arXiv subjects

Solon P. Pissis

Publications and source records attributed to Solon P. Pissis.

At least 19 recordsLinked to original sources

arXiv2026

ZigZag Trie: A Novel Index for Contextual Queries

There is increasing interest in queries about the context of a string $P$ in a longer text $T$, i.e., the set of all string pairs $(L,R)$, with $|L|=|R|=q$, for a given $q$, such that the string $LPR$ occurs in $T$. Such contextual queries are important in several domains but are challenging to answer efficiently. This is because the length of $T$ in applications is massive and existing indexes do not directly encode the context of a given $P$, which is key for answering retrieval queries efficiently. Our work introduces the ZigZag Trie (ZZT), a new full-text index to specifically address these challenges. This index reorganizes the text so that, for any $P$, all possible strings $L$ and $R$ growing symmetrically around $P$ are grouped into a common subtree of the index, allowing their efficient retrieval. We show how to construct the ZZT of $T$, which has size $\mathcal{O}(n)$ where $n=|T|$, in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(n)$ space. On top of ZZT, we design specialized indexes that, for a query pattern $P$, answer four new types of contextual queries: (I) finding the longest string $LPR$ that occurs at least $τ$ times in $T$, for a fixed $τ$; (II) finding the longest string $LPR$ that occurs in at least $τ$ texts of a text collection, for a fixed $τ$; (III) reporting the total number of distinct contexts of $P$ in $T$; and (IV) retrieving, for a given $q$, the $k$ pairs $(L,R)$ of $P$ with the highest scores according to a given scoring function. Our indexes answer queries of type I, II, and III in optimal time, and of type IV in near-optimal time. Moreover, their size, construction space, and construction time are linear or near-linear in $n$, given ZZT. Using real billion-letter datasets, we show that our indexes answer queries orders of magnitude faster than baselines and perform similarly or better in index size and construction space and time.

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arXiv2026

Text Indexing: From Reporting to Counting

We prove an elementary yet powerful combinatorial lemma: in any rooted tree with $L$ leaves, the number of nodes whose depth is smaller than the number of their leaf descendants is at most $L$. For any string $T$ of length $n$, a direct application of this lemma to the suffix trie of $T$ yields that the number of substrings of $T$ whose length is smaller than their number of occurrences in $T$ is at most $n$. This combinatorial insight leads to space-efficient data structures with optimal query times for string counting problems via the following algorithmic framework: store the counts for the at most $n$ ``frequent'' substrings of $T$ in a preprocessing step, and use a reporting query to count for the ``infrequent'' substrings. Our framework acts as a convenient black box, lifting indexes with reporting time $\mathcal{O}(|P|+|\textsf{Occ}_T(P)|)$ to support counting queries in time $\mathcal{O}(|P|)$, where $P$ is the queried pattern and $\textsf{Occ}_T(P)$ is the set of occurrences of $P$ in $T$. As applications, we show efficient indexes for consecutive occurrences, weighted sequences, strings with utilities, and non-overlapping occurrences.

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arXiv2026

Optimal Enumeration of Eulerian Trails in Directed Graphs

The BEST theorem, due to de Bruijn, van Aardenne-Ehrenfest, Smith, and Tutte, is a classical tool from graph theory that links the Eulerian trails in a directed graph $G=(V,E)$ with the arborescences in $G$. In particular, one can use the BEST theorem to count the Eulerian trails in $G$ in polynomial time. For enumerating the Eulerian trails in $G$, one could naturally resort to first enumerating the arborescences in $G$ and then exploiting the insight of the BEST theorem to enumerate the Eulerian trails in $G$: every arborescence in $G$ corresponds to at least one Eulerian trail in $G$. For over two decades, the fastest algorithm for enumerating arborescences in $G$ took $O(m\log n + n + z_A \log^2 n)$ time, where $n=|V|$, $m=|E|$, and $z_A$ is the number of arborescences in $G$ [Uno, ISAAC 1998]. Since Uno's algorithm does not lead to an optimal enumeration of Eulerian trails in directed graphs, we were motivated to develop a direct algorithm for this problem. Our central contribution is a remarkably simple algorithm to directly enumerate the $z_T$ Eulerian trails in $G$ in the optimal $O(m + z_T)$ time. As a consequence, our result improves on an implementation of the BEST theorem for counting Eulerian trails in $G$ when $z_T=o(n^2)$, and also unconditionally improves the combinatorial $O(m\cdot z_T)$-time algorithm of Conte et al. [FCT 2021] for the same task. Moreover, we show that, with some care, our algorithm can be extended to enumerate Eulerian trails in directed multigraphs in optimal time, enabling applications in bioinformatics and data privacy.

