36 Search Results for "Afshani, Peyman"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.80
Hardness of High-Dimensional Linear Classification

Authors: Alexander Munteanu, Simon Omlor, and Jeff M. Phillips

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only O(n^d) and respectively Õ(1/ε^d) upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and k-Sum problems. Our reductions yield matching lower bounds of Ω̃(n^d) and respectively Ω̃(1/ε^d) based on Affine Degeneracy testing, and Ω̃(n^{d/2}) and respectively Ω̃(1/ε^{d/2}) conditioned on k-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.
Cite as

Alexander Munteanu, Simon Omlor, and Jeff M. Phillips. Hardness of High-Dimensional Linear Classification. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 80:1-80:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{munteanu_et_al:LIPIcs.SoCG.2026.80,
  author =	{Munteanu, Alexander and Omlor, Simon and Phillips, Jeff M.},
  title =	{{Hardness of High-Dimensional Linear Classification}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{80:1--80:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.80},
  URN =		{urn:nbn:de:0030-drops-258871},
  doi =		{10.4230/LIPIcs.SoCG.2026.80},
  annote =	{Keywords: Conditional Hardness, k-Sum, Affine Degeneracy, Halfspace Discrepancy, Classification}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.13
Line Cover and Related Problems

Authors: Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
We study several extensions of the classic Line Cover problem of covering a set of n points in the plane with k lines. Line Cover is known to be NP-hard and our focus is on two natural generalizations: (1) Line Clustering, where the objective is to find k lines in the plane that minimize the sum of squares of distances of a given set of input points to the closest line, and (2) Hyperplane Cover, where the goal is to cover n points in ℝ^d by k hyperplanes. We also consider the more general Projective Clustering problem, which unifies both of these and has numerous applications in machine learning, data mining, and computational geometry. In this problem one seeks k affine subspaces of dimension r minimizing the sum of squares of distances of a given set of n points in ℝ^d to the closest point within one of the k affine subspaces. Our main contributions reveal interesting differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable parameterized by the number k of lines in the solution, we show that Line Clustering is W[1]-hard when parameterized by k and rule out algorithms of running time n^{o(k)} under the Exponential Time Hypothesis. Hyperplane Cover is known to be NP-hard even when d = 2 and by the work of Langerman and Morin [Discrete & Computational Geometry, 2005], it is FPT parameterized by k and d. We complement this result by establishing that Hyperplane Cover is W[2]-hard when parameterized by only k. We complement our hardness results by presenting an algorithm for Projective Clustering. We show that this problem is solvable in n^{𝒪(dk(r+1))} time. Not only does this yield an upper bound for Line Clustering that asymptotically matches our lower bound, but it also significantly extends the seminal work on k-Means Clustering (the special case r = 0) by Inaba, Katoh, and Imai [SoCG 1994].
Cite as

Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana. Line Cover and Related Problems. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{bentert_et_al:LIPIcs.STACS.2026.13,
  author =	{Bentert, Matthias and Fomin, Fedor V. and Golovach, Petr A. and Saha, Souvik and Seetharaman, Sanjay and Upasana, Anannya},
  title =	{{Line Cover and Related Problems}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.13},
  URN =		{urn:nbn:de:0030-drops-255023},
  doi =		{10.4230/LIPIcs.STACS.2026.13},
  annote =	{Keywords: Point Line Cover, Projective Clustering, W-hardness, XP algorithm}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.55
Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs

Authors: Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Sampling a random walk is a fundamental primitive in many graph applications. In the streaming model, it is known that sampling an L-step random walk on an n-vertex directed graph requires Ω(n L) space, implying that no sublinear-space streaming algorithm exists for general graphs. We show that sublinear algorithms are possible for the case of dense graphs, where every vertex has out-degree at least Ω(n). In particular, we give a one-pass turnstile streaming algorithm that uses only 𝒪̃(L) memory for such graphs. More broadly, for graphs with minimum out-degree at least d, our streaming algorithm samples a random walk using 𝒪̃(n/d ⋅ L) memory. We show that our algorithm is optimal in a strong "beyond worst-case" sense. To formalize this, we introduce the notion of universal optimality for graph streaming algorithms. Informally, a streaming algorithm is universally optimal if it performs (almost) as well as possible on every graph, assuming a worst-case choice of the streaming order. This notion of universal optimality is a key conceptual contribution of our work.
Cite as

Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang. Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 55:1-55:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{efremenko_et_al:LIPIcs.ITCS.2026.55,
  author =	{Efremenko, Klim and Kol, Gillat and Saxena, Raghuvansh R. and Zhang, Zhijun},
  title =	{{Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{55:1--55:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.55},
  URN =		{urn:nbn:de:0030-drops-253423},
  doi =		{10.4230/LIPIcs.ITCS.2026.55},
  annote =	{Keywords: Random Walk, streaming Algorithm, universal Optimality}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.87
Range Longest Increasing Subsequence and Its Relatives

Authors: Karthik C. S. and Saladi Rahul

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Longest increasing subsequence (LIS) is a classical textbook problem which is still actively studied in various computational models. In this work, we present a few results for the range longest increasing subsequence problem (Range-LIS) and its variants. The input to Range-LIS is a sequence 𝒮 of n real numbers and a collection 𝒬 of m query ranges and for each query in 𝒬, the goal is to report the LIS of the sequence 𝒮 restricted to that query. Our two main results are for the following generalizations of the Range-LIS problem: 2D Range Queries: In this variant of the Range-LIS problem, each query is a pair of ranges, one of indices and the other of values, and we provide a randomized algorithm with running time Õ(mn^{1/2}+ n^{3/2})+O(k), where k is the cumulative length of the m output subsequences. This improves on the elementary Õ(mn) runtime algorithm when m = Ω(√n). Previously, the only known result breaking the quadratic barrier was of Tiskin [SODA'10] which could only handle 1D range queries (i.e., each query was a range of indices) and also just outputted the length of the LIS (instead of reporting the subsequence achieving that length). Subsequent to our paper, Gawrychowski, Gorbachev, and Kociumaka in a preprint have extended Tiskin’s approach to handle reporting 1D range queries in O(n(log n)³+m+k) time. Colored Sequences: In this variant of the Range-LIS problem, each element in 𝒮 is colored and for each query in 𝒬, the goal is to report a monochromatic LIS contained in the sequence 𝒮 restricted to that query. For 2D queries, we provide a randomized algorithm for this colored version with running time Õ(mn^{2/3}+ n^{5/3})+O(k). Moreover, for 1D queries, we provide an improved algorithm with running time Õ(mn^{1/2}+ n^{3/2})+O(k). Thus, we again improve on the elementary Õ(mn) runtime algorithm. Additionally, we prove that assuming the well-known Combinatorial Boolean Matrix Multiplication Hypothesis, that the runtime for 1D queries is essentially tight for combinatorial algorithms. Our algorithms combine several tools such as dynamic programming (to precompute increasing subsequences with some desirable properties), geometric data structures (to efficiently compute the dynamic programming entries), random sampling (to capture elements which are part of the LIS), classification of query ranges into large LIS and small LIS, and classification of colors into light and heavy. We believe that our techniques will be of interest to tackle other variants of LIS problem and other range-searching problems.
Cite as

Karthik C. S. and Saladi Rahul. Range Longest Increasing Subsequence and Its Relatives. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 87:1-87:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{karthikc.s._et_al:LIPIcs.ITCS.2026.87,
  author =	{Karthik C. S. and Rahul, Saladi},
  title =	{{Range Longest Increasing Subsequence and Its Relatives}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{87:1--87:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.87},
  URN =		{urn:nbn:de:0030-drops-253740},
  doi =		{10.4230/LIPIcs.ITCS.2026.87},
  annote =	{Keywords: Longest Increasing Subsequence, Range Query, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.3
Circle-Segment Intersection Queries in Connected Geometric Graphs

