enumerate maximal closed substrings

This study addresses the problem of efficiently enumerating all maximal closed substrings (MCS) in run-length encoded (RLE) strings. The authors propose a unified approach that operates directly on the RLE representation, leveraging a compact family-based encoding to handle both periodic and aperiodic cases efficiently. Their key contributions include the first algorithm for directly enumerating MCS from RLE without decompression, a proof that all MCS can be fully represented using only O(m²) families—with this bound shown to be tight in certain instances—and an output-sensitive enumeration algorithm. By integrating techniques such as sparse suffix trees, height-driven three-sided range reporting, and McCreight’s balanced priority search trees, the method outputs a compact representation of all MCS in O(m log²m + |F| log m) time and O(m) space, where m is the length of the RLE string and |F| denotes the number of output families.

This study addresses the problem of efficiently computing all maximal closed substrings (MCS) during the online, character-by-character input of a string. To this end, the authors propose a novel data structure—the Link-Cut Suffix Tree (LCST)—which integrates an online suffix tree with a link-cut tree to dynamically maintain the rightmost occurrence information of every substring. This enables real-time detection of newly formed MCS after each character insertion. Based on the LCST, they design the first worst-case time-optimal online algorithm for MCS enumeration, achieving a total time complexity of $O(n \log n)$ and space complexity of $O(n)$. The approach further extends to related applications such as rightmost LZ77 factorization and recent match queries.

This work addresses the efficient enumeration of closed substrings and their maximal variants (maximal closed substrings, MCSs) in a string. We present the first algorithm achieving $O(n log n)$ time and space complexity for enumerating all $Theta(n^2)$ closed substrings. For MCS extraction, we propose a lightweight method based on the suffix array and LCP array, ensuring both theoretical optimality and practical efficiency. To mitigate output explosion, we introduce a compact representation that significantly reduces output size. Furthermore, we fully characterize the asymptotic behavior of MCS counts in Fibonacci words, deriving the exact asymptotic formula $sim 1.382,F_n$. This constitutes the first near-linear-time algorithm for complete closed substring enumeration.

Efficient enumeration and analysis of context-sensitive repeats in strings—such as maximal repeats, super-/near-super-maximal repeats—remain computationally challenging. Method: We propose a linear-time algorithm based on the run-length compressed Burrows–Wheeler transform (RLBWT), reducing the r-enum complexity from $O(n log log_w (n/r))$ to $O(n)$. Our approach enables, for the first time, joint computation of near-super-maximal repeats along with their net occurrences and net frequencies. We prove that the total number of net occurrences is strictly less than $2r$, and construct an $O(r)$-space data structure supporting efficient net-frequency queries for arbitrary patterns. Contributions/Results: (i) Full context-sensitive repeat enumeration in $O(n)$ time and $O(r)$ space; (ii) a new upper bound of $2r$ on the number of minimal unique substrings; (iii) a unified solution to key problems—including context diversity quantification of maximal repeats, net-frequency statistics, and dynamic net-frequency querying.

This paper studies the Longest Common Extension (LCE) problem for strings with wildcards, where each wildcard matches any character. Parameterizing by the number $G$ of wildcard groups—a natural structural parameter—we present the first smooth time–space trade-off LCE data structure: it supports $O(t)$-time queries using $O(nG/t)$ space and $O(n(G/t)log n)$ preprocessing time. Technically, our approach integrates group-parameterized design, kangaroo jumping, and sparse Boolean matrix multiplication, and establishes a tight reduction to Boolean matrix multiplication. Under the 3SUM and Set-Disjointness conjectures, we prove the conditional optimality of this trade-off. Our result yields the first combinatorial, parameter-sensitive, and asymptotically optimal primitive for approximate pattern matching and structural analysis of wildcard strings.

This study resolves a long-standing open problem concerning the attainability of the string repetitiveness measure χ(w): whether there always exists a string representation of size O(χ(w)). To this end, we introduce the Substring Equation System (SES), a novel theoretical framework, and combine it with combinatorial structures such as suffix-complete sets to construct the first compression scheme capable of representing any string w within O(χ(w)) space. Our work not only establishes, for the first time, the attainability of the χ measure but also provides a new modeling paradigm for string compression that leverages structural regularities in repetitive strings.

This study addresses the problem of efficiently identifying the longest palindromic substring in strings containing wildcards, where each wildcard can match any character and thereby disrupts the symmetry exploited by classical palindrome-finding algorithms. To overcome this challenge, we present the first non-trivial algorithm that operates within linear space, leveraging a novel wildcard-based longest common extension (wildcard-LCE) technique. Our approach establishes a continuous trade-off between time and memory usage, achieving substantial improvements in the time–memory product across a range of parameter settings. Furthermore, the method generalizes effectively to the more challenging setting allowing up to k mismatches, yielding both theoretical advances and practical performance gains.

We revisit two well-known algorithmic problems on strings: computing a shortest unique substring (SUS) and a shortest absent substring (SAS) of a string $S$ of length $n$. Both problems admit folklore $\mathcal{O}(n)$-time solutions using the suffix tree of $S$. However, for small alphabets, this complexity is not necessarily optimal in the word RAM model, where a string of length $n$ over alphabet $[0,σ)$ can be stored in $\mathcal{O}(n \log σ/\log n)$ space and read in $\mathcal{O}(n \log σ/\log n)$ time. We present an $\mathcal{O}(n \log σ/\sqrt{\log n})$-time algorithm for computing a SUS of $S$. This algorithm decomposes the problem according to the length and the period of the sought substring and uses several tools and techniques, such as synchronizing sets, the analysis of runs, and wavelet trees, to reduce the computation of a SUS to a simple geometric problem. Further, we adapt this algorithm and combine it with an efficient construction of de Bruijn sequences in order to obtain an $\mathcal{O}(n \log σ/\sqrt{\log n})$-time algorithm for computing a SAS of $S$.

This study investigates the instability of mainstream repetitiveness measures—such as the number of runs in the Burrows–Wheeler Transform (\(r\)), the size of the LZ77 parsing (\(z\)), and the LZ-end size (\(v\))—under string reversal. By constructing specific families of strings and employing combinatorial analysis, asymptotic methods, and information-theoretic tools, the authors establish the first tight additive sensitivity bound of \(\Theta(n)\) for \(r\) under reversal, substantially improving the previous \(\Omega(\log n)\) lower bound. They also demonstrate that the ratio \(z(w^R)/z(w)\) can approach 3 and that the multiplicative sensitivity of \(v\) is \(\Theta(\log n)\). All derived bounds are asymptotically tight, revealing the pronounced vulnerability of these measures to such a simple transformation.

This work addresses the problem of efficiently supporting substring frequency queries—returning occurrence counts rather than explicit positions. It introduces the first black-box framework that automatically transforms any reporting-based text index into a counting-based one. The approach leverages combinatorial lemmas to characterize the relationship between substring frequencies and lengths, precomputing frequencies for at most $n$ high-frequency substrings while handling low-frequency ones by converting reports from an existing index into counts. Requiring only linear space, the method achieves optimal $O(|P|)$ query time for a pattern $P$, and seamlessly extends to diverse settings—including consecutive occurrences, weighted sequences, strings with utilities, and non-overlapping matches—while maintaining optimal time and space efficiency in all cases.