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arXiv2026

String Matching in (Block) Graphs: A Full Classification by Walk Length

We consider directed graphs in which the nodes are labeled with strings. A walk in such a graph naturally corresponds to the concatenation of the visited nodes' labels. These graphs are widely used in bioinformatics to compactly describe large collections of highly similar genomes. Given such a graph $G=(V,E)$ and a pattern of length $m$, we seek a walk whose corresponding string has an occurrence of the pattern. We call this the SMLG problem. Amir et al. [J. Algorithms, 2000] showed that SMLG can be solved in $\mathcal{O}(m|E| + N)$ time, where $N$ is the total length of all node labels. Equi et al. [ACM Trans. Algorithms, 2023] showed that this is essentially optimal (under SETH). The existing lower bound assumes that the sought walk is of length $Θ(|V|)$. Thus, we might be able to bypass this lower bound by restricting the walk length to $b-1$, which naturally reduces to having as input a directed graph whose set of nodes is partitioned into $b$ blocks. Then, we seek a walk in this graph that starts in the first block and ends in the last block. We call this the $b$-SMBG problem. We provide a more fine-grained classification that essentially settles the complexity of $b$-SMBG parameterized by $b$: (1) We give a near-linear-time algorithm for $b=3$. (2) We show that there is no combinatorial algorithm improving over the state-of-the-art $\mathcal{O}(m|E| + N)$ bound for any $b\ge 4$. (3) We also present a fast matrix multiplication-based algorithm yielding an improvement for $b \in \mathcal{O}(1)$, which is conditionally optimal. (4) Finally, we show that under SETH, for any $b \in ω(\log |V|)$, no algorithm can improve over the state of the art.

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arXiv2026

Faster Algorithms for Shortest Unique or Absent Substrings

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 $S$.

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arXiv2026

Subtree Mode and Applications

The mode of a collection of values (i.e., the most frequent value in the collection) is a key summary statistic. Finding the mode in a given range of an array of values is thus of great importance, and constructing a data structure to solve this problem is in fact the well-known Range Mode problem. In this work, we introduce the Subtree Mode (SM) problem, the analogous problem in a leaf-colored tree, where the task is to compute the most frequent color in the leaves of the subtree of a given node. SM is motivated by several applications in domains such as text analytics and biology, where the data are hierarchical and can thus be represented as a (leaf-colored) tree. Our central contribution is a time-optimal algorithm for SM that computes the answer for every node of an input $N$-node tree in $O(N)$ time. We further show how our solution can be adapted for node-colored trees, or for computing the $k$ most frequent colors, for any given $k=O(1)$, in the optimal $O(N)$ time. Moreover, we prove that a similarly fast solution for when the input is a sink-colored directed acyclic graph instead of a leaf-colored tree is highly unlikely. Our experiments on real datasets with trees of up to $7.3$ billion nodes demonstrate that our algorithm is faster than baselines by at least one order of magnitude and much more space efficient. They also show that it is effective in pattern mining, sequence-to-database search, and biology applications.

