In this work, we study periodicity in strings with wildcards. A string T with at most k wildcards is called strongly (p,k)-periodic if the wildcards in T can be replaced with alphabet symbols to obtain a string with period p, and weakly (p,k)-periodic if T[i] matches T[i+p] for all i. Intuitively, both generalize to (≤ g, k)-periodicity, which is the property of being (p,k)-periodic for some p ∈ [1..g]. An ε-tester for a property 𝒫 is an algorithm that distinguishes between strings that satisfy 𝒫 and strings where one needs to change at least an ε-fraction of the symbols to obtain a string that satisfies 𝒫. We study one-sided error testers, where strings satisfying 𝒫 must always be accepted, while strings that are ε-far must be rejected with probability at least 2/3. The complexity of a tester is the worst-case number of symbols of an input of length n it must read to make the decision. We design the following testers for p,g ≤ n/2: 1) An ε-tester for strong (p,k)-periodicity with complexity Õ_ε(1) . 2) An ε-tester for strong (≤ g,k)-periodicity with complexity Õ_ε(√g). 3) An ε-tester for weak (p,k)-periodicity with complexity Õ_ε(min(k, n /(k+p))). 4) An ε-tester for weak (≤ g,k)-periodicity with complexity Õ_ε(min(k+ √{gk}, n/√k)). Additionally, we show a lower bound on the complexity of ε-testers for weak (≤ g,k)-periodicity, implying that our tester for weak (≤ g,k)-periodicity is optimal up to a multiplicative (ε^{-1} ln(gk))^O(1) factor for a wide range of g and k. Finally, our tester for strong (≤ g,k)-periodicity generalizes the one of [Lachish and Newman; Algorithmica 2011] for strings without wildcards, matching (up to polylogarithmic factors) the unconditional lower bound of ̃Ω(√g) in said work for constant ε.

We investigate the asymptotic growth of function CS(n), which maps n to the maximum number of distinct squares in a circular word of length n (that is, the maximum number of distinct squares of length at most n in a word ww of length 2n). We improve upon the lower bound of 1.25n established by Amit and Gawrychowski [SPIRE 2017] and the straightforward upper bound of 2n, which follows from the recent result of Brlek and Li [Comb. Theory, 2025] stating that there are fewer than n squares in standard (i.e., non-circular) words of length n. (Previously, Amit and Gawrychowski gave an upper bound of 32/15n using a weaker upper bound on squares in standard words.) Specifically, we show that CS(n) ≤ ⌈1.8 n⌉ and that, for infinitely many n, CS(n) ≥ 1.5n-𝒪(√n). For the lower bound, we exploit the combinatorial structure of Fibonacci words to construct a family of square-rich circular words. For the upper bound, we exploit density properties of the starting positions of long squares, adapting an approach of Amit and Gawrychowski.

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 𝒪(|P|+|Occ_T(P)|) to support counting queries in time 𝒪(|P|), where P is the queried pattern and 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.

In approximate circular pattern matching the goal is to compute all approximate occurrences of all rotations of a pattern P in a text T. We study this problem under the two most fundamental string distance metrics, the Hamming distance and the edit distance, in the setting where the text arrives online and the available space is limited. Specifically, we wish to report each ending position j of an approximate occurrence before symbol T[j+1] arrives, using sublinear space on top of having read-only access to P and (the seen prefix of) T. For both variants, we present algorithms that use O(poly(k)) extra space and process each arriving symbol in O(poly(k)) time. Notably, with an overhead, our algorithms can be lifted to the asymmetric streaming setting, where we only have read-only access to the pattern for free and account for all extra space.

Many string processing problems can be phrased in the streaming setting, where the input arrives symbol by symbol and we have sublinear working space. The area of streaming algorithms for string processing has flourished since the seminal work of Porat and Porat [FOCS 2009]. Unfortunately, problems with efficient solutions in the classical setting often do not admit efficient solutions in the streaming setting. As a bridge between these two settings, Saks and Seshadhri [SODA 2013] introduced the asymmetric streaming model (see also [Andoni, Krauthgamer, and Onak; FOCS 2010]). Here, one is given read-only access to a (typically short) reference string R of length m, while a (typically long) text T arrives as a stream. We provide a generic technique to reduce fundamental string problems in the asymmetric streaming model to the online read-only model, lifting several existing algorithms and generally improving upon the state of the art. Most notably, we obtain asymmetric streaming algorithms for exact and approximate pattern matching (under both the Hamming and edit distances), and for relative Lempel-Ziv compression, a popular scheme for measuring and exploiting redundancy in repetitive text collections. At the heart of our approach lies a novel tool that facilitates efficient computation in the asymmetric streaming model: the suffix random access data structure. In its simplest variant, it maintains constant-time random access to the longest suffix of (the seen prefix of) T that occurs in R. Let τ be a parameter that denotes the size of the data structure. A straightforward approach maintains the data structure in {O}(m/τ) time per arriving symbol of T. We drastically improve this tradeoff and reveal fundamental barriers via a bidirectional reduction between suffix random access and function inversion, a central problem in cryptography: - By leveraging Fiat and Naor’s function inversion data structure [SIAM J. Comput. 2000], we achieve Õ(1+m³/τ⁶) update time. In particular, for τ = √m, we obtain Õ(1) update time, improving over the Ω(√m) bound of the straightforward solution. - We establish an unconditional Ω̃(m/τ³) lower bound on the update time. Additionally, we show that achieving update time o(m³/τ⁷) would imply a breakthrough in function inversion. On the way to our upper bound, we propose a variant of the string synchronizing sets ([Kempa and Kociumaka; STOC 2019]) with a local sparsity condition that, as we show, admits an efficient streaming construction algorithm. We believe that our framework and techniques will find broad applications in the development of small-space string algorithms.

