38 Search Results for "Agarwal, Pankaj K."

Document
DOI: 10.4230/LIPIcs.ESA.2024.6
Lower Envelopes of Surface Patches in 3-Space

Authors: Pankaj K. Agarwal, Esther Ezra, and Micha Sharir

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
Let Σ be a collection of n surface patches, each being the graph of a partially defined semi-algebraic function of constant description complexity, and assume that any triple of them intersect in at most s = 2 points. We show that the complexity of the lower envelope of the surfaces in Σ is O(n² log^{6+ε} n), for any ε > 0. This almost settles a long-standing open problem posed by Halperin and Sharir, thirty years ago, who showed the nearly-optimal albeit weaker bound of O(n²⋅ 2^{c√{log n}}) on the complexity of the lower envelope, where c > 0 is some constant. Our approach is fairly simple and is based on hierarchical cuttings and gradations, as well as a simple charging scheme. We extend our analysis to the case s > 2, under a "favorable cross section" assumption, in which case we show that the bound on the complexity of the lower envelope is O(n² log^{11+ε} n), for any ε > 0. Incorporating these bounds with the randomized incremental construction algorithms of Boissonnat and Dobrindt, we obtain efficient constructions of lower envelopes of surface patches with the above properties, whose overall expected running time is O(n² polylog), as well as efficient data structures that support point location queries in their minimization diagrams in O(log²n) expected time.
Cite as

Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Lower Envelopes of Surface Patches in 3-Space. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.ESA.2024.6,
  author =	{Agarwal, Pankaj K. and Ezra, Esther and Sharir, Micha},
  title =	{{Lower Envelopes of Surface Patches in 3-Space}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.6},
  URN =		{urn:nbn:de:0030-drops-210772},
  doi =		{10.4230/LIPIcs.ESA.2024.6},
  annote =	{Keywords: Hierarchical cuttings, surface patches in 3-space, lower envelopes, charging scheme, gradation}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.7
Segment Proximity Graphs and Nearest Neighbor Queries Amid Disjoint Segments

Authors: Pankaj K. Agarwal, Haim Kaplan, Matthew J. Katz, and Micha Sharir

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In this paper we study a few proximity problems related to a set of pairwise-disjoint segments in {ℝ}². Let S be a set of n pairwise-disjoint segments in {ℝ}², and let r > 0 be a parameter. We define the segment proximity graph of S to be G_r(S) := (S,E), where E = {(e₁,e₂) ∣ dist(e₁,e₂) ≤ r} and dist (e₁,e₂) = min_{(p,q) ∈ e₁× e₂} ‖p-q‖ is the Euclidean distance between e₁ and e₂. We define the weight of an edge (e₁,e₂) ∈ E to be dist(e₁,e₂). We first present a simple grid-based O(nlog² n)-time algorithm for computing a BFS tree of G_r(S). We apply it to obtain an O^*(n^{6/5}) + O(nlog²nlogΔ)-time algorithm for the so-called reverse shortest path problem, in which we want to find the smallest value r^* for which G_{r^*}(S) contains a path of some specified length between two designated start and target segments (where the O^*(⋅) notation hides polylogarithmic factors). Here Δ = max_{e ≠ e' ∈ S} dist(e,e')/min_{e ≠ e' ∈ S} dist(e,e') is the spread of S. Next, we present a dynamic data structure that can maintain a set S of pairwise-disjoint segments in the plane under insertions/deletions, so that, for a query segment e from an unknown set Q of pairwise-disjoint segments, such that e does not intersect any segment in (the current version of) S, the segment of S closest to e can be computed in O(log⁵ n) amortized time. The amortized update time is also O(log⁵ n). We note that if the segments in S∪Q are allowed to intersect then the known lower bounds on halfplane range searching suggest that a sequence of n updates and queries may take at least close to Ω(n^{4/3}) time. One thus has to strongly rely on the non-intersecting property of S and Q to perform updates and queries in O(polylog(n)) (amortized) time each. Using these results on nearest-neighbor (NN) searching for disjoint segments, we show that a DFS tree (or forest) of G_r(S) can be computed in O^*(n) time. We also obtain an O^*(n)-time algorithm for constructing a minimum spanning tree of G_r(S). Finally, we present an O^*(n^{4/3})-time algorithm for computing a single-source shortest-path tree in G_r(S). This is the only result that does not exploit the disjointness of the input segments.
Cite as