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arXiv2026

Text Indexing and Pattern Matching with Ephemeral Edits

A sequence $e_0,e_1,\ldots$ of edit operations in a string $T$ is called ephemeral if operation $e_i$ constructing string $T^i$, for all $i=2k$ with $k\in\mathbb{N}$, is reverted by operation $e_{i+1}$ that reconstructs $T$. Such a sequence arises when processing a stream of independent edits or testing hypothetical edits. We introduce text indexing with ephemeral substring edits, a new version of text indexing. Our goal is to design a data structure over a given text that supports subsequent pattern matching queries with ephemeral substring insertions, deletions, or substitutions in the text; we require insertions and substitutions to be of constant length. In particular, we preprocess a text $T=T[0\mathinner{.\,.} n)$ over an integer alphabet $Σ=[0,σ)$ with $σ=n^{\mathcal{O}(1)}$ in $\mathcal{O}(n)$ time. Then, we can preprocess any arbitrary pattern $P=P[0\mathinner{.\,.} m)$ given online in $\mathcal{O}(m\log\log m)$ time and $\mathcal{O}(m)$ space and allow any ephemeral sequence of edit operations in $T$. Before reverting the $i$th operation, we report all Occ occurrences of $P$ in $T^i$ in $\mathcal{O}(\log\log n + \text{Occ})$ time. We also introduce pattern matching with ephemeral edits. In particular, we preprocess two strings $T$ and $P$, each of length at most $n$, over an integer alphabet $Σ=[0,σ)$ with $σ=n^{\mathcal{O}(1)}$ in $\mathcal{O}(n)$ time. Then, we allow any ephemeral sequence of edit operations in $T$. Before reverting the $i$th operation, we report all Occ occurrences of $P$ in $T^i$ in the optimal $\mathcal{O}(\text{Occ})$ time. Along our way to this result, we also give an optimal solution for pattern matching with ephemeral block deletions.

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arXiv2025

Faster Algorithms for Longest Common Substring

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.

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arXiv2025

Maximal Palindromes in MPC: Simple and Optimal

In the classical longest palindromic substring (LPS) problem, we are given a string $S$ of length $n$, and the task is to output a longest palindromic substring in $S$. Gilbert, Hajiaghayi, Saleh, and Seddighin [SPAA 2023] showed how to solve the LPS problem in the Massively Parallel Computation (MPC) model in $\mathcal{O}(1)$ rounds using $\mathcal{\widetilde{O}}(n)$ total memory, with $\mathcal{\widetilde{O}}(n^{1-ε})$ memory per machine, for any $ε\in (0,0.5]$. We present a simple and optimal algorithm to solve the LPS problem in the MPC model in $\mathcal{O}(1)$ rounds. The total time and memory are $\mathcal{O}(n)$, with $\mathcal{O}(n^{1-ε})$ memory per machine, for any $ε\in (0,0.5]$. A key attribute of our algorithm is its ability to compute all maximal palindromes in the same complexities. Furthermore, our new insights allow us to bypass the constraint $ε\in (0,0.5]$ in the Adaptive MPC model. Our algorithms and the one proposed by Gilbert et al. for the LPS problem are randomized and succeed with high probability.

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arXiv2025

Subsequence Covers of Words

We introduce subsequence covers (s-covers, in short), a new type of covers of a word. A word $C$ is an s-cover of a word $S$ if the occurrences of $C$ in $S$ as subsequences cover all the positions in $S$. The s-covers seem to be computationally much harder than standard covers of words (cf. Apostolico et al., Inf. Process. Lett. 1991), but, on the other hand, much easier than the related shuffle powers (Warmuth and Haussler, J. Comput. Syst. Sci. 1984). We give a linear-time algorithm for testing if a candidate word $C$ is an s-cover of a word $S$ over a polynomially-bounded integer alphabet. We also give an algorithm for finding a shortest s-cover of a word $S$, which in the case of a constant-sized alphabet, also runs in linear time. The words without proper s-cover are called s-primitive. We complement our algorithmic results with explicit lower and an upper bound on the length of a longest s-primitive word. Both bounds are exponential in the size of the alphabet. The upper bound presented here improves the bound given in the conference version of this paper [SPIRE 2022].

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arXiv2025

When is String Reconstruction using de Bruijn Graphs Hard?

