respectively.
Measuring sequence similarity and compressing texts are among the most fundamental tasks in string algorithms. In this work, we develop near-optimal quantum algorithms for the central problems in these two areas: computing the edit distance of two strings [Levenshtein, 1965] and building the Lempel-Ziv factorization of a string [Ziv & Lempel, 1977], respectively.
Classically, the edit distance of two length-n strings can be computed in O(n2) time, whereas an O(n2−ε)-time procedure would falsify the Orthogonal Vectors Hypothesis. If the edit distance does not exceed k, the running time can be improved to O(n+k2), which is near-optimal (conditioned on OVH) as a function of n and k. Our first main contribution is a quantum O~(nk+k2)-time algorithm that uses O~(nk) queries, where O~(⋅) hides polylogarithmic factors. This query complexity is unconditionally optimal, and any significant improvement in the time complexity would resolve a long-standing open question of whether edit distance admits an O(n2−ε)-time quantum algorithm. Our divide-and-conquer quantum algorithm reduces the edit distance problem to a case where the strings have small Lempel-Ziv factorizations. Then, it combines a quantum LZ compression algorithm with a classical edit-distance subroutine for compressed strings. The LZ factorization problem can be classically solved in O(n) time, which is unconditionally optimal in the quantum setting. We can, however, hope for a quantum speedup if we parameterize the complexity in terms of the factorization size z. Already a generic oracle identification algorithm yields the optimal query complexity of O~(nz) at the price of exponential running time. Our second main contribution is a quantum algorithm that achieves the optimal time complexity of O~(nz). The key tool is a novel LZ-like factorization of size O(zlog2n) whose subsequent factors can be efficiently computed through a combination of classical and quantum techniques. We can then obtain the string's run-length encoded Burrows-Wheeler Transform (BWT), construct the r-index, and solve many fundamental string processing problems in time O~(nz).
Approximate Pattern Matching is among the most fundamental string-processing tasks. Given a text T of length n, a pattern P of length m, and a threshold k, the task is to identify the fragments of T that are at distance at most k to P. We consider the two most common distances: Hamming distance (the number of mismatches or character substitutions) in Pattern Matching with Mismatches and edit distance (the minimum number of character insertions, deletions, and substitutions) in Pattern Matching with Edits. We revisit the complexity of these two problems in the quantum setting.
Sublinear time quantum algorithms have been established for many fundamental problems on strings. This work demonstrates that new, faster quantum algorithms can be designed when the string is highly compressible. We focus on two popular and theoretically significant compression algorithms -- the Lempel-Ziv77 algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT), and obtain the results below. We first provide a quantum algorithm running in O~(zn) time for finding the LZ77 factorization of an input string T[1..n] with z factors. Combined with multiple existing results, this yields an O~(rn) time quantum algorithm for finding the RL-BWT encoding with r BWT runs. Note that r=Θ~(z). We complement these results with lower bounds proving that our algorithms are optimal (up to polylog factors). Next, we study the problem of compressed indexing, where we provide a O~(rn) time quantum algorithm for constructing a recently designed O~(r) space structure with equivalent capabilities as the suffix tree. This data structure is then applied to numerous problems to obtain sublinear time quantum algorithms when the input is highly compressible. For example, we show that the longest common substring of two strings of total length n can be computed in O~(zn) time, where z is the number of factors in the LZ77 factorization of their concatenation. This beats the best known O~(n32) time quantum algorithm when z is sufficiently small.
The edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [21]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an O(n1.858) quantum algorithm that approximates the edit distance within a factor of 7. We further extend this result to an O(n1.781) quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of 3, with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.
Abstract Longest common substring (), longest palindrome substring (), and Ulam distance () are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for and are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact O~(n) O~(n) time algorithm for , we prove that needs at least time Ω~(n2/3) Ω~(n2/3) even for 0/1 strings.
Abstract 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 O~(n2/3) O ~ ( n 2 / 3 ) time, improving upon the recent O~(n5/6) O ~ ( n 5 / 6 ) -time algorithm by Le Gall and Seddighin (in: Proceedings of the 13th innovations in theoretical computer science conference (ITCS 2022), pp 97:1–97:23, 2022. https://doi.org/10.4230/LIPIcs.ITCS.2022.97 ). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, in: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing (STOC 2019), ACM, pp 756–767, 2019. https://doi.org/10.1145/3313276.3316368 ) and generalized difference covers. Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n1/2+o(1) n 1 / 2 + o ( 1 ) time, improving upon the recent O~(n3/4) O ~ ( n 3 / 4 ) -time algorithm by Wang and Ying (in: Quantum algorithm for lexicographically minimal string rotation. CoRR, 2020. arXiv:2012.09376 ). 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 O~(n) O ~ ( n ) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2022) 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.