Pankaj K. Agarwal, Haim Kaplan, Matthew J. Katz, and Micha Sharir. Segment Proximity Graphs and Nearest Neighbor Queries Amid Disjoint Segments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.ESA.2024.7,
  author =	{Agarwal, Pankaj K. and Kaplan, Haim and Katz, Matthew J. and Sharir, Micha},
  title =	{{Segment Proximity Graphs and Nearest Neighbor Queries Amid Disjoint Segments}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.7},
  URN =		{urn:nbn:de:0030-drops-210782},
  doi =		{10.4230/LIPIcs.ESA.2024.7},
  annote =	{Keywords: segment proximity graphs, nearest neighbor searching, dynamic data structures, BFS, DFS, unit-disk graphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.4
Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane

Authors: Pankaj K. Agarwal, Esther Ezra, and Micha Sharir

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

Abstract
Let P be a set of m points in ℝ², let Σ be a set of n semi-algebraic sets of constant complexity in ℝ², let (S,+) be a semigroup, and let w: P → S be a weight function on the points of P. We describe a randomized algorithm for computing w(P∩σ) for every σ ∈ Σ in overall expected time O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n), where s > 0 is a constant that bounds the maximum complexity of the regions of Σ, and where the O^*(⋅) notation hides subpolynomial factors. For s ≥ 3, surprisingly, this bound is smaller than the best-known bound for answering m such queries in an on-line manner. The latter takes O^*(m^{s/(2s-1)}n^{(2s-2)/(2s-1)} + m + n) time. Let Φ: Σ × P → {0,1} be the Boolean predicate (of constant complexity) such that Φ(σ,p) = 1 if p ∈ σ and 0 otherwise, and let Σ_Φ P = {(σ,p) ∈ Σ× P ∣ Φ(σ,p) = 1}. Our algorithm actually computes a partition ℬ_Φ of Σ_Φ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n). It is straightforward to compute w(P∩σ) for all σ ∈ Σ from ℬ_Φ. Similarly, if η: Σ → S is a weight function on the regions of Σ, ∑_{σ ∈ Σ: p ∈ σ} η(σ), for every point p ∈ P, can be computed from ℬ_Φ in a straightforward manner. We also mention a few other applications of computing ℬ_Φ.
Cite as

Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.SoCG.2024.4,
  author =	{Agarwal, Pankaj K. and Ezra, Esther and Sharir, Micha},
  title =	{{Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{4:1--4:15},
  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.2024.4},
  URN =		{urn:nbn:de:0030-drops-199497},
  doi =		{10.4230/LIPIcs.SoCG.2024.4},
  annote =	{Keywords: Range-searching, semi-algebraic sets, pseudo-lines, duality, geometric cuttings}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.18
Dynamic L-Budget Clustering of Curves

Authors: Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Lukas Plätz, Lea Thiel, and Sampson Wong

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
A key goal of clustering is data reduction. In center-based clustering of complex objects therefore not only the number of clusters but also the complexity of the centers plays a crucial role. We propose L-Budget Clustering as unifying perspective on this task, optimizing the clustering under the constraint that the summed complexity of all centers is at most L. We present algorithms for clustering planar curves under the Fréchet distance, but note that our algorithms more generally apply to objects in metric spaces if a notion of simplification of objects is applicable. A scenario in which data reduction is of particular importance is when the space is limited. Our main result is an efficient (8 + ε)-approximation algorithm with a (1 + ε)-resource augmentation that maintains an L-budget clustering under insertion of curves using only O(Lε^{-1}) space and O^*(L³log(L) + L²log(r^*/r₀)) time where O^* hides factors of ε^{-1}.
Cite as

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Lukas Plätz, Lea Thiel, and Sampson Wong. Dynamic L-Budget Clustering of Curves. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{buchin_et_al:LIPIcs.SWAT.2024.18,
  author =	{Buchin, Kevin and Buchin, Maike and Gudmundsson, Joachim and Pl\"{a}tz, Lukas and Thiel, Lea and Wong, Sampson},
  title =	{{Dynamic L-Budget Clustering of Curves}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.18},
  URN =		{urn:nbn:de:0030-drops-200588},
  doi =		{10.4230/LIPIcs.SWAT.2024.18},
  annote =	{Keywords: clustering, streaming algorithm, polygonal curves, Fr\'{e}chet distance, storage efficiency, simplification, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2024.18
Computing Data Distribution from Query Selectivities