The reduction of the fragment assembly problem to (variations of) the classical Eulerian trail problem [Pevzner et al., PNAS 2001] has led to remarkable progress in genome assembly. This reduction employs the notion of de Bruijn graph $G=(V,E)$ of order $k$ over an alphabet $Σ$. A single Eulerian trail in $G$ represents a candidate genome reconstruction. Bernardini et al. have also introduced the complementary idea in data privacy [ALENEX 2020] based on $z$-anonymity. The pressing question is: How hard is it to reconstruct a best string from a de Bruijn graph given a function that models domain knowledge? Such a function maps every length-$k$ string to an interval of positions where it may occur in the reconstructed string. By the above reduction to de Bruijn graphs, the latter function translates into a function $c$ mapping every edge to an interval where it may occur in an Eulerian trail. This gives rise to the following basic problem on graphs: Given an instance $(G,c)$, can we efficiently compute an Eulerian trail respecting $c$? Hannenhalli et al.~[CABIOS 1996] formalized this problem and showed that it is NP-complete. We focus on parametrization aiming to capture the quality of our domain knowledge in the complexity. Ben-Dor et al. developed an algorithm to solve the problem on de Bruijn graphs in $O(m \cdot w^{1.5} 4^{w})$ time, where $m=|E|$ and $w$ is the maximum interval length over all edges. Bumpus and Meeks [Algorithmica 2023] rediscovered the same algorithm on temporal graphs, highlighting the relevance of this problem in other contexts. We give combinatorial insights that lead to exponential-time improvements over the state-of-the-art. For the important class of de Bruijn graphs, we develop an algorithm parametrized by $w (\log w+1) /(k-1)$. Our improved algorithm shows that it is enough when the range of positions is small relative to $k$.

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arXiv2025

Testing Quasiperiodicity

A cover (or quasiperiod) of a string $S$ is a shorter string $C$ such that every position of $S$ is contained in some occurrence of $C$ as a substring. The notion of covers was introduced by Apostolico and Ehrenfeucht over 30 years ago [Theor. Comput. Sci. 1993] and it has received significant attention from the combinatorial pattern matching community. In this note, we show how to efficiently test whether $S$ admits a cover. Our tester can also be translated into a streaming algorithm.

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arXiv2025

Contextual Pattern Mining and Counting

Given a string $P$ of length $m$, a longer string $T$ of length $n>m$, and two integers $l\geq 0$ and $r\geq 0$, the context of $P$ in $T$ is the set of all string pairs $(L,R)$, with $|L|=l$ and $|R|=r$, such that the string $LPR$ occurs in $T$. We introduce two problems related to the notion of context: (1) the Contextual Pattern Mining (CPM) problem, which given $T$, $(m,l,r)$, and an integer $τ>0$, asks for outputting the context of each substring $P$ of length $m$ of $T$, provided that the size of the context of $P$ is at least $τ$; and (2) the Contextual Pattern Counting (CPC) problem, which asks for preprocessing $T$ so that the size of the context of a given query string $P$ of length $m$ can be found efficiently. For CPM, we propose a linear-work algorithm that either uses only internal memory, or a bounded amount of internal memory and external memory, which allows much larger datasets to be handled. For CPC, we propose an $\widetilde{\mathcal{O}}(n)$-space index that can be constructed in $\widetilde{\mathcal{O}}n)$ time and answers queries in $\mathcal{O}(m)+\widetilde{\mathcal{O}}(1)$ time. We further improve the practical performance of the CPC index by optimizations that exploit the LZ77 factorization of $T$ and an upper bound on the query length. Using billion-letter datasets from different domains, we show that the external memory version of our CPM algorithm can deal with very large datasets using a small amount of internal memory while its runtime is comparable to that of the internal memory version. Interestingly, we also show that our optimized index for CPC outperforms an approach based on the state of the art for the reporting version of CPC [Navarro, SPIRE 2020] in terms of query time, index size, construction time, and construction space, often by more than an order of magnitude.

