Quantum Speed-ups for String Synchronizing Sets and Longest Common Substring (Ce Jin)
Channel: IRIF
Date: 2022-11-30

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 where d is much larger than 1 and much smaller than 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 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). Our quantum string synchronizing set also yields a near-optimal LCE data structure in the quantum setting, and an algorithm for the k-mismatch matching problem with k^{3/4}n^{1/2+o(1)} quantum query complexity.

Based on a SODA'23 paper joint with Jakob Nogler (ETH Zurich) and a SODA'22 paper joint with Shyan Akmal (MIT).