Authors: Pankaj K. Agarwal, Rahul Raychaudhury, Stavros Sintos, and Jun Yang

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
We are given a set 𝒵 = {(R_1,s_1), …, (R_n,s_n)}, where each R_i is a range in ℝ^d, such as rectangle or ball, and s_i ∈ [0,1] denotes its selectivity. The goal is to compute a small-size discrete data distribution 𝒟 = {(q₁,w₁),…, (q_m,w_m)}, where q_j ∈ ℝ^d and w_j ∈ [0,1] for each 1 ≤ j ≤ m, and ∑_{1≤j≤m} w_j = 1, such that 𝒟 is the most consistent with 𝒵, i.e., err_p(𝒟,𝒵) = 1/n ∑_{i = 1}ⁿ |s_i - ∑_{j=1}^m w_j⋅1(q_j ∈ R_i)|^p is minimized. In a database setting, 𝒵 corresponds to a workload of range queries over some table, together with their observed selectivities (i.e., fraction of tuples returned), and 𝒟 can be used as compact model for approximating the data distribution within the table without accessing the underlying contents. In this paper, we obtain both upper and lower bounds for this problem. In particular, we show that the problem of finding the best data distribution from selectivity queries is NP-complete. On the positive side, we describe a Monte Carlo algorithm that constructs, in time O((n+δ^{-d}) δ^{-2} polylog n), a discrete distribution 𝒟̃ of size O(δ^{-2}), such that err_p(𝒟̃,𝒵) ≤ min_𝒟 err_p(𝒟,𝒵)+δ (for p = 1,2,∞) where the minimum is taken over all discrete distributions. We also establish conditional lower bounds, which strongly indicate the infeasibility of relative approximations as well as removal of the exponential dependency on the dimension for additive approximations. This suggests that significant improvements to our algorithm are unlikely.
Cite as

Pankaj K. Agarwal, Rahul Raychaudhury, Stavros Sintos, and Jun Yang. Computing Data Distribution from Query Selectivities. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.ICDT.2024.18,
  author =	{Agarwal, Pankaj K. and Raychaudhury, Rahul and Sintos, Stavros and Yang, Jun},
  title =	{{Computing Data Distribution from Query Selectivities}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.18},
  URN =		{urn:nbn:de:0030-drops-198007},
  doi =		{10.4230/LIPIcs.ICDT.2024.18},
  annote =	{Keywords: selectivity queries, discrete distributions, Multiplicative Weights Update, eps-approximation, learnable functions, depth problem, arrangement}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.6
Facility Location in the Sublinear Geometric Model

Authors: Morteza Monemizadeh

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
In the sublinear geometric model, we are provided with an oracle access to a point set P of n points in a bounded discrete space [Δ]², where Δ = n^O(1) is a polynomially bounded number in n. That is, we do not have direct access to the points, but we can make certain types of queries and there is an oracle that responds to our queries. The type of queries that we assume we can make in this paper, are range counting queries where ranges are axis-aligned rectangles (that are basic primitives in database [Srikanta Tirthapura and David P. Woodruff, 2012; Bentley, 1975; Mark de Berg et al., 2008], computational geometry [Pankaj K. Agarwal, 2004; Pankaj K. Agarwal et al., 1996; Boris Aronov et al., 2010; Boris Aronov et al., 2009], and machine learning [Menachem Sadigurschi and Uri Stemmer, 2021; Long and Tan, 1998; Michael J. Kearns and Umesh V. Vazirani, 1995; Michael J. Kearns and Umesh V. Vazirani, 1994]). The oracle then answers these queries by returning the number of points that are in queried ranges. Let {Alg} be an algorithm that (exactly or approximately) solves a problem 𝒫 in the sublinear geometric model. The query complexity of Alg is measured in terms of the number of queries that Alg makes to solve 𝒫. In this paper, we study the complexity of the (uniform) Euclidean facility location problem in the sublinear geometric model. We develop a randomized sublinear algorithm that with high probability, (1+ε)-approximates the cost of the Euclidean facility location problem of the point set P in the sublinear geometric model using Õ(√n) range counting queries. We complement this result by showing that approximating the cost of the Euclidean facility location problem within o(log(n))-factor in the sublinear geometric model using the sampling strategy that we propose for our sublinear algorithm needs Ω̃(n^{1/4}) RangeCount queries. We leave it as an open problem whether such a polynomial lower bound on the number of RangeCount queries exists for any randomized sublinear algorithm that approximates the cost of the facility location problem within a constant factor.
Cite as

Morteza Monemizadeh. Facility Location in the Sublinear Geometric Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{monemizadeh:LIPIcs.APPROX/RANDOM.2023.6,
  author =	{Monemizadeh, Morteza},
  title =	{{Facility Location in the Sublinear Geometric Model}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{6:1--6:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.6},
  URN =		{urn:nbn:de:0030-drops-188315},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.6},
  annote =	{Keywords: Facility Location, Sublinear Geometric Model, Range Counting Queries}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.7
On Range Summary Queries

Authors: Peyman Afshani, Pingan Cheng, Aniket Basu Roy, and Zhewei Wei

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

Abstract
We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter ε and a set P of n points in ℝ^d where each point is assigned a color from a set C, and the goal is to build a structure such that given any geometric range γ, we can efficiently find a list of approximate heavy hitters in γ∩P, i.e., colors that appear at least ε |γ∩P| times in γ∩P, as well as their frequencies with an additive error of ε |γ∩P|. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence S of 1+1/ε weights such that the i-th weight in S has approximate rank iε|γ∩P|, meaning, rank iε|γ∩P| up to an additive error of ε|γ∩P|. Previously, optimal results were only known in 1D [Wei and Yi, 2011] but a few sub-optimal methods were available in higher dimensions [Peyman Afshani and Zhewei Wei, 2017; Pankaj K. Agarwal et al., 2012]. We study the problems for two important classes of geometric ranges: 3D halfspace and 3D dominance queries. It is known that many other important queries can be reduced to these two, e.g., 1D interval stabbing or interval containment, 2D three-sided queries, 2D circular as well as 2D k-nearest neighbors queries. We consider the real RAM model of computation where integer registers of size w bits, w = Θ(log n), are also available. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time O(log n + 1/ε). Note that as the output size is 1/ε, after investing the initial O(log n) searching time, our structure takes on average O(1) time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra log_w(1/ε) (resp. log log_w(1/ε)) factor in 3D (resp. 2D). By spending extra log^O(1)(1/ε) factors in both time and space, we can also support quantile queries. We remark that it is hopeless to achieve a similar query bound for dimensions 4 or higher unless significant advances are made in the data structure side of theory of geometric approximations.
Cite as

Peyman Afshani, Pingan Cheng, Aniket Basu Roy, and Zhewei Wei. On Range Summary Queries. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{afshani_et_al:LIPIcs.ICALP.2023.7,
  author =	{Afshani, Peyman and Cheng, Pingan and Basu Roy, Aniket and Wei, Zhewei},
  title =	{{On Range Summary Queries}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.7},
  URN =		{urn:nbn:de:0030-drops-180590},
  doi =		{10.4230/LIPIcs.ICALP.2023.7},
  annote =	{Keywords: Computational Geometry, Range Searching, Random Sampling, Geometric Approximation, Data Structures and Algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.4
Computing Instance-Optimal Kernels in Two Dimensions

Authors: Pankaj K. Agarwal and Sariel Har-Peled

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

Abstract
Let P be a set of n points in ℝ². For a parameter ε ∈ (0,1), a subset C ⊆ P is an ε-kernel of P if the projection of the convex hull of C approximates that of P within (1-ε)-factor in every direction. The set C is a weak ε-kernel of P if its directional width approximates that of P in every direction. Let 𝗄_ε(P) (resp. 𝗄^𝗐_ε(P)) denote the minimum-size of an ε-kernel (resp. weak ε-kernel) of P. We present an O(n 𝗄_ε(P)log n)-time algorithm for computing an ε-kernel of P of size 𝗄_ε(P), and an O(n²log n)-time algorithm for computing a weak ε-kernel of P of size 𝗄^𝗐_ε(P). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of ε-core, a convex polygon lying inside ch(P), prove that it is a good approximation of the optimal ε-kernel, present an efficient algorithm for computing it, and use it to compute an ε-kernel of small size.
Cite as

Pankaj K. Agarwal and Sariel Har-Peled. Computing Instance-Optimal Kernels in Two Dimensions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.SoCG.2023.4,
  author =	{Agarwal, Pankaj K. and Har-Peled, Sariel},
  title =	{{Computing Instance-Optimal Kernels in Two Dimensions}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{4:1--4:15},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.4},
  URN =		{urn:nbn:de:0030-drops-178544},
  doi =		{10.4230/LIPIcs.SoCG.2023.4},
  annote =	{Keywords: Coreset, approximation, kernel}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.5
Line Intersection Searching Amid Unit Balls in 3-Space

Authors: Pankaj K. Agarwal and Esther Ezra

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

Abstract
Let ℬ be a set of n unit balls in ℝ³. We present a linear-size data structure for storing ℬ that can determine in O^*(n^{1/2}) time whether a query line intersects any ball of ℬ and report all k such balls in additional O(k) time. The data structure can be constructed in O(n log n) time. (The O^*(⋅) notation hides subpolynomial factors, e.g., of the form O(n^ε), for arbitrarily small ε > 0, and their coefficients which depend on ε.) We also consider the dual problem: Let ℒ be a set of n lines in ℝ³. We preprocess ℒ, in O^*(n²) time, into a data structure of size O^*(n²) that can determine in O^*(1) time whether a query unit ball intersects any line of ℒ, or report all k such lines in additional O(k) time.
Cite as

Pankaj K. Agarwal and Esther Ezra. Line Intersection Searching Amid Unit Balls in 3-Space. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.SoCG.2023.5,
  author =	{Agarwal, Pankaj K. and Ezra, Esther},
  title =	{{Line Intersection Searching Amid Unit Balls in 3-Space}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{5:1--5:14},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.5},
  URN =		{urn:nbn:de:0030-drops-178559},
  doi =		{10.4230/LIPIcs.SoCG.2023.5},
  annote =	{Keywords: Intersection searching, cylindrical range searching, partition trees, union of cylinders}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.35
Multi-Robot Motion Planning for Unit Discs with Revolving Areas

Authors: Pankaj K. Agarwal, Tzvika Geft, Dan Halperin, and Erin Taylor

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot R in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As R passes through a revolving area, a robot R' that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time (1+ε)-approximation algorithm. On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an O(1) factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time O(log n log log n)-approximation algorithm for this problem.
Cite as

Pankaj K. Agarwal, Tzvika Geft, Dan Halperin, and Erin Taylor. Multi-Robot Motion Planning for Unit Discs with Revolving Areas. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.ISAAC.2022.35,
  author =	{Agarwal, Pankaj K. and Geft, Tzvika and Halperin, Dan and Taylor, Erin},
  title =	{{Multi-Robot Motion Planning for Unit Discs with Revolving Areas}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.35},
  URN =		{urn:nbn:de:0030-drops-173204},
  doi =		{10.4230/LIPIcs.ISAAC.2022.35},
  annote =	{Keywords: motion planning, optimal motion planning, approximation, complexity, NP-hardness}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.42
On Reverse Shortest Paths in Geometric Proximity Graphs

Authors: Pankaj K. Agarwal, Matthew J. Katz, and Micha Sharir

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in ℝ², and let ϱ: S× S → ℝ_{≥ 0} be a distance function on S. For a parameter r ≥ 0, we define the proximity graph G(r) = (S,E) where E = {(e₁,e₂) ∈ S×S ∣ e₁≠e₂, ϱ(e₁,e₂) ≤ r}. Given S, s,t ∈ S, and an integer k ≥ 1, the reverse-shortest-path (RSP) problem asks for computing the smallest value r^* ≥ 0 such that G(r^*) contains a path from s to t of length at most k. In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value r ≥ 0, determine whether G(r) contains a path from s to t of length at most k. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute r^*, by efficiently performing a binary search over an implicit set of O(n²) candidate values that contains r^*. We illustrate the versatility of our general technique by applying it to a variety of geometric proximity graphs. For example, we obtain (i) an O^*(n^{4/3}) expected-time randomized algorithm (where O^*(⋅) hides polylog(n) factors) for the case where S is a set of pairwise-disjoint line segments in ℝ² and ϱ(e₁,e₂) = min_{x ∈ e₁, y ∈ e₂} ‖x-y‖ (where ‖⋅‖ is the Euclidean distance), and (ii) an O^*(n+m^{4/3}) expected-time randomized algorithm for the case where S is a set of m points lying on an x-monotone polygonal chain T with n vertices, and ϱ(p,q), for p,q ∈ S, is the smallest value h such that the points p' := p+(0,h) and q' := q+(0,h) are visible to each other, i.e., all points on the segment p'q' lie above or on the polygonal chain T.
Cite as