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arXiv2025

Approximate Circular Pattern Matching

We consider approximate circular pattern matching (CPM, in short) under the Hamming and edit distance, in which we are given a length-$n$ text $T$, a length-$m$ pattern $P$, and a threshold $k>0$, and we are to report all starting positions of fragments of $T$ (called occurrences) that are at distance at most $k$ from some cyclic rotation of $P$. In the decision version of the problem, we are to check if any such occurrence exists. All previous results for approximate CPM were either average-case upper bounds or heuristics, except for the work of Charalampopoulos et al. [CKP$^+$, JCSS'21], who considered only the Hamming distance. For the reporting version of the approximate CPM problem, under the Hamming distance we improve upon the main algorithm of [CKP$^+$, JCSS'21] from ${\cal O}(n+(n/m)\cdot k^4)$ to ${\cal O}(n+(n/m)\cdot k^3)$ time; for the edit distance, we give an ${\cal O}(nk^2)$-time algorithm. We also consider the decision version of the approximate CPM problem. Under the Hamming distance, we obtain an ${\cal O}(n+(n/m)\cdot k^2\log k/\log\log k)$-time algorithm, which nearly matches the algorithm by Chan et al. [CGKKP, STOC'20] for the standard counterpart of the problem. Under the edit distance, the ${\cal O}(nk\log^2 k)$ running time of our algorithm nearly matches the ${\cal O}(nk)$ running time of the Landau-Vishkin algorithm [LV, J. Algorithms'89]. As a stepping stone, we propose an ${\cal O}(nk\log^2 k)$-time algorithm for the Longest Prefix $k'$-Approximate Match problem, proposed by Landau et al. [LMS, SICOMP'98], for all $k'\in \{1,\dots,k\}$. We give a conditional lower bound that suggests a polynomial separation between approximate CPM under the Hamming distance over the binary alphabet and its non-circular counterpart. We also show that a strongly subquadratic-time algorithm for the decision version of approximate CPM under edit distance would refute SETH.

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arXiv2025

Resilient Pattern Mining

Frequent pattern mining is a flagship problem in data mining. In its most basic form, it asks for the set of substrings of a given string $S$ of length $n$ that occur at least $τ$ times in $S$, for some integer $τ\in[1,n]$. We introduce a resilient version of this classic problem, which we term the $(τ, k)$-Resilient Pattern Mining (RPM) problem. Given a string $S$ of length $n$ and two integers $τ, k\in[1,n]$, RPM asks for the set of substrings of $S$ that occur at least $τ$ times in $S$, even when the letters at any $k$ positions of $S$ are substituted by other letters. Unlike frequent substrings, resilient ones account for the fact that changes to string $S$ are often expensive to handle or are unknown. We propose an exact $\mathcal{O}(n\log n)$-time and $\mathcal{O}(n)$-space algorithm for RPM, which employs advanced data structures and combinatorial insights. We then present experiments on real large-scale datasets from different domains demonstrating that: (I) The notion of resilient substrings is useful in analyzing genomic data and is more powerful than that of frequent substrings, in scenarios where resilience is required, such as in the case of versioned datasets; (II) Our algorithm is several orders of magnitude faster and more space-efficient than a baseline algorithm that is based on dynamic programming; and (III) Clustering based on resilient substrings is effective.

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arXiv2025

U-index: A Universal Indexing Framework for Matching Long Patterns

Text indexing is a fundamental and well-studied problem. Classic solutions either replace the original text with a compressed representation, e.g., the FM-index and its variants, or keep it uncompressed but attach some redundancy - an index - to accelerate matching. The former solutions thus retain excellent compressed space, but are slow in practice. The latter approaches, like the suffix array, instead sacrifice space for speed. We show that efficient text indexing can be achieved using just a small extra space on top of the original text, provided that the query patterns are sufficiently long. More specifically, we develop a new indexing paradigm in which a sketch of a query pattern is first matched against a sketch of the text. Once candidate matches are retrieved, they are verified using the original text. This paradigm is thus universal in the sense that it allows us to use any solution to index the sketched text, like a suffix array, FM-index, or r-index. We explore both the theory and the practice of this universal framework. With an extensive experimental analysis, we show that, surprisingly, universal indexes can be constructed much faster than their unsketched counterparts and take a fraction of the space, as a direct consequence of (i) having a lower bound on the length of patterns and (ii) working in sketch space. Furthermore, these data structures have the potential of retaining or even improving query time, because matching against the sketched text is faster and verifying candidates can be theoretically done in constant time per occurrence (or, in practice, by short and cache-friendly scans of the text). Finally, we discuss some important applications of this novel indexing paradigm to computational biology. We hypothesize that such indexes will be particularly effective when the queries are sufficiently long, and so demonstrate applications in long-read mapping.