Authors: Peyman Afshani, Yannick Bosch, and Sabine Storandt

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
In this paper, we study the problem of efficiently reporting all intersections between a given set of line segments in the plane and a query circle, focusing on the case where the segments form the edges of a connected geometric graph. While previous data structures for circle-segment intersection queries on general segment sets incur high space or query time costs, we exploit the connectivity of the input to obtain significantly improved performance. In fact, we propose a new circle-segment intersection data structure that can be constructed in 𝒪((n + C) log³ n) time and space on connected graphs with n edges and C edge crossings. It answers intersection queries in 𝒪(k log³ n) time, where k denotes the output size. Our method relies on the construction of efficient circle-graph intersection oracles as well as a novel linear-time algorithm to partition the edges of the graph into balanced, connected components, which might be of independent interest. In a proof-of-concept experimental study on real-world road networks, we show that our novel data structure also performs well in practice. Even on networks with millions of edges, the construction time is within minutes and queries are answered in a few milliseconds.
Cite as

Peyman Afshani, Yannick Bosch, and Sabine Storandt. Circle-Segment Intersection Queries in Connected Geometric Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{afshani_et_al:LIPIcs.ISAAC.2025.3,
  author =	{Afshani, Peyman and Bosch, Yannick and Storandt, Sabine},
  title =	{{Circle-Segment Intersection Queries in Connected Geometric Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.3},
  URN =		{urn:nbn:de:0030-drops-249114},
  doi =		{10.4230/LIPIcs.ISAAC.2025.3},
  annote =	{Keywords: Intersection data structure, Graph partitioning, Dobkin-Kirkpatrick hierarchy}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.102
A Combinatorial Proof of Universal Optimality for Computing a Planar Convex Hull

Authors: Ivor van der Hoog, Eva Rotenberg, and Daniel Rutschmann

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
For a planar point set P, its convex hull is the smallest convex polygon that encloses all points in P. The construction of the convex hull from an array I_P containing P is a fundamental problem in computational geometry. By sorting I_P in lexicographical order, one can construct the convex hull of P in O(n log n) time which is worst-case optimal. Standard worst-case analysis, however, has been criticized as overly coarse or pessimistic, and researchers search for more refined analyses. For an algorithm A, worst-case analysis fixes n, and considers the maximum running time of A across all size-n point sets P and permutations I_P of P. Output-sensitive analysis fixes n and k, and considers the maximum running time across all size-n points sets P with k hull points and permutations I_P of P. Universal analysis provides an even stronger guarantee. It fixes a point set P and considers the maximum running time across all permutations I_P of P. Kirkpatrick, McQueen, and Seidel [SICOMP'86] consider output-sensitive analysis. If the convex hull of P contains k points, then their algorithm runs in O(n log k) time. Afshani, Barbay, Chan [FOCS'07] prove that the algorithm by Kirkpatrick, McQueen, and Seidel is also universally optimal. Their proof restricts the model of computation to any algebraic decision tree model where the test functions have at most constant degree and at most a constant number of arguments. They rely upon involved algebraic arguments to construct a lower bound for each point set P that matches the universal running time of [SICOMP'86]. We provide a different proof of universal optimality. Instead of restricting the computational model, we further specify the output. We require as output (1) the convex hull, and (2) for each internal point of P a witness for it being internal. Our argument is shorter, perhaps simpler, and applicable in more general models of computation.
Cite as

Ivor van der Hoog, Eva Rotenberg, and Daniel Rutschmann. A Combinatorial Proof of Universal Optimality for Computing a Planar Convex Hull. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 102:1-102:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2025.102,
  author =	{van der Hoog, Ivor and Rotenberg, Eva and Rutschmann, Daniel},
  title =	{{A Combinatorial Proof of Universal Optimality for Computing a Planar Convex Hull}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{102:1--102:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.102},
  URN =		{urn:nbn:de:0030-drops-245715},
  doi =		{10.4230/LIPIcs.ESA.2025.102},
  annote =	{Keywords: Convex hull, Combinatorial proofs, Universal optimality}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.25
Instance-Optimal Imprecise Convex Hull

Authors: Sarita de Berg, Ivor van der Hoog, Eva Rotenberg, Daniel Rutschmann, and Sampson Wong