Pankaj K. Agarwal, Matthew J. Katz, and Micha Sharir. On Reverse Shortest Paths in Geometric Proximity Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 42:1-42:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.ISAAC.2022.42,
  author =	{Agarwal, Pankaj K. and Katz, Matthew J. and Sharir, Micha},
  title =	{{On Reverse Shortest Paths in Geometric Proximity Graphs}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{42:1--42:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.42},
  URN =		{urn:nbn:de:0030-drops-173277},
  doi =		{10.4230/LIPIcs.ISAAC.2022.42},
  annote =	{Keywords: Geometric optimization, proximity graphs, semi-algebraic range searching, reverse shortest path}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2022.6
An Improved ε-Approximation Algorithm for Geometric Bipartite Matching

Authors: Pankaj K. Agarwal, Sharath Raghvendra, Pouyan Shirzadian, and Rachita Sowle

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
For two point sets A, B ⊂ ℝ^d, with |A| = |B| = n and d > 1 a constant, and for a parameter ε > 0, we present a randomized algorithm that, with probability at least 1/2, computes in O(n(ε^{-O(d³)}log log n + ε^{-O(d)}log⁴ nlog⁵log n)) time, an ε-approximate minimum-cost perfect matching under any L_p-metric. All previous algorithms take n(ε^{-1}log n)^{Ω(d)} time. We use a randomly-shifted tree, with a polynomial branching factor and O(log log n) height, to define a tree-based distance function that ε-approximates the L_p metric as well as to compute the matching hierarchically. Then, we apply the primal-dual framework on a compressed representation of the residual graph to obtain an efficient implementation of the Hungarian-search and augment operations.
Cite as

Pankaj K. Agarwal, Sharath Raghvendra, Pouyan Shirzadian, and Rachita Sowle. An Improved ε-Approximation Algorithm for Geometric Bipartite Matching. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.SWAT.2022.6,
  author =	{Agarwal, Pankaj K. and Raghvendra, Sharath and Shirzadian, Pouyan and Sowle, Rachita},
  title =	{{An Improved \epsilon-Approximation Algorithm for Geometric Bipartite Matching}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.6},
  URN =		{urn:nbn:de:0030-drops-161660},
  doi =		{10.4230/LIPIcs.SWAT.2022.6},
  annote =	{Keywords: Euclidean bipartite matching, approximation algorithms, primal dual method}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2022.11
Dynamic Approximate Multiplicatively-Weighted Nearest Neighbors

Authors: Boris Aronov and Matthew J. Katz

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
We describe a dynamic data structure for approximate nearest neighbor (ANN) queries with respect to multiplicatively weighted distances with additive offsets. Queries take polylogarithmic time, while the cost of updates is amortized polylogarithmic. The data structure requires near-linear space and construction time. The approach works not only for the Euclidean norm, but for other norms in ℝ^d, for any fixed d. We employ our ANN data structure to construct a faster dynamic structure for approximate SINR queries, ensuring polylogarithmic query and polylogarithmic amortized update for the case of non-uniform power transmitters, thus closing a gap in previous state of the art. To obtain the latter result, we needed a data structure for dynamic approximate halfplane range counting in the plane. Since we could not find such a data structure in the literature, we also show how to dynamize one of the known static data structures.
Cite as

Boris Aronov and Matthew J. Katz. Dynamic Approximate Multiplicatively-Weighted Nearest Neighbors. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{aronov_et_al:LIPIcs.SWAT.2022.11,
  author =	{Aronov, Boris and Katz, Matthew J.},
  title =	{{Dynamic Approximate Multiplicatively-Weighted Nearest Neighbors}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.11},
  URN =		{urn:nbn:de:0030-drops-161710},
  doi =		{10.4230/LIPIcs.SWAT.2022.11},
  annote =	{Keywords: Nearest neighbors, Approximate nearest neighbors, Weighted nearest neighbors, Nearest neighbor queries, SINR queries, Dynamic data structures}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.4
Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems

Authors: Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir

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

Abstract
Let 𝒯 be a set of n planar semi-algebraic regions in ℝ³ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess 𝒯 into a data structure so that for a query object γ, which is also a plate, we can quickly answer various intersection queries, such as detecting whether γ intersects any plate of 𝒯, reporting all the plates intersected by γ, or counting them. We focus on two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in ℝ³ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in ℝ³. These interesting special cases form the building blocks for the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if 𝒯 is a set of plates and the query objects are arcs, we obtain a data structure that uses O^*(n^{4/3}) storage (where the O^*(⋅) notation hides subpolynomial factors) and answers an intersection query in O^*(n^{2/3}) time. Alternatively, by increasing the storage to O^*(n^{3/2}), the query time can be decreased to O^*(n^{ρ}), where ρ = (2t-3)/3(t-1) < 2/3 and t ≥ 3 is the number of parameters needed to represent the query arcs.
Cite as

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{agarwal_et_al:LIPIcs.SoCG.2022.4,
  author =	{Agarwal, Pankaj K. and Aronov, Boris and Ezra, Esther and Katz, Matthew J. and Sharir, Micha},
  title =	{{Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{4:1--4:14},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.4},
  URN =		{urn:nbn:de:0030-drops-160126},
  doi =		{10.4230/LIPIcs.SoCG.2022.4},
  annote =	{Keywords: Intersection searching, Semi-algebraic range searching, Point-enclosure queries, Ray-shooting queries, Polynomial partitions, Cylindrical algebraic decomposition, Multi-level partition trees, Collision detection}
}
Document
DOI: 10.4230/LIPIcs.ITC.2021.18
Differentially Private Approximations of a Convex Hull in Low Dimensions

Authors: Yue Gao and Or Sheffet

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

Abstract
We give the first differentially private algorithms that estimate a variety of geometric features of points in the Euclidean space, such as diameter, width, volume of convex hull, min-bounding box, min-enclosing ball, etc. Our work relies heavily on the notion of Tukey-depth. Instead of (non-privately) approximating the convex-hull of the given set of points P, our algorithms approximate the geometric features of D_{P}(κ) - the κ-Tukey region induced by P (all points of Tukey-depth κ or greater). Moreover, our approximations are all bi-criteria: for any geometric feature μ our (α,Δ)-approximation is a value "sandwiched" between (1-α)μ(D_P(κ)) and (1+α)μ(D_P(κ-Δ)). Our work is aimed at producing a (α,Δ)-kernel of D_P(κ), namely a set 𝒮 such that (after a shift) it holds that (1-α)D_P(κ) ⊂ CH(𝒮) ⊂ (1+α)D_P(κ-Δ). We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by [Pankaj K. Agarwal et al., 2004], fails to give a kernel, and so we result to subtler notions of approximations of projections that do yield a kernel. First, we give differentially private algorithms that find (α,Δ)-kernels for a "fat" Tukey-region. Then, based on a private approximation of the min-bounding box, we find a transformation that does turn D_P(κ) into a "fat" region but only if its volume is proportional to the volume of D_P(κ-Δ). Lastly, we give a novel private algorithm that finds a depth parameter κ for which the volume of D_P(κ) is comparable to the volume of D_P(κ-Δ). We hope our work leads to the further study of the intersection of differential privacy and computational geometry.
Cite as

Yue Gao and Or Sheffet. Differentially Private Approximations of a Convex Hull in Low Dimensions. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{gao_et_al:LIPIcs.ITC.2021.18,
  author =	{Gao, Yue and Sheffet, Or},
  title =	{{Differentially Private Approximations of a Convex Hull in Low Dimensions}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.18},
  URN =		{urn:nbn:de:0030-drops-143377},
  doi =		{10.4230/LIPIcs.ITC.2021.18},
  annote =	{Keywords: Differential Privacy, Computational Geometry, Tukey Depth}
}
  • Refine by Author
  • 28 Agarwal, Pankaj K.
  • 9 Sharir, Micha
  • 5 Ezra, Esther
  • 5 Katz, Matthew J.
  • 4 Aronov, Boris
  • Show More...

  • Refine by Classification

  • Refine by Keyword
  • 3 approximation algorithms
  • 2 Computational Geometry
  • 2 Dynamic data structures
  • 2 Geometric optimization
  • 2 Intersection searching
  • Show More...

  • Refine by Type
  • 38 document

  • Refine by Publication Year
  • 5 2018
  • 5 2021
  • 5 2022
  • 5 2024
  • 4 2019
  • Show More...

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact