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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.01388">arXiv:2503.01388</a> <span> [<a href="https://arxiv.org/pdf/2503.01388">pdf</a>, <a href="https://arxiv.org/format/2503.01388">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Faster ED-String Matching with $k$ Mismatches </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gawrychowski%2C+P">Pawe艂 Gawrychowski</a>, <a href="/search/cs?searchtype=author&query=G%C3%B3rkiewicz%2C+A">Adam G贸rkiewicz</a>, <a href="/search/cs?searchtype=author&query=Marciniak%2C+P">Pola Marciniak</a>, <a href="/search/cs?searchtype=author&query=Pissis%2C+S+P">Solon P. Pissis</a>, <a href="/search/cs?searchtype=author&query=Pokorski%2C+K">Karol Pokorski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.01388v1-abstract-short" style="display: inline;"> 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 sequence… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.01388v1-abstract-full').style.display = 'inline'; document.getElementById('2503.01388v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.01388v1-abstract-full" style="display: none;"> 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 $\tilde{\mathcal{O}}(nm^{蠅-1})+\mathcal{O}(N)$-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 $m^{蠅-1}$ 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.01388v1-abstract-full').style.display = 'none'; document.getElementById('2503.01388v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.01373">arXiv:2302.01373</a> <span> [<a href="https://arxiv.org/pdf/2302.01373">pdf</a>, <a href="https://arxiv.org/ps/2302.01373">ps</a>, <a href="https://arxiv.org/format/2302.01373">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Optimal Heaviest Induced Ancestors </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Charalampopoulos%2C+P">Panagiotis Charalampopoulos</a>, <a href="/search/cs?searchtype=author&query=Dudek%2C+B">Bart艂omiej Dudek</a>, <a href="/search/cs?searchtype=author&query=Gawrychowski%2C+P">Pawe艂 Gawrychowski</a>, <a href="/search/cs?searchtype=author&query=Pokorski%2C+K">Karol Pokorski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.01373v1-abstract-short" style="display: inline;"> We revisit the Heaviest Induced Ancestors (HIA) problem that was introduced by Gagie, Gawrychowski, and Nekrich [CCCG 2013] and has a number of applications in string algorithms. Let $T_1$ and $T_2$ be two rooted trees whose nodes have weights that are increasing in all root-to-leaf paths, and labels on the leaves, such that no two leaves of a tree have the same label. A pair of nodes… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.01373v1-abstract-full').style.display = 'inline'; document.getElementById('2302.01373v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.01373v1-abstract-full" style="display: none;"> We revisit the Heaviest Induced Ancestors (HIA) problem that was introduced by Gagie, Gawrychowski, and Nekrich [CCCG 2013] and has a number of applications in string algorithms. Let $T_1$ and $T_2$ be two rooted trees whose nodes have weights that are increasing in all root-to-leaf paths, and labels on the leaves, such that no two leaves of a tree have the same label. A pair of nodes $(u, v)\in T_1 \times T_2$ is \emph{induced} if and only if there is a label shared by leaf-descendants of $u$ and $v$. In an HIA query, given nodes $x \in T_1$ and $y \in T_2$, the goal is to find an induced pair of nodes $(u, v)$ of the maximum total weight such that $u$ is an ancestor of~$x$ and $v$ is an ancestor of $y$. Let $n$ be the upper bound on the sizes of the two trees. It is known that no data structure of size $\tilde{\mathcal{O}}(n)$ can answer HIA queries in $o(\log n / \log \log n)$ time [Charalampopoulos, Gawrychowski, Pokorski; ICALP 2020]. This (unconditional) lower bound is a $\operatorname{polyloglog} n$ factor away from the query time of the fastest $\tilde{\mathcal{O}}(n)$-size data structure known to date for the HIA problem [Abedin, Hooshmand, Ganguly, Thankachan; Algorithmica 2022]. In this work, we resolve the query-time complexity of the HIA problem for the near-linear space regime by presenting a data structure that can be built in $\tilde{\mathcal{O}}(n)$ time and answers HIA queries in $\mathcal{O}(\log n/\log\log n)$ time. As a direct corollary, we obtain an $\tilde{\mathcal{O}}(n)$-size data structure that maintains the LCS of a static string and a dynamic string, both of length at most $n$, in time optimal for this space regime. The main ingredients of our approach are fractional cascading and the utilization of an $\mathcal{O}(\log n/ \log\log n)$-depth tree decomposition. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.01373v1-abstract-full').style.display = 'none'; document.getElementById('2302.01373v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.14310">arXiv:2203.14310</a> <span> [<a href="https://arxiv.org/pdf/2203.14310">pdf</a>, <a href="https://arxiv.org/format/2203.14310">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Sublinear Dynamic Interval Scheduling (on one or multiple machines) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gawrychowski%2C+P">Pawe艂 Gawrychowski</a>, <a href="/search/cs?searchtype=author&query=Pokorski%2C+K">Karol Pokorski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.14310v2-abstract-short" style="display: inline;"> We revisit the complexity of the classical Interval Scheduling in the dynamic setting. In this problem, the goal is to maintain a set of intervals under insertions and deletions and report the size of the maximum size subset of pairwise disjoint intervals after each update. Nontrivial approximation algorithms are known for this problem, for both the unweighted and weighted versions [Henzinger, Neu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.14310v2-abstract-full').style.display = 'inline'; document.getElementById('2203.14310v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.14310v2-abstract-full" style="display: none;"> We revisit the complexity of the classical Interval Scheduling in the dynamic setting. In this problem, the goal is to maintain a set of intervals under insertions and deletions and report the size of the maximum size subset of pairwise disjoint intervals after each update. Nontrivial approximation algorithms are known for this problem, for both the unweighted and weighted versions [Henzinger, Neumann, Wiese, SoCG 2020]. Surprisingly, it was not known if the general exact version admits an exact solution working in sublinear time, that is, without recomputing the answer after each update. Our first contribution is a structure for Dynamic Interval Scheduling with amortized $\tilde{\mathcal{O}}(n^{1/3})$ update time. Then, building on the ideas used for the case of one machine, we design a sublinear solution for any constant number of machines: we describe a structure for Dynamic Interval Scheduling on $m\geq 2$ machines with amortized $\tilde{\mathcal{O}}(n^{1 - 1/m})$ update time. We complement the above results by considering Dynamic Weighted Interval Scheduling on one machine, that is maintaining (the weight of) the maximum weight subset of pairwise disjoint intervals. We show an almost linear lower bound (conditioned on the hardness of Minimum Weight $k$-Clique) for the update/query time of any structure for this problem. Hence, in the weighted case one should indeed seek approximate solutions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.14310v2-abstract-full').style.display = 'none'; document.getElementById('2203.14310v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.03978">arXiv:2101.03978</a> <span> [<a href="https://arxiv.org/pdf/2101.03978">pdf</a>, <a href="https://arxiv.org/format/2101.03978">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> </div> <p class="title is-5 mathjax"> Strictly In-Place Algorithms for Permuting and Inverting Permutations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Dudek%2C+B">Bart艂omiej Dudek</a>, <a href="/search/cs?searchtype=author&query=Gawrychowski%2C+P">Pawe艂 Gawrychowski</a>, <a href="/search/cs?searchtype=author&query=Pokorski%2C+K">Karol Pokorski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.03978v1-abstract-short" style="display: inline;"> We revisit the problem of permuting an array of length $n$ according to a given permutation in place, that is, using only a small number of bits of extra storage. Fich, Munro and Poblete [FOCS 1990, SICOMP 1995] obtained an elegant $\mathcal{O}(n\log n)$-time algorithm using only $\mathcal{O}(\log^{2}n)$ bits of extra space for this basic problem by designing a procedure that scans the permutation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.03978v1-abstract-full').style.display = 'inline'; document.getElementById('2101.03978v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.03978v1-abstract-full" style="display: none;"> We revisit the problem of permuting an array of length $n$ according to a given permutation in place, that is, using only a small number of bits of extra storage. Fich, Munro and Poblete [FOCS 1990, SICOMP 1995] obtained an elegant $\mathcal{O}(n\log n)$-time algorithm using only $\mathcal{O}(\log^{2}n)$ bits of extra space for this basic problem by designing a procedure that scans the permutation and outputs exactly one element from each of its cycles. However, in the strict sense in place should be understood as using only an asymptotically optimal $\mathcal{O}(\log n)$ bits of extra space, or storing a constant number of indices. The problem of permuting in this version is, in fact, a well-known interview question, with the expected solution being a quadratic-time algorithm. Surprisingly, no faster algorithm seems to be known in the literature. Our first contribution is a strictly in-place generalisation of the method of Fich et al. that works in $\mathcal{O}_{\varepsilon}(n^{1+\varepsilon})$ time, for any $\varepsilon > 0$. Then, we build on this generalisation to obtain a strictly in-place algorithm for inverting a given permutation on $n$ elements working in the same complexity. This is a significant improvement on a recent result of Gu艣piel [arXiv 2019], who designed an $\mathcal{O}(n^{1.5})$-time algorithm. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.03978v1-abstract-full').style.display = 'none'; document.getElementById('2101.03978v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.02408">arXiv:2006.02408</a> <span> [<a href="https://arxiv.org/pdf/2006.02408">pdf</a>, <a href="https://arxiv.org/format/2006.02408">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Dynamic Longest Common Substring in Polylogarithmic Time </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Charalampopoulos%2C+P">Panagiotis Charalampopoulos</a>, <a href="/search/cs?searchtype=author&query=Gawrychowski%2C+P">Pawe艂 Gawrychowski</a>, <a href="/search/cs?searchtype=author&query=Pokorski%2C+K">Karol Pokorski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.02408v1-abstract-short" style="display: inline;"> 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. Rece… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.02408v1-abstract-full').style.display = 'inline'; document.getElementById('2006.02408v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.02408v1-abstract-full" style="display: none;"> 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 $\tilde{\mathcal{O}}(n^{2/3})$ 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 $\log^{\mathcal{O}(1)} n$ 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 $惟(\log n/ \log\log n)$ for the update time of any polynomial-size data structure that maintains the LCS of two dynamic strings, and the same lower bound for the update time of any data structure of size $\tilde{\mathcal{O}}(n)$ that maintains the LCS of a static and a dynamic string. Both lower bounds hold even allowing amortization and randomization. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.02408v1-abstract-full').style.display = 'none'; document.getElementById('2006.02408v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Full version of a paper that is to appear in the ICALP 2020 proceedings</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div 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