Document

**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

In Gapped String Indexing, the goal is to compactly represent a string S of length n such that for any query consisting of two strings P₁ and P₂, called patterns, and an integer interval [α, β], called gap range, we can quickly find occurrences of P₁ and P₂ in S with distance in [α, β]. Gapped String Indexing is a central problem in computational biology and text mining and has thus received significant research interest, including parameterized and heuristic approaches. Despite this interest, the best-known time-space trade-offs for Gapped String Indexing are the straightforward 𝒪(n) space and 𝒪(n+ occ) query time or Ω(n²) space and Õ(|P₁| + |P₂| + occ) query time.
We break through this barrier obtaining the first interesting trade-offs with polynomially subquadratic space and polynomially sublinear query time. In particular, we show that, for every 0 ≤ δ ≤ 1, there is a data structure for Gapped String Indexing with either Õ(n^{2-δ/3}) or Õ(n^{3-2δ}) space and Õ(|P₁| + |P₂| + n^{δ}⋅ (occ+1)) query time, where occ is the number of reported occurrences.
As a new fundamental tool towards obtaining our main result, we introduce the Shifted Set Intersection problem: preprocess a collection of sets S₁, …, S_k of integers such that for any query consisting of three integers i,j,s, we can quickly output YES if and only if there exist a ∈ S_i and b ∈ S_j with a+s = b. We start by showing that the Shifted Set Intersection problem is equivalent to the indexing variant of 3SUM (3SUM Indexing) [Golovnev et al., STOC 2020]. We then give a data structure for Shifted Set Intersection with gaps, which entails a solution to the Gapped String Indexing problem. Furthermore, we enhance our data structure for deciding Shifted Set Intersection, so that we can support the reporting variant of the problem, i.e., outputting all certificates in the affirmative case. Via the obtained equivalence to 3SUM Indexing, we thus give new improved data structures for the reporting variant of 3SUM Indexing, and we show how this improves upon the state-of-the-art solution for Jumbled Indexing [Chan and Lewenstein, STOC 2015] for any alphabet of constant size σ > 5.

Philip Bille, Inge Li Gørtz, Moshe Lewenstein, Solon P. Pissis, Eva Rotenberg, and Teresa Anna Steiner. Gapped String Indexing in Subquadratic Space and Sublinear Query Time. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bille_et_al:LIPIcs.STACS.2024.16, author = {Bille, Philip and G{\o}rtz, Inge Li and Lewenstein, Moshe and Pissis, Solon P. and Rotenberg, Eva and Steiner, Teresa Anna}, title = {{Gapped String Indexing in Subquadratic Space and Sublinear Query Time}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {16:1--16:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.16}, URN = {urn:nbn:de:0030-drops-197262}, doi = {10.4230/LIPIcs.STACS.2024.16}, annote = {Keywords: data structures, string indexing, indexing with gaps, two patterns} }

Document

**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

In this paper we give a new, efficient algorithm for computing curve skeletons, based on local separators. Our efficiency stems from a multilevel approach, where we solve small problems across levels of detail and combine these in order to quickly obtain a skeleton. We do this in a highly modular fashion, ensuring complete flexibility in adapting the algorithm for specific types of input or for otherwise targeting specific applications.
Separator based skeletonization was first proposed by Bærentzen and Rotenberg in [ACM Tran. Graphics'21], showing high quality output at the cost of running times which become prohibitive for large inputs. Our new approach retains the high quality output, and applicability to any spatially embedded graph, while being orders of magnitude faster for all practical purposes.
We test our skeletonization algorithm for efficiency and quality in practice, comparing it to local separator skeletonization on the University of Groningen Skeletonization Benchmark [Telea'16].