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Imprecise measurements of a point set P = (p₁, …, p_n) can be modelled by a family of regions F = (R₁, …, R_n), where each imprecise region R_i ∈ F contains a unique point p_i ∈ P. A retrieval models an accurate measurement by replacing an imprecise region R_i with its corresponding point p_i. We construct the convex hull of an imprecise point set in the plane, by determining the cyclic ordering of the convex hull vertices of P as efficiently as possible. Efficiency is interpreted in two ways: (i) minimising the number of retrievals, and (ii) the computation time to determine the set of regions that must be retrieved. Previous works focused on only one of these two aspects: either minimising retrievals or optimising algorithmic runtime. Our contribution is the first to simultaneously achieve both. Let r(F, P) denote the minimal number of retrievals required by any algorithm to determine the convex hull of P for a given instance (F, P). For a family F of n constant-complexity polygons, our main result is a reconstruction algorithm that performs Θ(r(F, P)) retrievals in O(r(F, P) log³ n) time. Compared to previous approaches that achieve optimal retrieval counts, we improve the runtime per retrieval from polynomial to polylogarithmic. We extend the generality of previous results to simple k-gons, to pairwise disjoint disks with radii in [1,k], and to unit disks where at most k disks overlap in a single point. Our runtime scales linearly with k.
Cite as

Sarita de Berg, Ivor van der Hoog, Eva Rotenberg, Daniel Rutschmann, and Sampson Wong. Instance-Optimal Imprecise Convex Hull. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{deberg_et_al:LIPIcs.ESA.2025.25,
  author =	{de Berg, Sarita and van der Hoog, Ivor and Rotenberg, Eva and Rutschmann, Daniel and Wong, Sampson},
  title =	{{Instance-Optimal Imprecise Convex Hull}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{25:1--25:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.25},
  URN =		{urn:nbn:de:0030-drops-244932},
  doi =		{10.4230/LIPIcs.ESA.2025.25},
  annote =	{Keywords: convex hull, imprecise geometry preprocessing model, partial information}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.64
A Dynamic Piecewise-Linear Geometric Index with Worst-Case Guarantees

Authors: Emil Toftegaard Gæde, Ivor van der Hoog, Eva Rotenberg, and Tord Stordalen

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Indexing data is a fundamental problem in computer science. The input is a set S of n distinct integers from a universe 𝒰. Indexing queries take a value q ∈ 𝒰 and return the membership, predecessor or rank of q in S. A range query takes two values q, r ∈ 𝒰 and returns the set S ∩ [q,r]. Recently, various papers study a special case where the the input data behaves in an approximately piece-wise linear way. Given the sorted (rank,value) pairs, and given some constant ε, one wants to maintain a small number of axis-disjoint line-segments such that, for each rank, the value is within ± ε of the corresponding line-segment. Ferragina and Vinciguerra (VLDB 2020) observe that this geometric problem is useful for solving indexing problems, particularly when the number of line-segments is small compared to the size of the dataset. We study the dynamic version of this geometric problem. In the dynamic setting, inserting or deleting just one data point may cause up to three line-segments to be merged, or one line-segment to be split at most three-way. To determine and compute this, we use techniques from dynamic maintenance of convex hulls, and provide new algorithms with worst-case guarantees, including an O(log n) algorithm to compute a separating line between two non-intersecting convex hulls - an operation previously missing from the literature. We then use our fully-dynamic geometry-based subroutine in an indexing data structure, combining it with a natural hashing technique. The resulting indexing data structure has theoretically efficient worst-case guarantees in expectation. We compare its practical performance to the solution of Ferragina and Vinciguerra, which was shown to perform better in certain structured settings [Sun, Zhou, Li VLDB 2023]. Our empirical analysis shows that our solution supports more efficient range queries in the special case where the update sequence contains many deletions.
Cite as

