Faster Algorithms for Longest Common Substring

In the classic Longest Common Substring (LCS) problem, we are given two strings \( S \) and \( T \) of total length \( n \) over an alphabet of size \(\sigma\) , 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 \(\mathcal{O}(n\log\sigma)\) -time algorithm for the LCS problem [SWAT 1973]. For polynomially-bounded integer alphabets, the linear-time construction of suffix trees by Farach yielded an \(\mathcal{O}(n)\) -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 \(\mathcal{O}(n\log\sigma/\log n)\) space and read in \(\mathcal{O}(n\log\sigma/\log n)\) time. We show that we can compute an LCS of two strings in time \(\mathcal{O}(n\log\sigma/\sqrt{\log n})\) in the word RAM model, which is sublinear in \( n \) if \(\sigma=2^{o(\sqrt{\log n})}\) (in particular, if \(\sigma=\mathcal{O}(1)\) ), using optimal space \(\mathcal{O}(n\log\sigma/\log n)\) . In fact, it was recently shown that this result is conditionally optimal [Kempa and Kociumaka, STOC 2025]. The same complexity can be achieved for computing an LCS of \(\lambda=\mathcal{O}(\sqrt{\log n}/\log\log n)\) input strings of total length \( n \) . We 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 \(\mathcal{O}(n\log n)\) time [IPL 2015]. Thankachan et al. extended this result to computing a \( k \) -mismatch LCS in \(\mathcal{O}(n\log^{k}n)\) time for \(k=\mathcal{O}(1)\) [J. Comput. Biol. 2016]. We show an \(\mathcal{O}(n\log^{k-0.5}n)\) -time algorithm, for any constant integer \(k > 0\) and irrespective of the alphabet size, using \(\mathcal{O}(n)\) 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. for string indexing with \( k \) errors [STOC 2004]. As a by-product, we improve upon the algorithm of Charalampopoulos et al. [CPM 2018] for computing a \( k \) -mismatch LCS in the case when the output \( k \) -mismatch LCS is sufficiently long.