J. Andreas Bærentzen, Rasmus Emil Christensen, Emil Toftegaard Gæde, and Eva Rotenberg. Multilevel Skeletonization Using Local Separators. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{brentzen_et_al:LIPIcs.SoCG.2023.13, author = {B{\ae}rentzen, J. Andreas and Christensen, Rasmus Emil and G{\ae}de, Emil Toftegaard and Rotenberg, Eva}, title = {{Multilevel Skeletonization Using Local Separators}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.13}, URN = {urn:nbn:de:0030-drops-178637}, doi = {10.4230/LIPIcs.SoCG.2023.13}, annote = {Keywords: Algorithm engineering, experimentation and implementation, shape skeletonization, curve skeletons, multilevel algorithm} }

Document

**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

We study dynamic planar graphs with n vertices, subject to edge deletion, edge contraction, edge insertion across a face, and the splitting of a vertex in specified corners. We dynamically maintain a combinatorial embedding of such a planar graph, subject to connectivity and 2-vertex-connectivity (biconnectivity) queries between pairs of vertices. Whenever a query pair is connected and not biconnected, we find the first and last cutvertex separating them.
Additionally, we allow local changes to the embedding by flipping the embedding of a subgraph that is connected by at most two vertices to the rest of the graph.
We support all queries and updates in deterministic, worst-case, O(log² n) time, using an O(n)-sized data structure.

Jacob Holm, Ivor van der Hoog, and Eva Rotenberg. Worst-Case Deterministic Fully-Dynamic Biconnectivity in Changeable Planar Embeddings. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{holm_et_al:LIPIcs.SoCG.2023.40, author = {Holm, Jacob and van der Hoog, Ivor and Rotenberg, Eva}, title = {{Worst-Case Deterministic Fully-Dynamic Biconnectivity in Changeable Planar Embeddings}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {40:1--40:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.40}, URN = {urn:nbn:de:0030-drops-178909}, doi = {10.4230/LIPIcs.SoCG.2023.40}, annote = {Keywords: dynamic graphs, planarity, connectivity} }

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Invited Talk

**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

In dynamic data structures, one is interested in efficiently facilitating queries to a data set, while being able to efficiently perform updates as the data set undergoes changes. Often, relaxing the efficiency measure to the amortised setting allows for simpler algorithms. A well-known example of a data structure with amortised guarantees is the splay tree by Sleator and Tarjan [Daniel D. Sleator and Robert E. Tarjan, 1985].
Similarly, in data structures for dynamic graphs, one is interested in efficiently maintaining some information about the graph, or facilitating queries, as the graph undergoes changes in the form of insertion and deletion of edges. Examples of such information include connectivity, planarity, and approximate sparsity of the graph: is the graph presently connected? Is it planar? Has its arboricity grossly exceeded some specified number α̃? The related queries could be: is a connected to b? Are the edges uv and uw consecutive in the ordering around u in its current planar embedding? Or, report the O(α) out-edges of vertex x.
In this talk, we will see Brodal and Fagerberg’s amortised algorithm for orienting sparse graphs (i.e. of arboricity ≤ α), so that each vertex has O(α) out-edges [Gerth Stølting Brodal and Rolf Fagerberg, 1999]. The algorithm itself is extremely simple, and uses an elegant amortised argument in its analysis. Then, we will visit the problem of dynamic planarity testing: is the graph presently planar? Here, we will see an elegant amortised reduction to the seemingly easier problem, where planarity-violating edges may be detected and rejected [Eppstein et al., 1996]. We will see a sketch of how the current state-of-the-art algorithm for efficient planarity testing [Jacob Holm and Eva Rotenberg, 2020] uses ideas similar to those in [Gerth Stølting Brodal and Rolf Fagerberg, 1999] to analyse the behaviour of a greedy algorithm via a possibly inefficient algorithm with provably low recourse [Jacob Holm and Eva Rotenberg, 2020]. If time permits, we will touch upon a recent simple amortised data structure for maintaining information in dynamic forests [Jacob Holm et al., 2023], which builds on ideas from splay trees.
The talk concludes with some open questions in the area.

Eva Rotenberg. Amortised Analysis of Dynamic Data Structures (Invited Talk). In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rotenberg:LIPIcs.STACS.2023.2, author = {Rotenberg, Eva}, title = {{Amortised Analysis of Dynamic Data Structures}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {2:1--2:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.2}, URN = {urn:nbn:de:0030-drops-176547}, doi = {10.4230/LIPIcs.STACS.2023.2}, annote = {Keywords: Amortised analysis, splaying, dynamic graphs, planarity testing} }