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.

The Cartesian tree of a sequence captures the relative order of the sequence’s elements. In recent years, Cartesian tree matching has attracted considerable attention, particularly due to its applications in time series analysis. Consider a text T of length n and a pattern P of length m. In the exact Cartesian tree matching problem, the task is to find all length-m fragments of T whose Cartesian tree coincides with the Cartesian tree CT(P) of the pattern. Although the exact version of the problem can be solved in linear time [Park et al., TCS 2020], it remains rather restrictive; for example, it is not robust to outliers in the pattern. To overcome this limitation, we consider the approximate setting, where the goal is to identify all fragments of T that are close to some string whose Cartesian tree matches CT(P). In this work, we quantify closeness via the widely used Hamming distance metric. For a given integer parameter k > 0, we present an algorithm that computes all fragments of T that are at Hamming distance at most k from a string whose Cartesian tree matches CT(P). Our algorithm runs in time 𝒪(n √m ⋅ k^{2.5}) for k ≤ m^{1/5} and in time 𝒪(nk⁵) for k ≥ m^{1/5}, thereby improving upon the state-of-the-art 𝒪(nmk)-time algorithm of Kim and Han [TCS 2025] in the regime k = o(m^{1/4}). On the way to our solution, we develop a toolbox of independent interest. First, we introduce a new notion of periodicity in Cartesian trees. Then, we lift multiple well-known combinatorial and algorithmic results for string matching and periodicity in strings to Cartesian tree matching and periodicity in Cartesian trees.

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 this 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. showed how 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 $k>0$ 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. [STOC 2004] for string indexing with $k$ errors [STOC 2004].

In Pattern Matching with Weighted Edits (PMwWE), we are given a pattern P of length m, a text T of length n, a positive threshold k, and oracle access to a weight function that specifies the costs of edits (depending on the involved characters, and normalized so that the cost of each edit is at least 1). The goal is to compute the starting positions of all fragments of T that can be obtained from P with edits of total cost at most k. PMwWE captures typical real-world applications more accurately than its unweighted variant (PMwE), where all edits have unit costs.Indeed, the textbook $\mathcal{O}\left( {nm} \right)$-time algorithm of Sellers [J. Algorithms'80], devised in the context of bioinformatics, already accounts for weights. Surprisingly, the understanding of PMWWE has not advanced in the last 45 years. In contrast, significant milestones for PMwE include an $\mathcal{O}\left( {nk} \right)$-time algorithm by Landau and Vishkin [STOC'86, J. Algorithms'89], an $\mathcal{O}(n + {k^4} \cdot n/m)$-time algorithm by Cole and Hariharan [SODA'98, SICOMP'02], and a recent $\tilde {\mathcal{O}}(n + {k^{3.5}} \cdot n/m)$-time solution by Charalampopoulos, Kociumaka, and Wellnitz [FOCS'22].In this work, we examine whether these results can be lifted to PMWWE even though (1) the underlying algorithms rely on combinatorial properties specific to the unweighted edit distance, and (2) under standard fine-grained complexity assumptions, computing the weighted edit distance is strictly harder than computing the unweighted edit distance [Cassis, Kociumaka, and Wellnitz; FOCS'23]. We obtain three main results:•a conceptually simple $\tilde {\mathcal{O}}\left( {nk} \right)$-time algorithm for PMWWE, very different from that of Landau and Vishkin;•a significantly more complicated $\tilde {\mathcal{O}}(n + {k^{3.5}} \cdot {W^4} \cdot n/m)$ time algorithm for PMWWE under the assumption that the weight function is a metric with integer values between 0 and W; and•an $\tilde {\mathcal{O}}(n + {k^4} \cdot n/m)$-time algorithm for PMWWE for the case of arbitrary weights. In the setting of metrics with small integer values, we nearly match the state of the art for PMwE where W=1.

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$\tau$times in$S$, for some integer$\tau\epsilon[1,n]$. We introduce a resilient version of this classic problem, which we term the$(\tau,\ k)$-Resilient Pattern Mining (rpm) problem. Given a string$S$of length$n$and two integers$\tau, k\in[1, n\vert$, RPM asks for the set of substrings of$S$that occur at least$\tau$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 make the following contributions. First, we present RPM-DP, a simple exact$\mathrm{O}(n^{{3}}k\log n)$-time and$\mathrm{O}(n^{2})$-space algorithm for RPM that is based on an existing dynamic programming algorithm. Second, we propose RPM-ESA, an exact$\mathrm{O}(n\log n)$-time and$\mathrm{O}(n)$-space algorithm for RPM, which employs advanced data structures and combinatorial insights. Third, we conduct experiments on real large-scale datasets from different domains demonstrating that: (I) The notion of resilient substrings is useful in analyzing genomic data and fundamentally different from that of frequent substrings, as frequent substrings are often not resilient and thus do not remain frequent for long in versioned datasets; (II) RPM-ESA is several orders of magnitude faster and more space-efficient than RPM-DP; and (III) Clustering based on resilient substrings is effective.

