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Documents authored by Cáceres, Manuel

Document
DOI: 10.4230/LIPIcs.ESA.2023.15
Sorting Finite Automata via Partition Refinement

Authors: Ruben Becker, Manuel Cáceres, Davide Cenzato, Sung-Hwan Kim, Bojana Kodric, Francisco Olivares, and Nicola Prezza

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Wheeler nondeterministic finite automata (WNFAs) were introduced in (Gagie et al., TCS 2017) as a powerful generalization of prefix sorting from strings to labeled graphs. WNFAs admit optimal solutions to classic hard problems on labeled graphs and languages such as compression and regular expression matching. The problem of deciding whether a given NFA is Wheeler is known to be NP-complete (Gibney and Thankachan, ESA 2019). Recently, however, Alanko et al. (Information and Computation 2021) showed how to side-step this complexity by switching to preorders: letting Q be the set of states and δ the set of transitions, they provided a O(|δ|⋅|Q|²)-time algorithm computing a totally-ordered partition (i.e. equivalence relation) of the WNFA’s states such that (1) equivalent states recognize the same regular language, and (2) the order of (the classes of) non-equivalent states is consistent with any Wheeler order, when one exists. As a result, the output is a preorder of the states as useful for pattern matching as standard Wheeler orders. Further extensions of this line of work (Cotumaccio et al., SODA 2021 and DCC 2022) generalized these concepts to arbitrary NFAs by introducing co-lex partial preorders: in general, any NFA admits a partial preorder of its states reflecting the co-lexicographic order of their accepted strings; the smaller the width of such preorder is, the faster regular expression matching queries can be performed. To date, the fastest algorithm for computing the smallest-width partial preorder on NFAs runs in O(|δ|² + |Q|^{5/2}) time (Cotumaccio, DCC 2022), while on DFAs the same task can be accomplished in O(min(|Q|²log|Q|, |δ|⋅|Q|)) time (Kim et al., CPM 2023). In this paper, we provide much more efficient solutions to the co-lex order computation problem. Our results are achieved by extending a classic algorithm for the relational coarsest partition refinement problem of Paige and Tarjan to work with ordered partitions. More specifically, we provide a O(|δ|log|Q|)-time algorithm computing a co-lex total preorder when the input is a Wheeler NFA, and an algorithm with the same time complexity computing the smallest-width co-lex partial order of any DFA. In addition, we present implementations of our algorithms and show that they are very efficient also in practice.
Cite as

Ruben Becker, Manuel Cáceres, Davide Cenzato, Sung-Hwan Kim, Bojana Kodric, Francisco Olivares, and Nicola Prezza. Sorting Finite Automata via Partition Refinement. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{becker_et_al:LIPIcs.ESA.2023.15,
  author =	{Becker, Ruben and C\'{a}ceres, Manuel and Cenzato, Davide and Kim, Sung-Hwan and Kodric, Bojana and Olivares, Francisco and Prezza, Nicola},
  title =	{{Sorting Finite Automata via Partition Refinement}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.15},
  URN =		{urn:nbn:de:0030-drops-186684},
  doi =		{10.4230/LIPIcs.ESA.2023.15},
  annote =	{Keywords: Wheeler automata, prefix sorting, pattern matching, graph compression, sorting, partition refinement}
}
Document
DOI: 10.4230/LIPIcs.WABI.2023.10
Finding Maximal Exact Matches in Graphs

Authors: Nicola Rizzo, Manuel Cáceres, and Veli Mäkinen

Published in: LIPIcs, Volume 273, 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)

Abstract
We study the problem of finding maximal exact matches (MEMs) between a query string Q and a labeled graph G. MEMs are an important class of seeds, often used in seed-chain-extend type of practical alignment methods because of their strong connections to classical metrics. A principled way to speed up chaining is to limit the number of MEMs by considering only MEMs of length at least κ (κ-MEMs). However, on arbitrary input graphs, the problem of finding MEMs cannot be solved in truly sub-quadratic time under SETH (Equi et al., ICALP 2019) even on acyclic graphs. In this paper we show an O(n⋅ L ⋅ d^{L-1} + m + M_{κ,L})-time algorithm finding all κ-MEMs between Q and G spanning exactly L nodes in G, where n is the total length of node labels, d is the maximum degree of a node in G, m = |Q|, and M_{κ,L} is the number of output MEMs. We use this algorithm to develop a κ-MEM finding solution on indexable Elastic Founder Graphs (Equi et al., Algorithmica 2022) running in time O(nH² + m + M_κ), where H is the maximum number of nodes in a block, and M_κ is the total number of κ-MEMs. Our results generalize to the analysis of multiple query strings (MEMs between G and any of the strings). Additionally, we provide some preliminary experimental results showing that the number of graph MEMs is an order of magnitude smaller than the number of string MEMs of the corresponding concatenated collection.
Cite as

Nicola Rizzo, Manuel Cáceres, and Veli Mäkinen. Finding Maximal Exact Matches in Graphs. In 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 273, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rizzo_et_al:LIPIcs.WABI.2023.10,
  author =	{Rizzo, Nicola and C\'{a}ceres, Manuel and M\"{a}kinen, Veli},
  title =	{{Finding Maximal Exact Matches in Graphs}},
  booktitle =	{23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-294-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{273},
  editor =	{Belazzougui, Djamal and Ouangraoua, A\"{i}da},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2023.10},
  URN =		{urn:nbn:de:0030-drops-186364},
  doi =		{10.4230/LIPIcs.WABI.2023.10},
  annote =	{Keywords: Sequence to graph alignment, bidirectional BWT, r-index, suffix tree, founder graphs}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.31
Minimum Chain Cover in Almost Linear Time