Emil Toftegaard Gæde, Ivor van der Hoog, Eva Rotenberg, and Tord Stordalen. A Dynamic Piecewise-Linear Geometric Index with Worst-Case Guarantees. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 64:1-64:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{gaede_et_al:LIPIcs.ESA.2025.64,
  author =	{G{\ae}de, Emil Toftegaard and van der Hoog, Ivor and Rotenberg, Eva and Stordalen, Tord},
  title =	{{A Dynamic Piecewise-Linear Geometric Index with Worst-Case Guarantees}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{64:1--64:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.64},
  URN =		{urn:nbn:de:0030-drops-245323},
  doi =		{10.4230/LIPIcs.ESA.2025.64},
  annote =	{Keywords: Algorithms Engineering, Data Structures, Indexing, Convex Hulls}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.84
Property Testing of Curve Similarity

Authors: Peyman Afshani, Maike Buchin, Anne Driemel, Marena Richter, and Sampson Wong

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We propose sublinear algorithms for probabilistic testing of the discrete and continuous Fréchet distance - a standard similarity measure for curves. We assume the algorithm is given access to the input curves via a query oracle: a query returns the set of vertices of the curve that lie within a radius δ of a specified vertex of the other curve. The goal is to use a small number of queries to determine with constant probability whether the two curves are similar (i.e., their discrete Fréchet distance is at most δ) or they are "ε-far" (for 0 < ε < 2) from being similar, i.e., more than an ε-fraction of the two curves must be ignored for them to become similar. We present two algorithms which are sublinear assuming that the curves are t-approximate shortest paths in the ambient metric space, for some t ≪ n. The first algorithm uses O(t/ε log t/ε) queries and is given the value of t in advance. The second algorithm does not have explicit knowledge of the value of t and therefore needs to gain implicit knowledge of the straightness of the input curves through its queries. We show that the discrete Fréchet distance can still be tested using roughly O({t³+t² log n}/ε) queries ignoring logarithmic factors in t. Our algorithms work in a matrix representation of the input and may be of independent interest to matrix testing. Our algorithms use a mild uniform sampling condition that constrains the edge lengths of the curves, similar to a polynomially bounded aspect ratio. Applied to testing the continuous Fréchet distance of t-straight curves, our algorithms can be used for (1+ε')-approximate testing using essentially the same bounds as stated above with an additional factor of poly(1/(ε')).
Cite as

Peyman Afshani, Maike Buchin, Anne Driemel, Marena Richter, and Sampson Wong. Property Testing of Curve Similarity. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{afshani_et_al:LIPIcs.ESA.2025.84,
  author =	{Afshani, Peyman and Buchin, Maike and Driemel, Anne and Richter, Marena and Wong, Sampson},
  title =	{{Property Testing of Curve Similarity}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{84:1--84:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.84},
  URN =		{urn:nbn:de:0030-drops-245522},
  doi =		{10.4230/LIPIcs.ESA.2025.84},
  annote =	{Keywords: Fr\'{e}chet distance, Trajectory Analysis, Curve Similarity, Property Testing, Monotonicity Testing}
}
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)

Copy BibTex To Clipboard

@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.MFCS.2025.87
Lazy B-Trees

Authors: Casper Moldrup Rysgaard and Sebastian Wild

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Lazy search trees (Sandlund & Wild FOCS 2020, Sandlund & Zhang SODA 2022) are sorted dictionaries whose update and query performance smoothly interpolates between that of efficient priority queues and binary search trees - automatically, depending on actual use; no adjustments are necessary to the data structure to realize the cost savings. In this paper, we design lazy B-trees, a variant of lazy search trees suitable for external memory that generalizes the speedup of B-trees over binary search trees wrt. input/output operations to the same smooth interpolation regime. A key technical difficulty to overcome is the lack of a (fully satisfactory) external variant of biased search trees, on which lazy search trees crucially rely. We give a construction for a subset of performance guarantees sufficient to realize external-memory lazy search trees, which we deem of independent interest. As one special case, lazy B-trees can be used as an external-memory priority queue, in which case they are competitive with some tailor-made heaps; indeed, they offer faster decrease-key and insert operations than known data structures.
Cite as