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. (1991) [1] ), but, on the other hand, much easier than the related shuffle powers (Warmuth and Haussler (1984) [6] ). 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].

A Longest Common Extension (LCE) query on a text $T$ of length $N$ asks for the length of the longest common prefix of suffixes starting at given two positions. We show that the signature encoding $\mathcal{G}$ of size $w = O(\min(z \log N \log^* M, N))$ [Mehlhorn et al., Algorithmica 17(2):183-198, 1997] of $T$, which can be seen as a compressed representation of $T$, has a capability to support LCE queries in $O(\log N + \log \ell \log^* M)$ time, where $\ell$ is the answer to the query, $z$ is the size of the Lempel-Ziv77 (LZ77) factorization of $T$, and $M \geq 4N$ is an integer that can be handled in constant time under word RAM model. In compressed space, this is the fastest deterministic LCE data structure in many cases. Moreover, $\mathcal{G}$ can be enhanced to support efficient update operations: After processing $\mathcal{G}$ in $O(w f_{\mathcal{A}})$ time, we can insert/delete any (sub)string of length $y$ into/from an arbitrary position of $T$ in $O((y+ \log N\log^* M) f_{\mathcal{A}})$ time, where $f_{\mathcal{A}} = O(\min \{ \frac{\log\log M \log\log w}{\log\log\log M}, \sqrt{\frac{\log w}{\log\log w}} \})$. This yields the first fully dynamic LCE data structure. We also present efficient construction algorithms from various types of inputs: We can construct $\mathcal{G}$ in $O(N f_{\mathcal{A}})$ time from uncompressed string $T$; in $O(n \log\log n \log N \log^* M)$ time from grammar-compressed string $T$ represented by a straight-line program of size $n$; and in $O(z f_{\mathcal{A}} \log N \log^* M)$ time from LZ77-compressed string $T$ with $z$ factors. On top of the above contributions, we show several applications of our data structures which improve previous best known results on grammar-compressed string processing.

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International audience

In the $k$-Edit Circular Pattern Matching ($k$-Edit CPM) problem, we are given a length-$n$ text $T$, a length-$m$ pattern $P$, and a positive integer threshold $k$, and we are to report all starting positions of the substrings of $T$ that are at edit distance at most $k$ from some cyclic rotation of $P$. In the decision version of the problem, we are to check if any such substring exists. Very recently, Charalampopoulos et al. [ESA 2022] presented $O(nk^2)$-time and $O(nk \log^3 k)$-time solutions for the reporting and decision versions of $k$-Edit CPM, respectively. Here, we show that the reporting and decision versions of $k$-Edit CPM can be solved in $O(n+(n/m) k^6)$ time and $O(n+(n/m) k^5 \log^3 k)$ time, respectively, thus obtaining the first algorithms with a complexity of the type $O(n+(n/m) \mathrm{poly}(k))$ for this problem. Notably, our algorithms run in $O(n)$ time when $m=Ω(k^6)$ and are superior to the previous respective solutions when $m=ω(k^4)$. We provide a meta-algorithm that yields efficient algorithms in several other interesting settings, such as when the strings are given in a compressed form (as straight-line programs), when the strings are dynamic, or when we have a quantum computer. We obtain our solutions by exploiting the structure of approximate circular occurrences of $P$ in $T$, when $T$ is relatively short w.r.t. $P$. Roughly speaking, either the starting positions of approximate occurrences of rotations of $P$ form $O(k^4)$ intervals that can be computed efficiently, or some rotation of $P$ is almost periodic (is at a small edit distance from a string with small period). Dealing with the almost periodic case is the most technically demanding part of this work; we tackle it using properties of locked fragments (originating from [Cole and Hariharan, SICOMP 2002]).

Abstract Data masking is a common technique for sanitizing sensitive data maintained in database systems which is becoming increasingly important in various application areas, such as in record linkage of personal data. This work formalizes the Pattern Masking for Dictionary Matching (PMDM) problem: given a dictionary $$\mathscr {D}$$ D of d strings, each of length $$\ell $$ ℓ , a query string q of length $$\ell $$ ℓ , and a positive integer z, we are asked to compute a smallest set $$K\subseteq \{1,\ldots ,\ell \}$$ K ⊆ { 1 , … , ℓ } , so that if q[i] is replaced by a wildcard for all $$i\in K$$ i ∈ K , then q matches at least z strings from $$\mathscr {D}$$ D . Solving PMDM allows providing data utility guarantees as opposed to existing approaches. We first show, through a reduction from the well-known k-Clique problem, that a decision version of the PMDM problem is NP-complete, even for binary strings. We thus approach the problem from a more practical perspective. We show a combinatorial $$\mathscr {O}((d\ell )^{|K|/3}+d\ell )$$ O ( ( d ℓ ) | K | / 3 + d ℓ ) -time and $$\mathscr {O}(d\ell )$$ O ( d ℓ ) -space algorithm for PMDM for $$|K|=\mathscr {O}(1)$$ | K | = O ( 1 ) . In fact, we show that we cannot hope for a faster combinatorial algorithm, unless the combinatorial k-Clique hypothesis fails (Abboud et al. in SIAM J Comput 47:2527–2555, 2018; Lincoln et al., in: 29th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2018). Our combinatorial algorithm, executed with small |K|, is the backbone of a greedy heuristic that we propose. Our experiments on real-world and synthetic datasets show that our heuristic finds nearly-optimal solutions in practice and is also very efficient. We also generalize this algorithm for the problem of masking multiple query strings simultaneously so that every string has at least z matches in $$\mathscr {D}$$ D . PMDM can be viewed as a generalization of the decision version of the dictionary matching with mismatches problem: by querying a PMDM data structure with string q and $$z=1$$ z = 1 , one obtains the minimal number of mismatches of q with any string from $$\mathscr {D}$$ D . The query time or space of all known data structures for the more restricted problem of dictionary matching with at most k mismatches incurs some exponential factor with respect to k. A simple exact algorithm for PMDM runs in time $$\mathscr {O}(2^\ell d)$$ O ( 2 ℓ d ) . We present a data structure for PMDM that answers queries over $$\mathscr {D}$$ D in time $$\mathscr {O}(2^{\ell /2}(2^{\ell /2}+\tau )\ell )$$ O ( 2 ℓ / 2 ( 2 ℓ / 2 + τ ) ℓ ) and requires space