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Complete Volume

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

LIPIcs, Volume 244, ESA 2022, Complete Volume

30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 1-1406, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{chechik_et_al:LIPIcs.ESA.2022, title = {{LIPIcs, Volume 244, ESA 2022, Complete Volume}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {1--1406}, 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}, URN = {urn:nbn:de:0030-drops-169374}, doi = {10.4230/LIPIcs.ESA.2022}, annote = {Keywords: LIPIcs, Volume 244, ESA 2022, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 0:i-0:xxii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chechik_et_al:LIPIcs.ESA.2022.0, author = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {0:i--0:xxii}, 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.2022.0}, URN = {urn:nbn:de:0030-drops-169382}, doi = {10.4230/LIPIcs.ESA.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

A graph has arboricity α if its edges can be partitioned into α forests. The dynamic arboricity decomposition problem is to update a partitioning of the graph’s edges into forests, as a graph undergoes insertions and deletions of edges. We present an algorithm for maintaining partitioning into α+1 forests, provided the arboricity of the dynamic graph never exceeds α. Our algorithm has an update time of Õ(n^{3/4}) when α is at most polylogarithmic in n.
Similarly, the dynamic bounded out-orientation problem is to orient the edges of the graph such that the out-degree of each vertex is at all times bounded. For this problem, we give an algorithm that orients the edges such that the out-degree is at all times bounded by α+1, with an update time of Õ(n^{5/7}), when α is at most polylogarithmic in n. Here, the choice of α+1 should be viewed in the light of the well-known lower bound by Brodal and Fagerberg which establishes that, for general graphs, maintaining only α out-edges would require linear update time.
However, the lower bound by Brodal and Fagerberg is non-planar. In this paper, we give a lower bound showing that even for planar graphs, linear update time is needed in order to maintain an explicit three-out-orientation. For planar graphs, we show that the dynamic four forest decomposition and four-out-orientations, can be updated in Õ(n^{1/2}) time.

Aleksander B. G. Christiansen, Jacob Holm, Eva Rotenberg, and Carsten Thomassen. On Dynamic α + 1 Arboricity Decomposition and Out-Orientation. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{christiansen_et_al:LIPIcs.MFCS.2022.34, author = {Christiansen, Aleksander B. G. and Holm, Jacob and Rotenberg, Eva and Thomassen, Carsten}, title = {{On Dynamic \alpha + 1 Arboricity Decomposition and Out-Orientation}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.34}, URN = {urn:nbn:de:0030-drops-168320}, doi = {10.4230/LIPIcs.MFCS.2022.34}, annote = {Keywords: Dynamic graphs, bounded arboricity, data structures} }

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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)

The arboricity α of a graph is the smallest number of forests necessary to cover its edges, and an arboricity decomposition of a graph is a decomposition of its edges into forests. The best near-linear time algorithm for arboricity decomposition guarantees at most α +2 forests if the graph has arboricity α (Blumenstock and Fischer [Markus Blumenstock and Frank Fischer, 2020]).
In this paper, we study arboricity decomposition for dynamic graphs, that is, graphs that are subject to insertions and deletions of edges. We give an algorithm that, provided the arboricity of the dynamic graph never exceeds α, maintains an α+2 arboricity decomposition of the graph in poly(log n,α) update time, thus matching the number of forests currently obtainable in near-linear time for static (non-changing) graphs.
Our construction goes via dynamic bounded out-degree orientations, and we present a fully-dynamic algorithm that explicitly orients the edges of the dynamic graph, such that no vertex has an out-degree exceeding ⌊ (1+ε)α ⌋ + 2. Our algorithm is deterministic and has a worst-case update time of O(ε^{-6}α² log³ n). The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a β⋅ α + log_β n out-orientation in O(β²α²+βαlog_β n) time [Tsvi Kopelowitz et al., 2014].
As a consequence, we get an algorithm that maintains an implicit vertex colouring with 4⋅ 2^α colours, in amortised poly-log n update time, and with O(α log n) worst-case query time. Thus, at the expense of log n-factors in the update time, we improve on the number of colours from 2^O(α) to O(2^α) compared to the state-of-the-art for implicit dynamic colouring [Monika Henzinger et al., 2020].

