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**Published in:** LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

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 T₁ and T₂ 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) ∈ T₁ × T₂ is induced if and only if there is a label shared by leaf-descendants of u and v. In an HIA query, given nodes x ∈ T₁ and y ∈ T₂, 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 𝒪̃(n) can answer HIA queries in o(log n / log log n) time [Charalampopoulos, Gawrychowski, Pokorski; ICALP 2020]. This (unconditional) lower bound is a polyloglog n factor away from the query time of the fastest 𝒪̃(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 𝒪̃(n) time and answers HIA queries in 𝒪(log n/log log n) time. As a direct corollary, we obtain an 𝒪̃(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 𝒪(log n/ log log n)-depth tree decomposition. The latter allows us to break through the Ω(log n) barrier faced by previous works, due to the depth of the considered heavy-path decompositions.

Panagiotis Charalampopoulos, Bartłomiej Dudek, Paweł Gawrychowski, and Karol Pokorski. Optimal Near-Linear Space Heaviest Induced Ancestors. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{charalampopoulos_et_al:LIPIcs.CPM.2023.8, author = {Charalampopoulos, Panagiotis and Dudek, Bart{\l}omiej and Gawrychowski, Pawe{\l} and Pokorski, Karol}, title = {{Optimal Near-Linear Space Heaviest Induced Ancestors}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-276-1}, ISSN = {1868-8969}, year = {2023}, volume = {259}, editor = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.8}, URN = {urn:nbn:de:0030-drops-179624}, doi = {10.4230/LIPIcs.CPM.2023.8}, annote = {Keywords: data structures, string algorithms, fractional cascading} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

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 𝒪̃(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 ≥ 2 machines with amortized 𝒪̃(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.

Paweł Gawrychowski and Karol Pokorski. Sublinear Dynamic Interval Scheduling (On One or Multiple Machines). In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gawrychowski_et_al:LIPIcs.ICALP.2022.67, author = {Gawrychowski, Pawe{\l} and Pokorski, Karol}, title = {{Sublinear Dynamic Interval Scheduling (On One or Multiple Machines)}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {67:1--67:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.67}, URN = {urn:nbn:de:0030-drops-164086}, doi = {10.4230/LIPIcs.ICALP.2022.67}, annote = {Keywords: interval scheduling, dynamic problems, data structures, greedy algorithms} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

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.

Panagiotis Charalampopoulos, Paweł Gawrychowski, and Karol Pokorski. Dynamic Longest Common Substring in Polylogarithmic Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{charalampopoulos_et_al:LIPIcs.ICALP.2020.27, author = {Charalampopoulos, Panagiotis and Gawrychowski, Pawe{\l} and Pokorski, Karol}, title = {{Dynamic Longest Common Substring in Polylogarithmic Time}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {27:1--27:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.27}, URN = {urn:nbn:de:0030-drops-124340}, doi = {10.4230/LIPIcs.ICALP.2020.27}, annote = {Keywords: string algorithms, dynamic algorithms, longest common substring} }