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arXiv2025

Faster ED-String Matching with $k$ Mismatches

We revisit the complexity of approximate pattern matching in an elastic-degenerate string. Such a string is a sequence of $n$ finite sets of strings of total length $N$, and compactly describes a collection of strings obtained by first choosing exactly one string in every set, and then concatenating them together. This is motivated by the need of storing a collection of highly similar DNA sequences. The basic algorithmic question on elastic-degenerate strings is pattern matching: given such an elastic-degenerate string and a standard pattern of length $m$, check if the pattern occurs in one of the strings in the described collection. Bernardini et al.~[SICOMP 2022] showed how to leverage fast matrix multiplication to obtain an $\tilde{\mathcal{O}}(nm^{ω-1})+\mathcal{O}(N)$-time complexity for this problem, where $w$ is the matrix multiplication exponent. However, the best result so far for finding occurrences with $k$ mismatches, where $k$ is a constant, is the $\tilde{\mathcal{O}}(nm^{2}+N)$-time algorithm of Pissis et al.~[CPM 2025]. This brings the question whether increasing the dependency on $m$ from $m^{ω-1}$ to quadratic is necessary when moving from $k=0$ to larger (but still constant) $k$. We design an $\tilde{\mathcal{O}}(nm^{1.5}+N)$-time algorithm for pattern matching with $k$ mismatches in an elastic-degenerate string, for any constant $k$. To obtain this time bound, we leverage the structural characterization of occurrences with $k$ mismatches of Charalampopoulos et al.~[FOCS 2020] together with fast Fourier transform. We need to work with multiple patterns at the same time, instead of a single pattern, which requires refining the original characterization. This might be of independent interest.

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arXiv2025

Minimizers in Semi-Dynamic Strings

Minimizers sampling is one of the most widely-used mechanisms for sampling strings. Let $S=S[0]\ldots S[n-1]$ be a string over an alphabet $Σ$. In addition, let $w\geq 2$ and $k\geq 1$ be two integers and $ρ=(Σ^k,\leq)$ be a total order on $Σ^k$. The minimizer of window $X=S[i\mathinner{.\,.} i+w+k-2]$ is the smallest position in $[i,i+w-1]$ where the smallest length-$k$ substring of $S[i\mathinner{.\,.} i+w+k-2]$ based on $ρ$ starts. The set of minimizers for all $i\in[0,n-w-k+1]$ is the set $\mathcal{M}_{w,k,ρ}(S)$ of the minimizers of $S$. The set $\mathcal{M}_{w,k,ρ}(S)$ can be computed in $\mathcal{O}(n)$ time. The folklore algorithm for this computation computes the minimizer of every window in $\mathcal{O}(1)$ amortized time using $\mathcal{O}(w)$ working space. It is thus natural to pose the following two questions: Question 1: Can we efficiently support other dynamic updates on the window? Question 2: Can we improve on the $\mathcal{O}(w)$ working space? We answer both questions in the affirmative: 1. We term a string $X$ semi-dynamic when one is allowed to insert or delete a letter at any of its ends. We show a data structure that maintains a semi-dynamic string $X$ and supports minimizer queries in $X$ in $\mathcal{O}(1)$ time with amortized $\mathcal{O}(1)$ time per update operation. 2. We show that this data structure can be modified to occupy strongly sublinear space without increasing the asymptotic complexity of its operations. To the best of our knowledge, this yields the first algorithm for computing $\mathcal{M}_{w,k,ρ}(S)$ in $\mathcal{O}(n)$ time using $\mathcal{O}(\sqrt{w})$ working space. We complement our theoretical results with a concrete application and an experimental evaluation.

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