We consider the problem of preprocessing a weighted directed planar graph in order to quickly answer exact distance queries. The main tension in this problem is between space S and query time Q , and since the mid-1990s all results had polynomial time-space tradeoffs, e.g., Q = ~ Θ( n/√ S ) or Q = ~Θ( n 5/2 /S 3/2 ). In this article we show that there is no polynomial tradeoff between time and space and that it is possible to simultaneously achieve almost optimal space n 1+ o (1) and almost optimal query time n o (1) . More precisely, we achieve the following space-time tradeoffs: n 1+ o (1) space and log 2+ o (1) n query time, n log 2+ o (1) n space and n o (1) query time, n 4/3+ o (1) space and log 1+ o (1) n query time. We reduce a distance query to a variety of point location problems in additively weighted Voronoi diagrams and develop new algorithms for the point location problem itself using several partially persistent dynamic tree data structures.

A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings. We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting.

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.

We consider the approximate pattern matching problem under the edit distance. Given a text T of length n, a pattern P of length m, and a threshold k, the task is to find the starting positions of all substrings of T that can be transformed to P with at most k edits. More than 20 years ago, Cole and Hariharan [SODA’98, J. Comput.’02] gave an $\mathcal{O}(n+k^{4}\cdot n/m)$ time algorithm for this classic problem, and this runtime has not been improved since.Here, we present an algorithm that runs in time $\mathcal{O}\left(n+ k^{3.5}\sqrt{\log m\log k}\cdot n/m\right)$, thus breaking through this longstanding barrier. In the case where $n^{1/4+\varepsilon}\leq k\leq n^{2/5-\varepsilon}$ for some arbitrarily small positive constant $\varepsilon$, our algorithm improves over the state-of-the-art by polynomial factors: it is polynomially faster than both the algorithm of Cole and Hariharan and the classic $\mathcal{O}(kn)$-time algorithm of Landau and Vishkin [STOC’86, J. Algorithms’89].We observe that the bottleneck case of the alternative $\mathcal{O}(n+k^4 \cdot n / m$-time algorithm of Charalampopoulos, Kociumaka, and Wellnitz [FOCS’20] is when the text and the pattern are (almost) periodic. Our new algorithm reduces this case to a new Dynamic Puzzle Matching problem, which we solve by building on tools developed by Tiskin [SODA’10, Algorithmica’15] for the so-called seaweed monoid of permutation matrices. Our algorithm relies only on a small set of primitive operations on strings and thus also applies to the fully-compressed setting (where text and pattern are given as straight-line programs) and to the dynamic setting (where we maintain a collection of strings under creation, splitting, and concatenation), improving over the state of the art.

Abstract Sequence mappability is an important task in genome resequencing. In the (k, m)-mappability problem, for a given sequence T of length n, the goal is to compute a table whose ith entry is the number of indices $$j \ne i$$ j ≠ i such that the length-m substrings of T starting at positions i and j have at most k mismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of $$k=1$$ k = 1 . We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for $$k=O(1)$$ k = O ( 1 ) , works in $$O(n)$$ O ( n ) space and, with high probability, in $$O(n \cdot \min \{m^k,\log ^k n\})$$ O ( n · min { m k , log k n } ) time. Our algorithm requires a careful adaptation of the k-errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the all-pairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop $$O(n^2)$$ O ( n 2 ) -time algorithms to compute all (k, m)-mappability tables for a fixed m and all $$k\in \{0,\ldots ,m\}$$ k ∈ { 0 , … , m } or a fixed k and all $$m\in \{k,\ldots ,n\}$$ m ∈ { k , … , n } . Finally, we show that, for $$k,m = \Theta (\log n)$$ k , m = Θ ( log n ) , the (k, m)-mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018.

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u , a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u -to- v path that avoids all vertices in X . We propose oracles that can handle any number k of failures. We show several tradeoffs between space, query time, and preprocessing time. In particular, for a directed weighted planar graph with n vertices and any constant k , we show an Õ( n )-size, Õ(√ n )-query-time oracle. 1 We then present a space vs. query time tradeoff: for any q ε [ 1,√ n ], we propose an oracle of size n k+1+o(1) / q 2k that answers queries in Õ( q ) time. For single vertex failures ( k = 1), our n 2+o(1) / q 2 -size, Õ( q )-query-time oracle improves over the previously best known tradeoff of Baswana et al. SODA 2012 by polynomial factors for q ≥ n t , for any t ∈ (0,1/2]. For multiple failures, no planarity exploiting results were previously known.

