We give a sublinear quantum algorithm for the longest common substring (LCS) problem on the run-length encoded (RLE) inputs, under the assumption that the prefix-sums of the runs are given. Our algorithm costs O~(n5/6)⋅O(polylog(n~)) time, where n and n~ are the encoded and decoded length of the inputs, respectively. We justify the use of the prefix-sum oracles by showing that, without the oracles, there is a Ω(n/log2n) lower-bound on the quantum query complexity of finding LCS given two RLE strings due to a reduction of PARITY to the problem.
@article{arxiv.2310.00966,
title = {A sublinear time quantum algorithm for longest common substring problem between run-length encoded strings},
author = {Tzu-Ching Lee and Han-Hsuan Lin},
journal= {arXiv preprint arXiv:2310.00966},
year = {2023}
}