10 Search Results for "Seybold, Martin P."

Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.2
Covering Simple Orthogonal Polygons with Rectangles

Authors: Aniket Basu Roy

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We study the problem of Covering Orthogonal Polygons with Rectangles, focusing on three variants: covering the interior, the boundary, and the corners. While previous work provided constant-factor approximation algorithms for these problems, significant improvements had not been achieved for over two decades. The main contribution of this work is the development of a Polynomial Time Approximation Scheme (PTAS) for both the Boundary Cover and Corner Cover problems on simple polygons, using a local search algorithm. Our work advances the state of the art, improving upon the previous best-known 4-approximation for the Boundary Cover and 2-approximation for the Corner Cover problems. The technical core of our work lies in proving the existence of planar support graphs for certain geometric hypergraphs defined by the polygon and its containment-maximal rectangles. This structural insight enables the application of the local search framework to achieve the PTAS results. We also demonstrate the limitations of this approach by constructing instances where local search fails for the Interior Cover and certain dual problems, such as the Maximum Antirectangle and Hitting Set problems. Additionally, the methods yield a PTAS for a special case of the Discrete Independent Set problem for rectangles. These results not only settle longstanding open questions but also introduce new techniques that may be of independent interest within computational geometry.
Cite as

Aniket Basu Roy. Covering Simple Orthogonal Polygons with Rectangles. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{basuroy:LIPIcs.APPROX/RANDOM.2025.2,
  author =	{Basu Roy, Aniket},
  title =	{{Covering Simple Orthogonal Polygons with Rectangles}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{2:1--2:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.2},
  URN =		{urn:nbn:de:0030-drops-243686},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.2},
  annote =	{Keywords: Polygon Covering, Approximation Algorithms, Orthogonal Polygons, Rectangles, Local Search, Planar Supports}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.24
Computational Geometry with Probabilistically Noisy Primitive Operations

Authors: David Eppstein, Michael T. Goodrich, and Vinesh Sridhar

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Much prior work has been done on designing computational geometry algorithms that handle input degeneracies, data imprecision, and arithmetic round-off errors. We take a new approach, inspired by the noisy sorting literature, and study computational geometry algorithms subject to noisy Boolean primitive operations in which, e.g., the comparison "is point q above line 𝓁?" returns the wrong answer with some fixed probability. We propose a novel technique called path-guided pushdown random walks that generalizes the results of noisy sorting. We apply this technique to solve point-location, plane-sweep, convex hulls in 2D and 3D, and Delaunay triangulations for noisy primitives in optimal time with high probability.
Cite as

David Eppstein, Michael T. Goodrich, and Vinesh Sridhar. Computational Geometry with Probabilistically Noisy Primitive Operations. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eppstein_et_al:LIPIcs.WADS.2025.24,
  author =	{Eppstein, David and Goodrich, Michael T. and Sridhar, Vinesh},
  title =	{{Computational Geometry with Probabilistically Noisy Primitive Operations}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.24},
  URN =		{urn:nbn:de:0030-drops-242552},
  doi =		{10.4230/LIPIcs.WADS.2025.24},
  annote =	{Keywords: Computational geometry, noisy comparisons, random walks}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.47
B-Treaps Revised: Write Efficient Randomized Block Search Trees with High Load

Authors: Roodabeh Safavi and Martin P. Seybold

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Uniquely represented (UR) data structures represent each logical state with a unique storage state. We study the problem of maintaining a dynamic set of n keys from a totally ordered universe in this context. UR structures are also called "strongly history independent" structures in the literature. We introduce a two-layer data structure called (α,ε)-Randomized Block Search Tree (RBST) that is uniquely represented and suitable for external memory (EM). Though RBSTs naturally generalize the well-known binary Treaps, several new ideas are needed to analyze the expected search, update, and storage efficiency in terms of block-reads, block-writes, and blocks stored. We prove that searches have O(ε^{-1} + log_α n) block-reads, that dynamic updates perform O(ε^{-1} + log_α(n)/α) block-writes and O(ε^{-2}+(1+(ε^{-1}+log n)/α)log_α n) block-reads, and that (α, ε)-RBSTs have an asymptotic load-factor of at least (1-ε) for every ε ∈ (0,1/2]. Thus (α, ε)-RBSTs improve on the known, uniquely represented B-Treap [Golovin; ICALP'09]. Compared with non-UR structures, the RBST is also, to the best of our knowledge, the first external memory structure that is storage-efficient and has a non-amortized, write-efficient update bound.
Cite as

Roodabeh Safavi and Martin P. Seybold. B-Treaps Revised: Write Efficient Randomized Block Search Trees with High Load. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{safavi_et_al:LIPIcs.WADS.2025.47,
  author =	{Safavi, Roodabeh and Seybold, Martin P.},
  title =	{{B-Treaps Revised: Write Efficient Randomized Block Search Trees with High Load}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{47:1--47:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.47},
  URN =		{urn:nbn:de:0030-drops-242786},
  doi =		{10.4230/LIPIcs.WADS.2025.47},
  annote =	{Keywords: Unique Representation, Randomization, Top-Down Analysis, Block Search Tree, Write-Efficiency, Storage-Efficiency}
}
Document
DOI: 10.4230/LIPIcs.SEA.2025.7
Continuous Map Matching to Paths Under Travel Time Constraints

Authors: Yannick Bosch and Sabine Storandt

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract
In this paper, we study the problem of map matching with travel time constraints. Given a sequence of k spatio-temporal measurements and an embedded path graph with travel time costs, the goal is to snap each measurement to a close-by location in the graph, such that consecutive locations can be reached from one another along the path within the timestamp difference of the respective measurements. This problem arises in public transit data processing as well as in map matching of movement trajectories to general graphs. We show that the classical approach for this problem, which relies on selecting a finite set of candidate locations in the graph for each measurement, cannot guarantee to find a consistent solution. We propose a new algorithm that can deal with an infinite set of candidate locations per measurement. We prove that our algorithm always detects a consistent map matching path (if one exists). Despite the enlarged candidate set, we also demonstrate that our algorithm has superior running time in theory and practice. For a path graph with n nodes, we show that our algorithm runs in 𝒪(k² n log {nk}) and under mild assumptions in 𝒪(k n ^λ + n log³ n) for λ ≈ 0.695. This is a significant improvement over the baseline, which runs in 𝒪(k n²) and which might not even identify a correct solution. The performance of our algorithm hinges on an efficient segment-circle intersection data structure. We describe how to design and implement such a data structure for our application. In the experimental evaluation, we demonstrate the usefulness of our novel algorithm on a diverse set of generated measurements as well as GTFS data.
Cite as

Yannick Bosch and Sabine Storandt. Continuous Map Matching to Paths Under Travel Time Constraints. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bosch_et_al:LIPIcs.SEA.2025.7,
  author =	{Bosch, Yannick and Storandt, Sabine},
  title =	{{Continuous Map Matching to Paths Under Travel Time Constraints}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.7},
  URN =		{urn:nbn:de:0030-drops-232457},
  doi =		{10.4230/LIPIcs.SEA.2025.7},
  annote =	{Keywords: Map matching, Travel time, Segment-circle intersection data structure}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.117
Incremental Approximate Single-Source Shortest Paths with Predictions

Authors: Samuel McCauley, Benjamin Moseley, Aidin Niaparast, Helia Niaparast, and Shikha Singh

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
The algorithms-with-predictions framework has been used extensively to develop online algorithms with improved beyond-worst-case competitive ratios. Recently, there is growing interest in leveraging predictions for designing data structures with improved beyond-worst-case running times. In this paper, we study the fundamental data structure problem of maintaining approximate shortest paths in incremental graphs in the algorithms-with-predictions model. Given a sequence σ of edges that are inserted one at a time, the goal is to maintain approximate shortest paths from the source to each vertex in the graph at each time step. Before any edges arrive, the data structure is given a prediction of the online edge sequence σ̂ which is used to "warm start" its state. As our main result, we design a learned algorithm that maintains (1+ε)-approximate single-source shortest paths, which runs in Õ(m η log W/ε) time, where W is the weight of the heaviest edge and η is the prediction error. We show these techniques immediately extend to the all-pairs shortest-path setting as well. Our algorithms are consistent (performing nearly as fast as the offline algorithm) when predictions are nearly perfect, have a smooth degradation in performance with respect to the prediction error and, in the worst case, match the best offline algorithm up to logarithmic factors. That is, the algorithms are "ideal" in the algorithms-with-predictions model. As a building block, we study the offline incremental approximate single-source shortest-path (SSSP) problem. In the offline incremental SSSP problem, the edge sequence σ is known a priori and the goal is to construct a data structure that can efficiently return the length of the shortest paths in the intermediate graph G_t consisting of the first t edges, for all t. Note that the offline incremental problem is defined in the worst-case setting (without predictions) and is of independent interest.
Cite as

Samuel McCauley, Benjamin Moseley, Aidin Niaparast, Helia Niaparast, and Shikha Singh. Incremental Approximate Single-Source Shortest Paths with Predictions. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 117:1-117:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mccauley_et_al:LIPIcs.ICALP.2025.117,
  author =	{McCauley, Samuel and Moseley, Benjamin and Niaparast, Aidin and Niaparast, Helia and Singh, Shikha},
  title =	{{Incremental Approximate Single-Source Shortest Paths with Predictions}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{117:1--117:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.117},
  URN =		{urn:nbn:de:0030-drops-234946},
  doi =		{10.4230/LIPIcs.ICALP.2025.117},
  annote =	{Keywords: Algorithms with Predictions, Shortest Paths, Approximation Algorithms, Dynamic Graph Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.36
Faster Fréchet Distance Under Transformations

Authors: Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the problem of computing the Fréchet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves π and σ of total complexity n and a threshold δ ≥ 0, we present an 𝒪̃(n^{7 + 1/3}) time algorithm to determine whether there exists a translation t ∈ ℝ² such that the Fréchet distance between π and σ + t is at most δ. This improves on the previous best result, which is an 𝒪(n⁸) time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class T of rationally parametrized transformations with k degrees of freedom, we show that one can determine whether there is a transformation τ ∈ T such that the Fréchet distance between π and τ(σ) is at most δ in 𝒪̃(n^{3k+4/3}) time.
Cite as

Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong. Faster Fréchet Distance Under Transformations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{buchin_et_al:LIPIcs.ICALP.2025.36,
  author =	{Buchin, Kevin and Buchin, Maike and Huang, Zijin and Nusser, Andr\'{e} and Wong, Sampson},
  title =	{{Faster Fr\'{e}chet Distance Under Transformations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.36},
  URN =		{urn:nbn:de:0030-drops-234137},
  doi =		{10.4230/LIPIcs.ICALP.2025.36},
  annote =	{Keywords: Fr\'{e}chet distance, curve similarity, shape matching}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.44
Learning-Augmented Streaming Algorithms for Approximating MAX-CUT

Authors: Yinhao Dong, Pan Peng, and Ali Vakilian

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We study learning-augmented streaming algorithms for estimating the value of MAX-CUT in a graph. In the classical streaming model, while a 1/2-approximation for estimating the value of MAX-CUT can be trivially achieved with O(1) words of space, Kapralov and Krachun [STOC’19] showed that this is essentially the best possible: for any ε > 0, any (randomized) single-pass streaming algorithm that achieves an approximation ratio of at least 1/2 + ε requires Ω(n / 2^poly(1/ε)) space. We show that it is possible to surpass the 1/2-approximation barrier using just O(1) words of space by leveraging a (machine learned) oracle. Specifically, we consider streaming algorithms that are equipped with an ε-accurate oracle that for each vertex in the graph, returns its correct label in {-1, +1}, corresponding to an optimal MAX-CUT solution in the graph, with some probability 1/2 + ε, and the incorrect label otherwise. Within this framework, we present a single-pass algorithm that approximates the value of MAX-CUT to within a factor of 1/2 + Ω(ε²) with probability at least 2/3 for insertion-only streams, using only poly(1/ε) words of space. We also extend our algorithm to fully dynamic streams while maintaining a space complexity of poly(1/ε,log n) words.
Cite as

Yinhao Dong, Pan Peng, and Ali Vakilian. Learning-Augmented Streaming Algorithms for Approximating MAX-CUT. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 44:1-44:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dong_et_al:LIPIcs.ITCS.2025.44,
  author =	{Dong, Yinhao and Peng, Pan and Vakilian, Ali},
  title =	{{Learning-Augmented Streaming Algorithms for Approximating MAX-CUT}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{44:1--44:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.44},
  URN =		{urn:nbn:de:0030-drops-226728},
  doi =		{10.4230/LIPIcs.ITCS.2025.44},
  annote =	{Keywords: Learning-Augmented Algorithms, Graph Streaming Algorithms, MAX-CUT}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.62
Approximating Multiplicatively Weighted Voronoi Diagrams: Efficient Construction with Linear Size

Authors: Joachim Gudmundsson, Martin P. Seybold, and Sampson Wong

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Given a set of n sites from ℝ^d, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is minimal for all points in the cell. We give novel approximation algorithms that output a cube-based subdivision such that the weighted distance of a point with respect to the associated site is at most (1+ε) times the minimum weighted distance, for any fixed parameter ε ∈ (0,1). The diagram size is O_d(n log(1/ε)/ε^{d-1}) and the construction time is within an O_D(log(n)/ε^{(d+5)/2})-factor of the size bound. We also prove a matching lower bound for the size, showing that the proposed method is the first to achieve optimal size, up to Θ(1)^d-factors. In particular, the obscure log(1/ε) factor is unavoidable. As a by-product, we obtain a factor d^{O(d)} improvement in size for the unweighted case and O(d log(n) + d² log(1/ε)) point-location time in the subdivision, improving the known query bound by one d-factor. The key ingredients of our approximation algorithms are the study of convex regions that we call cores, an adaptive refinement algorithm to obtain optimal size, and a novel notion of bisector coresets, which may be of independent interest. In particular, we show that coresets with O_d(1/ε^{(d+3)/2}) worst-case size can be computed in near-linear time.
Cite as

Joachim Gudmundsson, Martin P. Seybold, and Sampson Wong. Approximating Multiplicatively Weighted Voronoi Diagrams: Efficient Construction with Linear Size. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gudmundsson_et_al:LIPIcs.SoCG.2024.62,
  author =	{Gudmundsson, Joachim and Seybold, Martin P. and Wong, Sampson},
  title =	{{Approximating Multiplicatively Weighted Voronoi Diagrams: Efficient Construction with Linear Size}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{62:1--62:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.62},
  URN =		{urn:nbn:de:0030-drops-200078},
  doi =		{10.4230/LIPIcs.SoCG.2024.62},
  annote =	{Keywords: Multiplicatively Weighted Voronoi Diagram, Compressed QuadTree, Adaptive Refinement, Bisector Coresets, Semi-Separated Pair Decomposition, Lower Bound}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.62
On the Complexity of Algorithms with Predictions for Dynamic Graph Problems

Authors: Monika Henzinger, Barna Saha, Martin P. Seybold, and Christopher Ye

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Algorithms with predictions is a new research direction that leverages machine learned predictions for algorithm design. So far a plethora of recent works have incorporated predictions to improve on worst-case bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike online algorithms, the goal in dynamic data structures is to maintain the solution efficiently with every update. We investigate three natural models of prediction: (1) δ-accurate predictions where each predicted request matches the true request with probability δ, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are a permutation of the predicted requests. We give general reductions among the prediction models, showing that bounded delay is the strongest prediction model, followed by list-accurate, and δ-accurate. Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that locally correctable dynamic problems have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, we show that locally reducible dynamic problems have time complexity that degrades gracefully with the quality of bounded delay predictions. We categorize problems with known OMv lower bounds accordingly and give several upper bounds in the delay model that show that our lower bounds are almost tight. We note that concurrent work by v.d.Brand et al. [SODA '24] and Liu and Srinivas [arXiv:2307.08890] independently study dynamic graph algorithms with predictions, but their work is mostly focused on showing upper bounds.
Cite as

Monika Henzinger, Barna Saha, Martin P. Seybold, and Christopher Ye. On the Complexity of Algorithms with Predictions for Dynamic Graph Problems. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 62:1-62:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{henzinger_et_al:LIPIcs.ITCS.2024.62,
  author =	{Henzinger, Monika and Saha, Barna and Seybold, Martin P. and Ye, Christopher},
  title =	{{On the Complexity of Algorithms with Predictions for Dynamic Graph Problems}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{62:1--62:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.62},
  URN =		{urn:nbn:de:0030-drops-195907},
  doi =		{10.4230/LIPIcs.ITCS.2024.62},
  annote =	{Keywords: Dynamic Graph Algorithms, Algorithms with Predictions}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.18
A Simple Dynamization of Trapezoidal Point Location in Planar Subdivisions

Authors: Milutin Brankovic, Nikola Grujic, André van Renssen, and Martin P. Seybold

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

Abstract
We study how to dynamize the Trapezoidal Search Tree (TST) - a well known randomized point location structure for planar subdivisions of kinetic line segments. Our approach naturally extends incremental leaf-level insertions to recursive methods and allows adaptation for the online setting. The dynamization carries over to the Trapezoidal Search DAG (TSD), which has linear size and logarithmic point location costs with high probability. On a set S of non-crossing segments, each TST update performs expected 𝒪(log²|S|) operations and each TSD update performs expected 𝒪(log |S|) operations. We demonstrate the practicality of our method with an open-source implementation, based on the Computational Geometry Algorithms Library, and experiments on the update performance.
Cite as

Milutin Brankovic, Nikola Grujic, André van Renssen, and Martin P. Seybold. A Simple Dynamization of Trapezoidal Point Location in Planar Subdivisions. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{brankovic_et_al:LIPIcs.ICALP.2020.18,
  author =	{Brankovic, Milutin and Grujic, Nikola and van Renssen, Andr\'{e} and Seybold, Martin P.},
  title =	{{A Simple Dynamization of Trapezoidal Point Location in Planar Subdivisions}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{18:1--18:18},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.18},
  URN =		{urn:nbn:de:0030-drops-124253},
  doi =		{10.4230/LIPIcs.ICALP.2020.18},
  annote =	{Keywords: Dynamization, Trapezoidal Search Tree, Trapezoidal Search DAG, Backward Analysis, Point Location, Planar Subdivision, Treap, Order-maintenance}
}
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