Given a string T of length n over an alphabet Σ ⊂ {1,2,…,n^{𝒪(1)}} of size σ, we are to preprocess T so that given a range [i,j], we can return a representation of a shortest string over Σ that is absent in the fragment T[i]⋯ T[j] of T. For any positive integer k ∈ [1,log log_σ n], we present an 𝒪((n/k)⋅ log log_σ n)-size data structure, which can be constructed in 𝒪(nlog_σ n) time, and answers queries in time 𝒪(log log_σ k).

We consider two notions of covers of a two-dimensional string T. A (rectangular) subarray P of T is a 2D-cover of T if each position of T is in an occurrence of P in T. A one-dimensional string S is a 1D-cover of T if its vertical and horizontal occurrences in T cover all positions of T. We show how to compute the smallest-area 2D-cover of an m × n array T in the optimal 𝒪(N) time, where N = mn, all aperiodic 2D-covers of T in 𝒪(N log N) time, and all 2D-covers of T in N^{4/3}⋅ log^{𝒪(1)}N time. Further, we show how to compute all 1D-covers in the optimal 𝒪(N) time. Along the way, we show that the Klee’s measure of a set of rectangles, each of width and height at least √n, on an n × n grid can be maintained in √n⋅ log^{𝒪(1)}n time per insertion or deletion of a rectangle, a result which could be of independent interest.

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.

We consider the problem of preprocessing two strings S and T, of lengths m and n, respectively, in order to be able to efficiently answer the following queries: Given positions i,j in S and positions a,b in T, return the optimal alignment score of S[i..j] and T[a..b]. Let N = mn. We present an oracle with preprocessing time N^{1+o(1)} and space N^{1+o(1)} that answers queries in log^{2+o(1)}N time. In other words, we show that we can efficiently query for the alignment score of every pair of substrings after preprocessing the input for almost the same time it takes to compute just the alignment of S and T. Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS). The best previously known oracle with construction time and size 𝒪(N) has slow Ω(√N) query time [Sakai, TCS 2019], and the one with size N^{1+o(1)} and query time log^{2+o(1)}N (using a planar graph distance oracle) has slow Ω(N^{3/2}) construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a √ N factor.

Data masking is a common technique for sanitizing sensitive data maintained in database systems, and it is also becoming increasingly important in various application areas, such as in record linkage of personal data. This work formalizes the Pattern Masking for Dictionary Matching (PMDM) problem. In PMDM, we are given a dictionary 𝒟 of d strings, each of length 𝓁, a query string q of length 𝓁, and a positive integer z, and we are asked to compute a smallest set K ⊆ {1,…,𝓁}, so that if q[i] is replaced by a wildcard for all i ∈ K, then q matches at least z strings from 𝒟. Solving PMDM allows providing data utility guarantees as opposed to existing approaches. We first show, through a reduction from the well-known k-Clique problem, that a decision version of the PMDM problem is NP-complete, even for strings over a binary alphabet. We thus approach the problem from a more practical perspective. We show a combinatorial 𝒪((d𝓁)^{|K|/3}+d𝓁)-time and 𝒪(d𝓁)-space algorithm for PMDM for |K| = 𝒪(1). In fact, we show that we cannot hope for a faster combinatorial algorithm, unless the combinatorial k-Clique hypothesis fails [Abboud et al., SIAM J. Comput. 2018; Lincoln et al., SODA 2018]. We also generalize this algorithm for the problem of masking multiple query strings simultaneously so that every string has at least z matches in 𝒟. Note that PMDM can be viewed as a generalization of the decision version of the dictionary matching with mismatches problem: by querying a PMDM data structure with string q and z = 1, one obtains the minimal number of mismatches of q with any string from 𝒟. The query time or space of all known data structures for the more restricted problem of dictionary matching with at most k mismatches incurs some exponential factor with respect to k. A simple exact algorithm for PMDM runs in time 𝒪(2^𝓁 d). We present a data structure for PMDM that answers queries over 𝒟 in time 𝒪(2^{𝓁/2}(2^{𝓁/2}+τ)𝓁) and requires space 𝒪(2^𝓁 d²/τ²+2^{𝓁/2}d), for any parameter τ ∈ [1,d]. We complement our results by showing a two-way polynomial-time reduction between PMDM and the Minimum Union problem [Chlamtáč et al., SODA 2017]. This gives a polynomial-time 𝒪(d^{1/4+ε})-approximation algorithm for PMDM, which is tight under a plausible complexity conjecture.

We introduce data structures answering queries concerning the occurrences of patterns from a given dictionary $$\mathsf {D}$$ in fragments of a given string T of length n. The dictionary is internal in the sense that each pattern in $$\mathsf {D}$$ is given as a fragment of T. This way, $$\mathsf {D}$$ takes space proportional to the number of patterns $$d=|\mathsf {D}|$$ rather than their total length, which could be $$\varTheta (n\cdot d)$$ . In particular, we consider the following types of queries: reporting and counting all occurrences of patterns from $$\mathsf {D}$$ in a fragment $$T[i \mathinner {.\,.}j]$$ and reporting distinct patterns from $$\mathsf {D}$$ that occur in $$T[i \mathinner {.\,.}j]$$ . We show how to construct, in $$O((n+d) \log ^{O(1)} n)$$ time, a data structure that answers each of these queries in time $$O(\log ^{O(1)} n+| output |)$$ . The case of counting patterns is much more involved and needs a combination of a locally consistent parsing with orthogonal range searching. Reporting distinct patterns, on the other hand, uses the structure of maximal repetitions in strings. Finally, we provide tight—up to subpolynomial factors—upper and lower bounds for the case of a dynamic dictionary.