Aleksander B. G. Christiansen and Eva Rotenberg. Fully-Dynamic α + 2 Arboricity Decompositions and Implicit Colouring. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{christiansen_et_al:LIPIcs.ICALP.2022.42, author = {Christiansen, Aleksander B. G. and Rotenberg, Eva}, title = {{Fully-Dynamic \alpha + 2 Arboricity Decompositions and Implicit Colouring}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {42:1--42:20}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.42}, URN = {urn:nbn:de:0030-drops-163835}, doi = {10.4230/LIPIcs.ICALP.2022.42}, annote = {Keywords: Dynamic graphs, bounded arboricity, graph colouring, data structures} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

The Fréchet distance is a well-studied similarity measure between curves that is widely used throughout computer science. Motivated by applications where curves stem from paths and walks on an underlying graph (such as a road network), we define and study the Fréchet distance for paths and walks on graphs. When provided with a distance oracle of G with O(1) query time, the classical quadratic-time dynamic program can compute the Fréchet distance between two walks P and Q in a graph G in O(|P|⋅|Q|) time. We show that there are situations where the graph structure helps with computing Fréchet distance: when the graph G is planar, we apply existing (approximate) distance oracles to compute a (1+ε)-approximation of the Fréchet distance between any shortest path P and any walk Q in O(|G|log|G|/√ε+|P|+|Q|/ε) time. We generalise this result to near-shortest paths, i.e. κ-straight paths, as we show how to compute a (1+ε)-approximation between a κ-straight path P and any walk Q in O(|G|log|G|/√ε+|P|+(κ|Q|)/ε) time. Our algorithmic results hold for both the strong and the weak discrete Fréchet distance over the shortest path metric in G.
Finally, we show that additional assumptions on the input, such as our assumption on path straightness, are indeed necessary to obtain truly subquadratic running time. We provide a conditional lower bound showing that the Fréchet distance, or even its 1.01-approximation, between arbitrary paths in a weighted planar graph cannot be computed in O((|P|⋅|Q|)^{1-δ}) time for any δ > 0 unless the Orthogonal Vector Hypothesis fails. For walks, this lower bound holds even when G is planar, unit-weight and has O(1) vertices.

Anne Driemel, Ivor van der Hoog, and Eva Rotenberg. On the Discrete Fréchet Distance in a Graph. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{driemel_et_al:LIPIcs.SoCG.2022.36, author = {Driemel, Anne and van der Hoog, Ivor and Rotenberg, Eva}, title = {{On the Discrete Fr\'{e}chet Distance in a Graph}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {36:1--36:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.36}, URN = {urn:nbn:de:0030-drops-160448}, doi = {10.4230/LIPIcs.SoCG.2022.36}, annote = {Keywords: Fr\'{e}chet, graphs, planar, complexity analysis} }

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Invited Talk

**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In graph algorithms, many questions about a graph can be answered in time proportional to the size of the input, and such linear time algorithms are considered the epitome of efficiency. However, when the graph changes slightly, e.g. by the insertion or deletion of an edge or a vertex, it is undesirable to consider the entire input again. Rather, one would wish to keep some of the partial answers to questions about the old graph, and re-use them when computing answers to questions about the resulting graph. The art of handling such changes is studied in dynamic graph algorithms.
In this talk, we will see some examples of ideas and techniques for efficiently maintaining knowledge about a dynamically changing graph. We will consider classical and natural graph properties such as connectivity and planarity, and we will focus on deterministic algorithms.

Eva Rotenberg. On Dynamic Graphs (Invited Talk). In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{rotenberg:LIPIcs.MFCS.2021.4, author = {Rotenberg, Eva}, title = {{On Dynamic Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {4:1--4:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.4}, URN = {urn:nbn:de:0030-drops-144445}, doi = {10.4230/LIPIcs.MFCS.2021.4}, annote = {Keywords: Graph algorithms, dynamic graphs, connectivity, planarity, matching, online algorithms} }

