Faster Algorithms for Longest Common Substring

TL;DR

This work advances longest common substring (LCS) and k-mismatch LCS algorithms in the word RAM model for small alphabets. It introduces a Two String Families LCP framework built on compacted tries, wavelet trees, and string synchronising sets, enabling sublinear-time LCS for total input length $n$ with space $O(n\log\sigma/\log n)$ and, for constant $k$, subquadratic k-LCS time $O(n\log^{k-1/2} n)$. A key methodological leap is reducing LCS to structured LCP problems via anchors and difference covers, and then solving special cases efficiently with wavelet-tree based data structures. The paper further extends these ideas to multiple input strings (up to $\lambda=O(\sqrt{\log n}/\log\log n)$) and develops a general k-LCS framework that breaks the longstanding $n\log^k n$ barrier, underpinned by bicomplete/complete families and advanced data-structural techniques. Overall, the results push the practical limits of sublinear LCS computations in compressed/packed representations, with conditional lower bounds indicating the near-optimality of the approach for general alphabets.

Abstract

In the classic longest common substring (LCS) problem, we are given two strings $S$ and $T$, each of length at most $n$, over an alphabet of size $σ$, 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 $O(n \log σ)$-time algorithm for this problem [SWAT 1973]. For polynomially-bounded integer alphabets, the linear-time construction of suffix trees by Farach yielded an $O(n)$-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 $O(n \log σ/\log n )$ space and read in $O(n \log σ/\log n )$ time. We show that, in this model, we can compute an LCS in time $O(n \log σ/ \sqrt{\log n})$, which is sublinear in $n$ if $σ=2^{o(\sqrt{\log n})}$ (in particular, if $σ=O(1)$), using optimal space $O(n \log σ/\log n)$. In fact, it was recently shown that this result is conditionally optimal [Kempa and Kociumaka, STOC 2025]. 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. Thankachan et al.~showed how to compute a $k$-mismatch LCS in $O(n \log^k n)$ time for $k=O(1)$ [J. Comput. Biol. 2016]. We show an $O(n \log^{k-1/2} n)$-time algorithm, for any constant $k>0$ and irrespective of the alphabet size, using $O(n)$ 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. [STOC 2004] for string indexing with $k$ errors.

Figures (9)

• Figure 1: Left: The LCS of the two strings has length 5. Right: The 1-LCS (1-mismatch LCS) of the same two strings has length 7; the mismatching letters are shown in red.
• Figure 2: An example of a $\tau$-synchronising set ($\tau=3$) of this string is $A=\{1,3,6,12,13,16\}$. Fragments $T[1 \mathinner{.\,.} 7)$ and $T[16 \mathinner{.\,.} 22)$ match and thus $1$, $16$ are both in $A$. Fragments $T[2 \mathinner{.\,.} 8)$ and $T[17 \mathinner{.\,.} 23)$ match and thus $2$, $17$ are both not in $A$. Among every 3 consecutive positions (that are sufficiently far from the end of the string) there is a synchronising position, except for the positions $7,\ldots,11$ which imply a long fragment with period $\frac{1}{3}\tau=1$ (a so-called $\tau$-run).
• Figure 3: Left: partitioning of $S$ and $T$ into fragments of length (up to) $2m$. All distinct substrings in $S$ and in $T$ are $X_1,\ldots,X_4$ and $Y_1,\ldots,Y_5$, respectively. The LCS of length $m$ (in blue) is a substring of $X_1$ in $S$ and of $Y_2$ in $T$. Right: strings $X$ and $Y$; their LCS (shown in blue) is the same as the LCS of $S$ and $T$.
• Figure 4: Strings from \ref{['fig:0-1-LCS']} with their LCS $S[3 \mathinner{.\,.} 8)=T[4 \mathinner{.\,.} 9)$ of length $\ell=5$. The dots correspond to elements of a 5-cover $(D,h)$ where $D=\{1,2,4,\,6,7,9,\,11,12,14,\ldots\}$. For $i=3$, $j=4$, we have $i'=i+h(i,j)=6$, $j'=j+h(i,j)=7$, and $\mathsf{LCP}((S[1 \mathinner{.\,.} i'))^R,(T[1 \mathinner{.\,.} j'))^R)+\mathsf{LCP}(S[i' \mathinner{.\,.} |S|], T[j' \mathinner{.\,.} |T|])=\ell=5$.
• Figure 5: The setting in Lemma \ref{['lem:suffix']} on list $\mathcal{R}$. With red color, we denote the elements of $\mathcal{P}$ and with blue color the elements of $\mathcal{Q}$. For element $e=(U,V)$ from $\mathcal{Q}$, we have $\textsf{lex-pred}(e) = (Y_1,Y_2)$ and $\textsf{lex-succ}(e) = (Z_1,Z_2)$.

Theorems & Definitions (92)

• theorem 1.1
• theorem 1.2
• Lemma 1.3: DBLP:conf/cpm/Charalampopoulos18
• Example 1
• Lemma 1.3
• theorem 1.4
• Lemma 2.2: Periodicity Lemma (weak version) FW:periodicity-lemma
• Lemma 2.3
• Proposition 2.3
• theorem 2.4: DBLP:conf/stoc/KempaK19