We consider the problem of preprocessing a text T of length n and a dictionary 𝒟 in order to be able to efficiently answer queries CountDistinct(i,j), that is, given i and j return the number of patterns from 𝒟 that occur in the fragment T[i..j]. The dictionary is internal in the sense that each pattern in 𝒟 is given as a fragment of T. This way, the dictionary takes space proportional to the number of patterns d=|𝒟| rather than their total length, which could be Θ(n⋅ d). An 𝒪̃(n+d)-size data structure that answers CountDistinct(i,j) queries 𝒪(log n)-approximately in 𝒪̃(1) time was recently proposed in a work that introduced internal dictionary matching [ISAAC 2019]. Here we present an 𝒪̃(n+d)-size data structure that answers CountDistinct(i,j) queries 2-approximately in 𝒪̃(1) time. Using range queries, for any m, we give an 𝒪̃(min(nd/m,n²/m²)+d)-size data structure that answers CountDistinct(i,j) queries exactly in 𝒪̃(m) time. We also consider the special case when the dictionary consists of all square factors of the string. We design an 𝒪(n log² n)-size data structure that allows us to count distinct squares in a text fragment T[i..j] in 𝒪(log n) time.

We consider the problem of dynamically maintaining an optimal alignment of two strings, each of length at most n, as they undergo insertions, deletions, and substitutions of letters. The string alignment problem generalizes the longest common subsequence (LCS) problem and the edit distance problem (also with non-unit costs, as long as insertions and deletions cost the same). The conditional lower bound of Backurs and Indyk [J. Comput. 2018] for computing the LCS in the static case implies that strongly sublinear update time for the dynamic string alignment problem would refute the Strong Exponential Time Hypothesis. We essentially match this lower bound when the alignment weights are constants, by showing how to process each update in 𝒪̃(n) time. When the weights are integers bounded in absolute value by some w=n^{𝒪(1)}, we can maintain the alignment in 𝒪̃(n ⋅ min {√ n,w}) time per update. For the 𝒪̃(nw)-time algorithm, we heavily rely on Tiskin’s work on semi-local LCS, and in particular, in an implicit way, on his algorithm for computing the (min,+)-product of two simple unit-Monge matrices [Algorithmica 2015]. As for the 𝒪̃(n√n)-time algorithm, we employ efficient data structures for computing distances in planar graphs.

Efficient algorithms for computing and processing additively weighted Voronoi diagrams on planar graphs have been instrumental in obtaining several recent breakthrough results, most notably the almost-optimal exact distance oracle for planar graphs [Charalampopoulos et al., STOC'19], and subquadratic algorithms for planar diameter [Cabello, SODA'17, Gawrychowski et al., SODA'18]. In this paper, we show how Voronoi diagrams can be useful in obtaining dynamic planar graph algorithms and apply them to classical problems such as dynamic single-source shortest paths and dynamic strongly connected components. First, we give a fully dynamic single-source shortest paths data structure for planar weighted digraphs with Õ(n^{4/5}) worst-case update time and O(log² n) query time. Here, a single update can either change the graph by inserting or deleting an edge, or reset the source s of interest. All known non-trivial planarity-exploiting exact dynamic single-source shortest paths algorithms to date had polynomial query time. Further, note that a data structure with strongly sublinear update time capable of answering distance queries between all pairs of vertices in polylogarithmic time would refute the APSP conjecture [Abboud and Dahlgaard, FOCS'16]. Somewhat surprisingly, the Voronoi diagram based approach we take for single-source shortest paths can also be used in the fully dynamic strongly connected components problem. In particular, we obtain a data structure maintaining a planar digraph under edge insertions and deletions, capable of returning the identifier of the strongly connected component of any query vertex. The worst-case update and query time bounds are the same as for our single-source distance oracle. To the best of our knowledge, this is the first fully dynamic strong-connectivity algorithm achieving both sublinear update time and polylogarithmic query time for an important class of digraphs.

The notions of periodicity and repetitions in strings, and hence these of runs and squares, naturally extend to two-dimensional strings. We consider two types of repetitions in 2D-strings: 2D-runs and quartics (quartics are a 2D-version of squares in standard strings). Amir et al. introduced 2D-runs, showed that there are 𝒪(n³) of them in an n × n 2D-string and presented a simple construction giving a lower bound of Ω(n²) for their number (Theoretical Computer Science, 2020). We make a significant step towards closing the gap between these bounds by showing that the number of 2D-runs in an n × n 2D-string is 𝒪(n² log² n). In particular, our bound implies that the 𝒪(n²log n + output) run-time of the algorithm of Amir et al. for computing 2D-runs is also 𝒪(n² log² n). We expect this result to allow for exploiting 2D-runs algorithmically in the area of 2D pattern matching. A quartic is a 2D-string composed of 2 × 2 identical blocks (2D-strings) that was introduced by Apostolico and Brimkov (Theoretical Computer Science, 2000), where by quartics they meant only primitively rooted quartics, i.e. built of a primitive block. Here our notion of quartics is more general and analogous to that of squares in 1D-strings. Apostolico and Brimkov showed that there are 𝒪(n² log² n) occurrences of primitively rooted quartics in an n × n 2D-string and that this bound is attainable. Consequently the number of distinct primitively rooted quartics is 𝒪(n² log² n). The straightforward bound for the maximal number of distinct general quartics is 𝒪(n⁴). Here, we prove that the number of distinct general quartics is also 𝒪(n² log² n). This extends the rich combinatorial study of the number of distinct squares in a 1D-string, that was initiated by Fraenkel and Simpson (Journal of Combinatorial Theory, Series A, 1998), to two dimensions. Finally, we show some algorithmic applications of 2D-runs. Specifically, we present algorithms for computing all occurrences of primitively rooted quartics and counting all general distinct quartics in 𝒪(n² log² n) time, which is quasi-linear with respect to the size of the input. The former algorithm is optimal due to the lower bound of Apostolico and Brimkov. The latter can be seen as a continuation of works on enumeration of distinct squares in 1D-strings using runs (Crochemore et al., Theoretical Computer Science, 2014). However, the methods used in 2D are different because of different properties of 2D-runs and quartics.