Document

**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time.
As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

Jacob Holm and Eva Rotenberg. Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{holm_et_al:LIPIcs.STACS.2021.42, author = {Holm, Jacob and Rotenberg, Eva}, title = {{Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {42:1--42:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.42}, URN = {urn:nbn:de:0030-drops-136875}, doi = {10.4230/LIPIcs.STACS.2021.42}, annote = {Keywords: Dynamic graphs, 2-connectivity, graph minors, r-divisions, graph separators} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string P, report all occurrences of P within S. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-k close consecutive occurrences problem (Sitcco). Here, a consecutive occurrence is a pair (i,j), i < j, such that P occurs at positions i and j in S and there is no occurrence of P between i and j, and their distance is defined as j-i. Given a pattern P and a parameter k, the goal is to report the top-k consecutive occurrences of P in S of minimal distance. The challenge is to compactly represent S while supporting queries in time close to the length of P and k. We give two time-space trade-offs for the problem. Let n be the length of S, m the length of P, and ε ∈ (0,1]. Our first result achieves O(nlog n) space and optimal query time of O(m+k), and our second result achieves linear space and query time O(m+k^{1+ε}). Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.

Philip Bille, Inge Li Gørtz, Max Rishøj Pedersen, Eva Rotenberg, and Teresa Anna Steiner. String Indexing for Top-k Close Consecutive Occurrences. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bille_et_al:LIPIcs.FSTTCS.2020.14, author = {Bille, Philip and G{\o}rtz, Inge Li and Pedersen, Max Rish{\o}j and Rotenberg, Eva and Steiner, Teresa Anna}, title = {{String Indexing for Top-k Close Consecutive Occurrences}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {14:1--14:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.14}, URN = {urn:nbn:de:0030-drops-132558}, doi = {10.4230/LIPIcs.FSTTCS.2020.14}, annote = {Keywords: String indexing, pattern matching, consecutive occurrences} }

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Track A: Algorithms, Complexity and Games

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

Given a string S of length n, its Lyndon array identifies for each suffix S[i..n] the next lexicographically smaller suffix S[j..n], i.e. the minimal index j > i with S[i..n] ≻ S[j..n]. Apart from its plain (n log₂ n)-bit array representation, the Lyndon array can also be encoded as a succinct parentheses sequence that requires only 2n bits of space. While linear time construction algorithms for both representations exist, it has previously been unknown if the same time bound can be achieved with less than Ω(n lg n) bits of additional working space. We show that, in fact, o(n) additional bits are sufficient to compute the succinct 2n-bit version of the Lyndon array in linear time. For the plain (n log₂ n)-bit version, we only need 𝒪(1) additional words to achieve linear time. Our space efficient construction algorithm makes the Lyndon array more accessible as a fundamental data structure in applications like full-text indexing.

Philip Bille, Jonas Ellert, Johannes Fischer, Inge Li Gørtz, Florian Kurpicz, J. Ian Munro, and Eva Rotenberg. Space Efficient Construction of Lyndon Arrays in Linear Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bille_et_al:LIPIcs.ICALP.2020.14, author = {Bille, Philip and Ellert, Jonas and Fischer, Johannes and G{\o}rtz, Inge Li and Kurpicz, Florian and Munro, J. Ian and Rotenberg, Eva}, title = {{Space Efficient Construction of Lyndon Arrays in Linear Time}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {14:1--14:18}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.14}, URN = {urn:nbn:de:0030-drops-124211}, doi = {10.4230/LIPIcs.ICALP.2020.14}, annote = {Keywords: String algorithms, string suffixes, succinct data structures, Lyndon word, Lyndon array, nearest smaller values, nearest smaller suffixes} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over Omega(n) operations, is O(log^2 n). Via SPQR-trees, we give a decremental data structure for maintaining 3-vertex connectivity in planar graphs. It answers queries in O(1) time and processes edge deletions and contractions in O(log^2 n) amortized time. The previous best supported deletions and insertions in O(sqrt{n}) time.

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, and Eva Rotenberg. Decremental SPQR-trees for Planar Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{holm_et_al:LIPIcs.ESA.2018.46, author = {Holm, Jacob and Italiano, Giuseppe F. and Karczmarz, Adam and Lacki, Jakub and Rotenberg, Eva}, title = {{Decremental SPQR-trees for Planar Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {46:1--46:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.46}, URN = {urn:nbn:de:0030-drops-95091}, doi = {10.4230/LIPIcs.ESA.2018.46}, annote = {Keywords: Graph embeddings, data structures, graph algorithms, planar graphs, SPQR-trees, triconnectivity} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set Gamma subseteq [1..n] is a k-attractor for a string S in Sigma^n if and only if every distinct substring of S of length at most k has an occurrence crossing at least one of the positions in Gamma. Finding the smallest k-attractor is NP-hard for k >= 3, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the k-attractor problem to a set-cover instance where the string's positions are interpreted as sets of substrings. The main result of this paper is a much more powerful reduction based on the truncated suffix tree. Our new characterization of the problem leads to more efficient algorithms for string attractors: we show how to check the validity and minimality of a k-attractor in near-optimal time and how to quickly compute exact solutions. For example, we prove that a minimum 3-attractor can be found in O(n) time when |Sigma| in O(sqrt[3+epsilon]{log n}) for some constant epsilon > 0, despite the problem being NP-hard for large Sigma.

Dominik Kempa, Alberto Policriti, Nicola Prezza, and Eva Rotenberg. String Attractors: Verification and Optimization. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 52:1-52:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kempa_et_al:LIPIcs.ESA.2018.52, author = {Kempa, Dominik and Policriti, Alberto and Prezza, Nicola and Rotenberg, Eva}, title = {{String Attractors: Verification and Optimization}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {52:1--52:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.52}, URN = {urn:nbn:de:0030-drops-95153}, doi = {10.4230/LIPIcs.ESA.2018.52}, annote = {Keywords: Dictionary compression, String attractors, Set cover} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math. Monthly, 1939] asserts that such an orientation exists if and only if the graph is 2-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph are partitioned into trails. Can the trails be oriented consistently such that the resulting directed graph is strongly connected?
We show that 2-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation.
The generalised Robbins' theorem [Boesch, Am. Math. Monthly, 1980] for mixed multigraphs asserts that the undirected edges of a mixed multigraph can be oriented to make the resulting directed graph strongly connected exactly when the mixed graph is strongly connected and the underlying graph is bridgeless.
We consider the natural extension where the undirected edges of a mixed multigraph are partitioned into trails. It turns out that in this case the condition of the generalised Robbin's Theorem is not sufficient. However, we show that as long as each cut either contains at least 2 undirected edges or directed edges in both directions, there exists an orientation of the trails such that the resulting directed graph is strongly connected. Moreover, if the condition is satisfied, we may start by orienting an arbitrary trail in an arbitrary direction. Using this result one obtains a very simple polynomial time algorithm for finding a strong trail orientation if it exists, both in the undirected and the mixed setting.

Anders Aamand, Niklas Hjuler, Jacob Holm, and Eva Rotenberg. One-Way Trail Orientations. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{aamand_et_al:LIPIcs.ICALP.2018.6, author = {Aamand, Anders and Hjuler, Niklas and Holm, Jacob and Rotenberg, Eva}, title = {{One-Way Trail Orientations}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {6:1--6:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.6}, URN = {urn:nbn:de:0030-drops-90109}, doi = {10.4230/LIPIcs.ICALP.2018.6}, annote = {Keywords: Graph algorithms, Robbins' theorem, Graph orientation} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We present a data structure that can maintain a simple planar graph under edge contractions in linear total time. The data structure supports adjacency queries and provides access to neighbor lists in O(1) time. Moreover, it can report all the arising self-loops and parallel edges.
By applying the data structure, we can achieve optimal running times for decremental bridge detection, 2-edge connectivity, maximal 3-edge connected components, and the problem of finding a unique perfect matching for a static planar graph. Furthermore, we improve the running times of algorithms for several planar graph problems, including decremental 2-vertex and 3-edge connectivity, and we show that using our data structure in a black-box manner, one obtains conceptually simple optimal algorithms for computing MST and 5-coloring in planar graphs.

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, and Piotr Sankowski. Contracting a Planar Graph Efficiently. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{holm_et_al:LIPIcs.ESA.2017.50, author = {Holm, Jacob and Italiano, Giuseppe F. and Karczmarz, Adam and Lacki, Jakub and Rotenberg, Eva and Sankowski, Piotr}, title = {{Contracting a Planar Graph Efficiently}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {50:1--50:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.50}, URN = {urn:nbn:de:0030-drops-78755}, doi = {10.4230/LIPIcs.ESA.2017.50}, annote = {Keywords: planar graphs, algorithms, data structures, connectivity, coloring} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

We answer the following question dating back to J.E. Littlewood (1885-1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to be points moving with at most unit speed. That the lakes are rectifiable means that their boundaries are finitely long. This requirement is to avoid pathological examples where the man survives forever because any path to the lions is infinitely long. We show that the answer to the question is not always "yes", by giving an example of a region R in the plane where the man has a strategy to survive forever. R is a polygonal region with holes and the exterior and interior boundaries are pairwise disjoint, simple polygons. Our construction is the first truly two-dimensional example where the man can survive.
Next, we consider the following game played on the entire plane instead of a bounded area: There is any finite number of unit speed lions and one fast man who can run with speed 1+epsilon for some value epsilon>0. Can the man always survive? We answer the question in the affirmative for any constant epsilon>0.

Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilsen. Best Laid Plans of Lions and Men. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2017.6, author = {Abrahamsen, Mikkel and Holm, Jacob and Rotenberg, Eva and Wulff-Nilsen, Christian}, title = {{Best Laid Plans of Lions and Men}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {6:1--6:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.6}, URN = {urn:nbn:de:0030-drops-72053}, doi = {10.4230/LIPIcs.SoCG.2017.6}, annote = {Keywords: Lion and man game, Pursuit evasion game, Winning strategy} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Graph reconstruction algorithms seek to learn a hidden graph by repeatedly querying a black-box oracle for information about the graph structure. Perhaps the most well studied and applied version of the problem uses a distance oracle, which can report the shortest path distance between any pair of nodes.
We introduce and study the betweenness oracle, where bet(a, m, z) is true iff m lies on a shortest path between a and z. This oracle is strictly weaker than a distance oracle, in the sense that a betweenness query can be simulated by a constant number of distance queries, but not vice versa. Despite this, we are able to develop betweenness reconstruction algorithms that match the current state of the art for distance reconstruction, and even improve it for certain types of graphs. We obtain the following algorithms: (1) Reconstruction of general graphs in O(n^2) queries, (2) Reconstruction of degree-bounded graphs in ~O(n^{3/2}) queries, (3) Reconstruction of geodetic degree-bounded graphs in ~O(n) queries
In addition to being a fundamental graph theoretic problem with some natural applications, our new results shed light on some avenues for progress in the distance reconstruction problem.

Mikkel Abrahamsen, Greg Bodwin, Eva Rotenberg, and Morten Stöckel. Graph Reconstruction with a Betweenness Oracle. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abrahamsen_et_al:LIPIcs.STACS.2016.5, author = {Abrahamsen, Mikkel and Bodwin, Greg and Rotenberg, Eva and St\"{o}ckel, Morten}, title = {{Graph Reconstruction with a Betweenness Oracle}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.5}, URN = {urn:nbn:de:0030-drops-57068}, doi = {10.4230/LIPIcs.STACS.2016.5}, annote = {Keywords: graph reconstruction, bounded degree graphs, query complexity} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We present an algorithm to support the dynamic embedding in the plane
of a dynamic graph. An edge can be inserted across a face between two vertices on the boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Given vertices u,v, linkable(u,v) decides whether u and v are linkable, and if so, returns a list of suggestions for the placement of (u,v) in the embedding. For non-linkable vertices u,v, we define a new query, one-flip-linkable(u,v) providing a suggestion for a flip that will make them linkable if one exists. We will support all updates and queries in O(log^2 n) time. Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph.
Our new algorithm is simpler, exploiting that the complement of a spanning tree of a connected plane graph is a spanning tree of the dual graph. The primal and dual trees are interpreted as having the same Euler tour, and a main idea of the new algorithm is an elegant
interaction between top trees over the two trees via their common Euler tour.

Jacob Holm and Eva Rotenberg. Dynamic Planar Embeddings of Dynamic Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 434-446, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{holm_et_al:LIPIcs.STACS.2015.434, author = {Holm, Jacob and Rotenberg, Eva}, title = {{Dynamic Planar Embeddings of Dynamic Graphs}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {434--446}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.434}, URN = {urn:nbn:de:0030-drops-49319}, doi = {10.4230/LIPIcs.STACS.2015.434}, annote = {Keywords: dynamic graphs, planar embeddings, data structures} }

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