Casper Moldrup Rysgaard and Sebastian Wild. Lazy B-Trees. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 87:1-87:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{rysgaard_et_al:LIPIcs.MFCS.2025.87,
  author =	{Rysgaard, Casper Moldrup and Wild, Sebastian},
  title =	{{Lazy B-Trees}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{87:1--87:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.87},
  URN =		{urn:nbn:de:0030-drops-241949},
  doi =		{10.4230/LIPIcs.MFCS.2025.87},
  annote =	{Keywords: B-tree, lazy search trees, lazy updates, external memory, deferred data structures, database cracking}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.23
On the Instance Optimality of Detecting Collisions and Subgraphs

Authors: Omri Ben-Eliezer, Tomer Grossman, and Moni Naor

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

Abstract
Suppose you are given a function f: [n] → [n] via (black-box) query access to the function. You are looking to find something local, like a collision (a pair x ≠ y s.t. f(x) = f(y)). The question is whether knowing the "shape" of the function helps you or not (by shape we mean that some permutation of the function is known). Formally, we investigate the unlabeled instance optimality of substructure detection problems in graphs and functions. A problem is g(n)-instance optimal if it admits an algorithm A satisfying that for any possible input, the (randomized) query complexity of A is at most g(n) times larger than the query complexity of any algorithm A' which solves the same problem while holding an unlabeled copy of the input (i.e., any A' that "knows the structure of the input"). Our results point to a trichotomy of unlabeled instance optimality among substructure detection problems in graphs and functions: - A few very simple properties have an O(1)-instance optimal algorithm. - Most properties of graphs and functions, with examples such as containing a fixed point or a 3-collision in functions, or a triangle in graphs, are n^{c}-far from instance optimal for some constant c > 0. - The problems of collision detection in functions and finding a claw in a graph serve as a middle ground between the two regimes. We show that these two properties are not Ω(log n)-instance optimal, and conjecture that this bound is tight. We provide evidence towards this conjecture, by proving that finding a claw in a graph is O(log(n))-instance optimal among all input graphs for which the query complexity of an algorithm holding an unlabeled certificate is O(√{n/(log n)}).
Cite as

Omri Ben-Eliezer, Tomer Grossman, and Moni Naor. On the Instance Optimality of Detecting Collisions and Subgraphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{beneliezer_et_al:LIPIcs.ICALP.2025.23,
  author =	{Ben-Eliezer, Omri and Grossman, Tomer and Naor, Moni},
  title =	{{On the Instance Optimality of Detecting Collisions and Subgraphs}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{23:1--23:14},
  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.2025.23},
  URN =		{urn:nbn:de:0030-drops-234002},
  doi =		{10.4230/LIPIcs.ICALP.2025.23},
  annote =	{Keywords: instance optimality, instance complexity, unlabeled certificate, subgraph detection, collision detection}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.3
Convexity Helps Iterated Search in 3D

Authors: Peyman Afshani, Yakov Nekrich, and Frank Staals

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Inspired by the classical fractional cascading technique [Bernard Chazelle and Leonidas J. Guibas, 1986; Bernard Chazelle and Leonidas J. Guibas, 1986], we introduce new techniques to speed up the following type of iterated search in 3D: The input is a graph 𝐆 with bounded degree together with a set H_v of 3D hyperplanes associated with every vertex of v of 𝐆. The goal is to store the input such that given a query point q ∈ ℝ³ and a connected subgraph 𝐇 ⊂ 𝐆, we can decide if q is below or above the lower envelope of H_v for every v ∈ 𝐇. We show that using linear space, it is possible to answer queries in roughly O(log n + |𝐇|√{log n}) time which improves trivial bound of O(|𝐇|log n) obtained by using planar point location data structures. Our data structure can in fact answer more general queries (it combines with shallow cuttings) and it even works when 𝐇 is given one vertex at a time. We show that this has a number of new applications and in particular, we give improved solutions to a set of natural data structure problems that up to our knowledge had not seen any improvements. We believe this is a very surprising result because obtaining similar results for the planar point location problem was known to be impossible [Chazelle and Liu, 2004].
Cite as