In the approximate pattern matching problem, given a text T, a pattern P, and a threshold k, the task is to find (the starting positions of) all substrings of T that are at distance at most k from P. We consider the two most fundamental string metrics: Under the Hamming distance, we search for substrings of T that have at most k mismatches with P, while under the edit distance, we search for substrings of T that can be transformed to P with at most k edits. Exact occurrences of P in T have a very simple structure: If we assume for simplicity that |P|2) k-mismatch occurrences of P in T, or both P and T are at Hamming distance O(k) from strings with a common string period of length O(m/k). We tighten this characterization by showing that there are O(k) k-mismatch occurrences in the non-periodic case, and we lift it to the edit distance setting, where we tightly bound the number of k-edit occurrences by O(k2) in the non-periodic case. Our proofs are constructive and let us obtain a unified framework for approximate pattern matching for both considered distances. In particular, we provide meta-algorithms that only rely on a small set of primitive operations. We showcase the generality of our meta-algorithms with results for the fully compressed setting, the dynamic setting, and the standard setting.

The longest common substring problem consists in finding a longest string that appears as a (contiguous) substring of two input strings. We consider the dynamic variant of this problem, in which we are to maintain two dynamic strings S and T, each of length at most n, that undergo substitutions of letters, in order to be able to return a longest common substring after each substitution. Recently, Amir et al. [ESA 2019] presented a solution for this problem that needs only 𝒪̃(n^(2/3)) time per update. This brought the challenge of determining whether there exists a faster solution with polylogarithmic update time, or (as is the case for other dynamic problems), we should expect a polynomial (conditional) lower bound. We answer this question by designing a significantly faster algorithm that processes each substitution in amortized log^𝒪(1) n time with high probability. Our solution relies on exploiting the local consistency of the parsing of a collection of dynamic strings due to Gawrychowski et al. [SODA 2018], and on maintaining two dynamic trees with labeled bicolored leaves, so that after each update we can report a pair of nodes, one from each tree, of maximum combined weight, which have at least one common leaf-descendant of each color. We complement this with a lower bound of Ω(log n/ log log n) for the update time of any polynomial-size data structure that maintains the LCS of two dynamic strings, even allowing amortization and randomization.

Abstract Given two strings S and T, each of length at most n, the longest common substring (LCS) problem is to find a longest substring common to S and T. This is a classical problem in computer science with an $$\mathcal {O}(n)$$ O ( n ) -time solution. In the fully dynamic setting, edit operations are allowed in either of the two strings, and the problem is to find an LCS after each edit. We present the first solution to the fully dynamic LCS problem requiring sublinear time in n per edit operation. In particular, we show how to find an LCS after each edit operation in $$\tilde{\mathcal {O}}(n^{2/3})$$ O ~ ( n 2 / 3 ) time, after $$\tilde{\mathcal {O}}(n)$$ O ~ ( n ) -time and space preprocessing. This line of research has been recently initiated in a somewhat restricted dynamic variant by Amir et al. [SPIRE 2017]. More specifically, the authors presented an $$\tilde{\mathcal {O}}(n)$$ O ~ ( n ) -sized data structure that returns an LCS of the two strings after a single edit operation (that is reverted afterwards) in $$\tilde{\mathcal {O}}(1)$$ O ~ ( 1 ) time. At CPM 2018, three papers (Abedin et al., Funakoshi et al., and Urabe et al.) studied analogously restricted dynamic variants of problems on strings; specifically, computing the longest palindrome and the Lyndon factorization of a string after a single edit operation. We develop dynamic sublinear-time algorithms for both of these problems as well. We also consider internal LCS queries, that is, queries in which we are to return an LCS of a pair of substrings of S and T. We show that answering such queries is hard in general and propose efficient data structures for several restricted cases.

SUMMARY: State-of-the-art repeat analysis tools rely on extending maximal repeated pairs to enumerate maximal k-mismatch repeats. These pairs can be quadratic in n, the length of the input sequence, and thus greedy heuristics are applied to speed up the extension. Here, we introduce supermaximal k-mismatch repeats, which are linear in n and capture all maximal k-mismatch repeats: every maximal k-mismatch repeat is a substring of some supermaximal k-mismatch repeat. We present SMART, a tool based on recent algorithmic advances implemented in C++ to compute supermaximal k-mismatch repeats directly, and show that these elements are statistically much more significant than the output of the state-of-the-art. AVAILABILITY AND IMPLEMENTATION: http://github.com/lorrainea/smart (GNU GPL v3.0). SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings. We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting.

Given two strings S and T, each of length at most n, the longest common substring (LCS) problem is to find a longest substring common to S and T. This is a classical problem in computer science with an O(n)-time solution. In the fully dynamic setting, edit operations are allowed in either of the two strings, and the problem is to find an LCS after each edit. We present the first solution to this problem requiring sublinear time in n per edit operation. In particular, we show how to find an LCS after each edit operation in O~(n^(2/3)) time, after O~(n)-time and space preprocessing. This line of research has been recently initiated in a somewhat restricted dynamic variant by Amir et al. [SPIRE 2017]. More specifically, they presented an O~(n)-sized data structure that returns an LCS of the two strings after a single edit operation (that is reverted afterwards) in O~(1) time. At CPM 2018, three papers (Abedin et al., Funakoshi et al., and Urabe et al.) studied analogously restricted dynamic variants of problems on strings. We show that the techniques we develop can be applied to obtain fully dynamic algorithms for all of these variants. The only previously known sublinear-time dynamic algorithms for problems on strings were for maintaining a dynamic collection of strings for comparison queries and for pattern matching, with the most recent advances made by Gawrychowski et al. [SODA 2018] and by Clifford et al. [STACS 2018]. As an intermediate problem we consider computing the solution for a string with a given set of k edits, which leads us, in particular, to answering internal queries on a string. The input to such a query is specified by a substring (or substrings) of a given string. Data structures for answering internal string queries that were proposed by Kociumaka et al. [SODA 2015] and by Gagie et al. [CCCG 2013] are used, along with new ones, based on ingredients such as the suffix tree, heavy-path decomposition, orthogonal range queries, difference covers, and string periodicity.

The k -mismatch problem consists in computing the Hamming distance between a pattern P of length m and every length- m substring of a text T of length n , if this distance is no more than k . In many real-world applications, any cyclic shift of P is a relevant pattern, and thus one is interested in computing the minimal distance of every length- m substring of T and any cyclic shift of P . This is the circular pattern matching with k mismatches ( k -CPM) problem. A multitude of papers have been devoted to solving this problem but, to the best of our knowledge, only average-case upper bounds are known. In this paper, we present the first non-trivial worst-case upper bounds for the k -CPM problem. Specifically, we show an \(\mathcal {O}(nk)\) -time algorithm and an \(\mathcal {O}(n+\frac{n}{m}\,{\small k^5})\) -time algorithm. The latter algorithm applies in an extended way a technique that was very recently developed for the k -mismatch problem [Bringmann et al., SODA 2019].

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u, a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u-to-v path that avoids all vertices in X. We propose oracles that can handle any number k of failures. More specifically, for a directed weighted planar graph with n vertices, any constant k, and for any q ∊ [1, ], we propose an oracle of size that answers queries in Õ(q) time.1 In particular, we show an Õ(n)-size, -query-time oracle for any constant k. This matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space. For single vertex failures (k = 1), our -size, Õ(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for q = Ω(nt), t ∊ (1/4, 1/2]. For multiple failures, no planarity exploiting results were previously known.

We present new tradeoffs between space and query-time for exact distance oracles in directed weighted planar graphs. These tradeoffs are almost optimal in the sense that they are within polylogarithmic, subpolynomial or arbitrarily small polynomial factors from the naïve linear space, constant query-time lower bound. These tradeoffs include: (i) an oracle with space O(n1+є) and query-time Õ(1) for any constant є>0, (ii) an oracle with space Õ(n) and query-time O(nє) for any constant є>0, and (iii) an oracle with space n1+o(1) and query-time no(1).

In the Longest Common Factor with k Mismatches (LCF_k) problem, we are given two strings X and Y of total length n, and we are asked to find a pair of maximal-length factors, one of X and the other of Y, such that their Hamming distance is at most k. Thankachan et al. [Thankachan et al. 2016] show that this problem can be solved in O(n log^k n) time and O(n) space for constant k. We consider the LCF_k(l) problem in which we assume that the sought factors have length at least l. We use difference covers to reduce the LCF_k(l) problem with l=Omega(log^{2k+2}n) to a task involving m=O(n/log^{k+1}n) synchronized factors. The latter can be solved in O(m log^{k+1}m) time, which results in a linear-time algorithm for LCF_k(l) with l=Omega(log^{2k+2}n). In general, our solution to the LCF_k(l) problem for arbitrary l takes O(n + n log^{k+1} n/sqrt{l}) time.

The observed frequency of the longest proper prefix, the longest proper suffix, and the longest infix of a word w in a given sequence x can be used for classifying w as avoided or overabundant. The definitions used for the expectation and deviation of w in this statistical model were described and biologically justified by Brendel et al. (J Biomol Struct Dyn 1986). We have very recently introduced a time-optimal algorithm for computing all avoided words of a given sequence over an integer alphabet (Algorithms Mol Biol 2017). In this article, we extend this study by presenting an O(n)-time and O(n)-space algorithm for computing all overabundant words in a sequence x of length n over an integer alphabet. Our main result is based on a new non-trivial combinatorial property of the suffix tree T of x: the number of distinct factors of x whose longest infix is the label of an explicit node of T is no more than 3n-4. We further show that the presented algorithm is time-optimal by proving that O(n) is a tight upper bound for the number of overabundant words. Finally, we present experimental results, using both synthetic and real data, which justify the effectiveness and efficiency of our approach in practical terms.