Authors: Manuel Cáceres

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
A minimum chain cover (MCC) of a k-width directed acyclic graph (DAG) G = (V, E) is a set of k chains (paths in the transitive closure) of G such that every vertex appears in at least one chain in the cover. The state-of-the-art solutions for MCC run in time Õ(k(|V|+|E|)) [Mäkinen et at., TALG], O(T_{MF}(|E|) + k|V|), O(k²|V| + |E|) [Cáceres et al., SODA 2022], Õ(|V|^{3/2} + |E|) [Kogan and Parter, ICALP 2022] and Õ(T_{MCF}(|E|) + √k|V|) [Kogan and Parter, SODA 2023], where T_{MF}(|E|) and T_{MCF}(|E|) are the running times for solving maximum flow (MF) and minimum-cost flow (MCF), respectively. In this work we present an algorithm running in time O(T_{MF}(|E|) + (|V|+|E|)log k). By considering the recent result for solving MF [Chen et al., FOCS 2022] our algorithm is the first running in almost linear time. Moreover, our techniques are deterministic and derive a deterministic near-linear time algorithm for MCC if the same is provided for MF. At the core of our solution we use a modified version of the mergeable dictionaries [Farach and Thorup, Algorithmica], [Iacono and Özkan, ICALP 2010] data structure boosted with the SIZE-SPLIT operation and answering queries in amortized logarithmic time, which can be of independent interest.
Cite as

Manuel Cáceres. Minimum Chain Cover in Almost Linear Time. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 31:1-31:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{caceres:LIPIcs.ICALP.2023.31,
  author =	{C\'{a}ceres, Manuel},
  title =	{{Minimum Chain Cover in Almost Linear Time}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{31:1--31:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.31},
  URN =		{urn:nbn:de:0030-drops-180834},
  doi =		{10.4230/LIPIcs.ICALP.2023.31},
  annote =	{Keywords: Minimum chain cover, directed acyclic graph, minimum flow, flow decomposition, mergeable dictionaries, amortized running time}
}
Document
DOI: 10.4230/LIPIcs.CPM.2023.7
Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond

Authors: Manuel Cáceres

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

Abstract
The problem of String Matching to Labeled Graphs (SMLG) asks to find all the paths in a labeled graph G = (V, E) whose spellings match that of an input string S ∈ Σ^m. SMLG can be solved in quadratic O(m|E|) time [Amir et al., JALG 2000], which was proven to be optimal by a recent lower bound conditioned on SETH [Equi et al., ICALP 2019]. The lower bound states that no strongly subquadratic time algorithm exists, even if restricted to directed acyclic graphs (DAGs). In this work we present the first parameterized algorithms for SMLG on DAGs. Our parameters capture the topological structure of G. All our results are derived from a generalization of the Knuth-Morris-Pratt algorithm [Park and Kim, CPM 1995] optimized to work in time proportional to the number of prefix-incomparable matches. To obtain the parameterization in the topological structure of G, we first study a special class of DAGs called funnels [Millani et al., JCO 2020] and generalize them to k-funnels and the class ST_k. We present several novel characterizations and algorithmic contributions on both funnels and their generalizations.
Cite as

Manuel Cáceres. Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{caceres:LIPIcs.CPM.2023.7,
  author =	{C\'{a}ceres, Manuel},
  title =	{{Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond}},
  booktitle =	{34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)},
  pages =	{7:1--7:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.7},
  URN =		{urn:nbn:de:0030-drops-179619},
  doi =		{10.4230/LIPIcs.CPM.2023.7},
  annote =	{Keywords: string matching, parameterized algorithms, FPT inside P, string algorithms, graph algorithms, directed acyclic graphs, labeled graphs, funnels}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.31
Width Helps and Hinders Splitting Flows

Authors: Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow X on directed graph G into weighted source-to-sink paths whose superposition equals X. We focus on a common formulation of the problem where the path weights must be non-negative integers and also on a new variant where these weights can be negative. We show that, for acyclic graphs, considering the width of the graph (the minimum number of s-t paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the non-negative version, we show that a popular heuristic is a O(log |X|)-approximation (|X| being the total flow of X) on graphs satisfying two properties related to the width (satisfied by e.g., series-parallel graphs), and strengthen its worst-case approximation ratio from Ω(√m) to Ω(m / log m) for sparse graphs, where m is the number of edges in the graph. For the negative version, we give a (⌈log ║X║⌉+1)-approximation (║X║ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary flows (║X║ ≤ 1) into at most width paths. We also disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.
Cite as

Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width Helps and Hinders Splitting Flows. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{caceres_et_al:LIPIcs.ESA.2022.31,
  author =	{C\'{a}ceres, Manuel and Cairo, Massimo and Grigorjew, Andreas and Khan, Shahbaz and Mumey, Brendan and Rizzi, Romeo and Tomescu, Alexandru I. and Williams, Lucia},
  title =	{{Width Helps and Hinders Splitting Flows}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{31:1--31:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.31},
  URN =		{urn:nbn:de:0030-drops-169695},
  doi =		{10.4230/LIPIcs.ESA.2022.31},
  annote =	{Keywords: Flow decomposition, approximation algorithms, graph width}
}
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