Quantum Speed-ups for String Synchronizing Sets, Longest Common Substring, and \(k\) -mismatch Matching
Author(s)
Jin, Ce
Nogler, Jakob
Date Issued
June 10, 2024
Journal
ACM Transactions on Algorithms
Publisher
Association for Computing Machinery (ACM)
Citation
Jin, Ce and Nogler, Jakob. 2024. "Quantum Speed-ups for String Synchronizing Sets, Longest Common Substring, and \(k\) -mismatch Matching." ACM Transactions on Algorithms.
Version
Final published version
Abstract
Longest Common Substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decisional version of this problem, LCS with threshold $d$, asks whether two length-$n$ input strings have a common substring of length $d$. The two extreme cases, $d=1$ and $d=n$, correspond respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case $1\ll d\ll n$ was not fully understood. We show that the complexity of LCS with threshold $d$ smoothly interpolates between the two extreme cases up to $n^{o(1)}$ factors: LCS with threshold $d$ has a quantum algorithm in $n^{2/3+o(1)}/d^{1/6}$ query complexity and time complexity, and requires at least $\Omega(n^{2/3}/d^{1/6})$ quantum query complexity. Our result improves upon previous upper bounds $\tilde O(\min \{n/d^{1/2}, n^{2/3}\})$ (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples $n/\tau^{1-o(1)}$ synchronizing positions in the string depending on their length-$\Theta(\tau)$ contexts, and each synchronizing position can be reported by a quantum algorithm in $\tilde O(\tau^{1/2+o(1)})$ time. As another application of our quantum string synchronizing set, we study the $k$-mismatch Matching problem under Hamming distance. Using a structural result of Charalampopoulos, Kociumaka, and Wellnitz (FOCS 2020), we obtain a quantum algorithm for k-mismatch matching with $k^{3/4} n^{1/2+o(1)}$ query complexity and $\tilde O(kn^{1/2})$ time complexity.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3672395