Peyman Afshani, Yakov Nekrich, and Frank Staals. Convexity Helps Iterated Search in 3D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{afshani_et_al:LIPIcs.SoCG.2025.3,
  author =	{Afshani, Peyman and Nekrich, Yakov and Staals, Frank},
  title =	{{Convexity Helps Iterated Search in 3D}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.3},
  URN =		{urn:nbn:de:0030-drops-231558},
  doi =		{10.4230/LIPIcs.SoCG.2025.3},
  annote =	{Keywords: Data structures, range searching}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.51
Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction

Authors: Shion Fukuzawa, Michael T. Goodrich, and Sandy Irani

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in Õ(√{nh}) time, w.h.p., where h is the size of the output.
Cite as

Shion Fukuzawa, Michael T. Goodrich, and Sandy Irani. Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{fukuzawa_et_al:LIPIcs.SoCG.2025.51,
  author =	{Fukuzawa, Shion and Goodrich, Michael T. and Irani, Sandy},
  title =	{{Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{51:1--51:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.51},
  URN =		{urn:nbn:de:0030-drops-232035},
  doi =		{10.4230/LIPIcs.SoCG.2025.51},
  annote =	{Keywords: quantum computing, computational geometry, divide and conquer, convex hulls, maxima sets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.42
Range Counting Oracles for Geometric Problems

Authors: Anne Driemel, Morteza Monemizadeh, Eunjin Oh, Frank Staals, and David P. Woodruff

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
In this paper, we study estimators for geometric optimization problems in the sublinear geometric model. In this model, we have oracle access to a point set with size n in a discrete space [Δ]^d, where queries can be made to an oracle that responds to orthogonal range counting requests. The query complexity of an optimization problem is measured by the number of oracle queries required to compute an estimator for the problem. We investigate two problems in this framework, the Euclidean Minimum Spanning Tree (MST) and Earth Mover Distance (EMD). For EMD, we show the existence of an estimator that approximates the cost of EMD with O(log Δ)-relative error and O(nΔ/(s^{1+1/d}))-additive error using O(s polylog Δ) range counting queries for any parameter s with 1 ≤ s ≤ n. Moreover, we prove that this bound is tight. For MST, we demonstrate that the weight of MST can be estimated within a factor of (1 ± ε) using Õ(√n) range counting queries.
Cite as

Anne Driemel, Morteza Monemizadeh, Eunjin Oh, Frank Staals, and David P. Woodruff. Range Counting Oracles for Geometric Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{driemel_et_al:LIPIcs.SoCG.2025.42,
  author =	{Driemel, Anne and Monemizadeh, Morteza and Oh, Eunjin and Staals, Frank and Woodruff, David P.},
  title =	{{Range Counting Oracles for Geometric Problems}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{42:1--42:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.42},
  URN =		{urn:nbn:de:0030-drops-231941},
  doi =		{10.4230/LIPIcs.SoCG.2025.42},
  annote =	{Keywords: Range counting oracles, minimum spanning trees, Earth Mover’s Distance}
}
  • Refine by Type
  • 36 Document/PDF
  • 17 Document/HTML

  • Refine by Publication Year
  • 4 2026
  • 14 2025
  • 1 2024
  • 2 2023
  • 5 2022
  • Show More...

  • Refine by Author
  • 17 Afshani, Peyman
  • 4 Cheng, Pingan
  • 3 Driemel, Anne
  • 3 Phillips, Jeff M.
  • 3 Rotenberg, Eva
  • Show More...

  • Refine by Series/Journal
  • 36 LIPIcs

  • Refine by Classification
  • 23 Theory of computation → Computational geometry
  • 5 Theory of computation → Design and analysis of algorithms
  • 4 Theory of computation → Data structures design and analysis
  • 4 Theory of computation → Randomness, geometry and discrete structures
  • 2 Theory of computation
  • Show More...

  • Refine by Keyword
  • 6 Computational Geometry
  • 4 Data Structures
  • 4 Range Searching
  • 3 Data Structures and Algorithms
  • 2 Collision detection
  • Show More...

Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact