43 Search Results for "Agarwal, Pankaj K."

Document
DOI: 10.4230/LIPIcs.MFCS.2024.9
Quantum Algorithms for Hopcroft’s Problem

Authors: Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O(n^{4/3}) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity Õ(n^{5/6}). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.
Cite as

Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{andrejevs_et_al:LIPIcs.MFCS.2024.9,
  author =	{Andrejevs, Vladimirs and Belovs, Aleksandrs and Vihrovs, Jevg\={e}nijs},
  title =	{{Quantum Algorithms for Hopcroft’s Problem}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.9},
  URN =		{urn:nbn:de:0030-drops-205653},
  doi =		{10.4230/LIPIcs.MFCS.2024.9},
  annote =	{Keywords: Quantum algorithms, Quantum walks, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.26
Capturing the Shape of a Point Set with a Line Segment

Authors: Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, and Jules Wulms

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group’s shape carries meaning as well. In this paper, we represent a group’s shape using a simple geometric object, a line segment. Specifically, given a radius r, we say a line segment is representative of a point set P of n points if it is within distance r of each point p ∈ P. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius r using the shortest line segment. We describe an algorithm to find the shortest representative segment in O(n log h + h log³h) time, where h is the size of the convex hull of P. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in P move.
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Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, and Jules Wulms. Capturing the Shape of a Point Set with a Line Segment. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanbeusekom_et_al:LIPIcs.MFCS.2024.26,
  author =	{van Beusekom, Nathan and van Kreveld, Marc and van Mulken, Max and Roeloffzen, Marcel and Speckmann, Bettina and Wulms, Jules},
  title =	{{Capturing the Shape of a Point Set with a Line Segment}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{26:1--26:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.26},
  URN =		{urn:nbn:de:0030-drops-205820},
  doi =		{10.4230/LIPIcs.MFCS.2024.26},
  annote =	{Keywords: Shape descriptor, Stabbing, Rotating calipers}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.68
Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems

Authors: Gang Liu and Haitao Wang

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Given a set P of n points and a set S of m disks in the plane, the disk hitting set problem asks for a smallest subset of P such that every disk of S contains at least one point in the subset. The problem is NP-hard. This paper considers a line-constrained version in which all disks have their centers on a line. We present an O(mlog²n+(n+m)log(n+m)) time algorithm for the problem. This improves the previous result of O(m²log m+(n+m)log(n+m)) time for the weighted case of the problem where every point of P has a weight and the objective is to minimize the total weight of the hitting set. Our algorithm also solves a more general line-separable problem with a single intersection property: The points of P and the disk centers are separated by a line 𝓁 and the boundary of every two disks intersect at most once on the side of 𝓁 containing P.
Cite as

Gang Liu and Haitao Wang. Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{liu_et_al:LIPIcs.MFCS.2024.68,
  author =	{Liu, Gang and Wang, Haitao},
  title =	{{Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{68:1--68:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.68},
  URN =		{urn:nbn:de:0030-drops-206240},
  doi =		{10.4230/LIPIcs.MFCS.2024.68},
  annote =	{Keywords: hitting set, line-constrained, line-separable, unit disks, half-planes, coverage}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.70
On Line-Separable Weighted Unit-Disk Coverage and Related Problems

Authors: Gang Liu and Haitao Wang

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Given a set P of n points and a set S of n weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of P. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of P by a line 𝓁. We present an O(n^{3/2}log² n) time algorithm for the problem. This improves the previously best work of O(n²log n) time. Our result leads to an algorithm of O(n^{7/2}log² n) time for the halfplane coverage problem (i.e., using n weighted halfplanes to cover n points), an improvement over the previous O(n⁴log n) time solution. If all halfplanes are lower ones, our algorithm runs in O(n^{3/2}log² n) time, while the previous best algorithm takes O(n²log n) time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.
Cite as

Gang Liu and Haitao Wang. On Line-Separable Weighted Unit-Disk Coverage and Related Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{liu_et_al:LIPIcs.MFCS.2024.70,
  author =	{Liu, Gang and Wang, Haitao},
  title =	{{On Line-Separable Weighted Unit-Disk Coverage and Related Problems}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{70:1--70:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.70},
  URN =		{urn:nbn:de:0030-drops-206265},
  doi =		{10.4230/LIPIcs.MFCS.2024.70},
  annote =	{Keywords: Line-separable, unit disks, halfplanes, geometric coverage, geometric hitting set}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.19
Algorithms for Gradual Polyline Simplification

Authors: Nick Krumbholz, Stefan Funke, Peter Schäfer, and Sabine Storandt

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
Displaying line data is important in many visualization applications, and especially in the context of interactive geographical and cartographic visualization. When rendering linear features as roads, rivers or movement data on zoomable maps, the challenge is to display the data in an appropriate level of detail. A too detailed representation results in slow rendering and cluttered maps, while a too coarse representation might miss important data aspects. In this paper, we propose the gradual line simplification (GLS) problem, which aims to compute a fine-grained succession of consistent simplifications of a given input polyline with certain quality guarantees. The core concept of gradual simplification is to iteratively remove points from the polyline to obtain increasingly coarser representations. We devise two objective functions to guide this simplification process and present dynamic programs that compute the optimal solutions in 𝒪(n³) for an input line with n points. For practical application to large inputs, we also devise significantly faster greedy algorithms that provide constant factor guarantees for both problem variants at once. In an extensive experimental study on real-world data, we demonstrate that our algorithms are capable of producing simplification sequences of high quality within milliseconds on polylines consisting of over half a million points.
Cite as

Nick Krumbholz, Stefan Funke, Peter Schäfer, and Sabine Storandt. Algorithms for Gradual Polyline Simplification. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{krumbholz_et_al:LIPIcs.SEA.2024.19,
  author =	{Krumbholz, Nick and Funke, Stefan and Sch\"{a}fer, Peter and Storandt, Sabine},
  title =	{{Algorithms for Gradual Polyline Simplification}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.19},
  URN =		{urn:nbn:de:0030-drops-203847},
  doi =		{10.4230/LIPIcs.SEA.2024.19},
  annote =	{Keywords: Polyline simplification, Progressive simplification, Fr\'{e}chet distance}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.9
Top-k Frequent Patterns in Streams and Parameterized-Space LZ Compression

Authors: Patrick Dinklage, Johannes Fischer, and Nicola Prezza

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
We present novel online approximations of the Lempel-Ziv 77 (LZ77) and Lempel-Ziv 78 (LZ78) compression schemes [Lempel & Ziv, 1977/1978] with parameterizable space usage based on estimating which k patterns occur the most frequently in the streamed input for parameter k. This new approach overcomes the issue of finding only local repetitions, which is a natural limitation of algorithms that compress using a sliding window or by partitioning the input into blocks. For this, we introduce the top-k trie, a summary for maintaining online the top-k frequent consecutive patterns in a stream of characters based on a combination of the Lempel-Ziv 78 compression scheme and the Misra-Gries algorithm for frequent item estimation in streams. Using straightforward encoding, our implementations yield compression ratios (output over input size) competitive with established general-purpose LZ-based compression utilities such as gzip or xz.
Cite as

Patrick Dinklage, Johannes Fischer, and Nicola Prezza. Top-k Frequent Patterns in Streams and Parameterized-Space LZ Compression. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dinklage_et_al:LIPIcs.SEA.2024.9,
  author =	{Dinklage, Patrick and Fischer, Johannes and Prezza, Nicola},
  title =	{{Top-k Frequent Patterns in Streams and Parameterized-Space LZ Compression}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{9:1--9:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.9},
  URN =		{urn:nbn:de:0030-drops-203748},
  doi =		{10.4230/LIPIcs.SEA.2024.9},
  annote =	{Keywords: compression, streaming, heavy hitters, algorithm engineering}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.27
The Discrepancy of Shortest Paths

Authors: Greg Bodwin, Chengyuan Deng, Jie Gao, Gary Hoppenworth, Jalaj Upadhyay, and Chen Wang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
The hereditary discrepancy of a set system is a quantitative measure of the pseudorandom properties of the system. Roughly speaking, hereditary discrepancy measures how well one can 2-color the elements of the system so that each set contains approximately the same number of elements of each color. Hereditary discrepancy has numerous applications in computational geometry, communication complexity and derandomization. More recently, the hereditary discrepancy of the set system of shortest paths has found applications in differential privacy [Chen et al. SODA 23]. The contribution of this paper is to improve the upper and lower bounds on the hereditary discrepancy of set systems of unique shortest paths in graphs. In particular, we show that any system of unique shortest paths in an undirected weighted graph has hereditary discrepancy O(n^{1/4}), and we construct lower bound examples demonstrating that this bound is tight up to polylog n factors. Our lower bounds hold even for planar graphs and bipartite graphs, and improve a previous lower bound of Ω(n^{1/6}) obtained by applying the trace bound of Chazelle and Lvov [SoCG'00] to a classical point-line system of Erdős. As applications, we improve the lower bound on the additive error for differentially-private all pairs shortest distances from Ω(n^{1/6}) [Chen et al. SODA 23] to Ω̃(n^{1/4}), and we improve the lower bound on additive error for the differentially-private all sets range queries problem to Ω̃(n^{1/4}), which is tight up to polylog n factors [Deng et al. WADS 23].
Cite as

Greg Bodwin, Chengyuan Deng, Jie Gao, Gary Hoppenworth, Jalaj Upadhyay, and Chen Wang. The Discrepancy of Shortest Paths. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bodwin_et_al:LIPIcs.ICALP.2024.27,
  author =	{Bodwin, Greg and Deng, Chengyuan and Gao, Jie and Hoppenworth, Gary and Upadhyay, Jalaj and Wang, Chen},
  title =	{{The Discrepancy of Shortest Paths}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{27:1--27:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.27},
  URN =		{urn:nbn:de:0030-drops-201705},
  doi =		{10.4230/LIPIcs.ICALP.2024.27},
  annote =	{Keywords: Discrepancy, hereditary discrepancy, shortest paths, differential privacy}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.30
Õptimal Dynamic Time Warping on Run-Length Encoded Strings

Authors: Itai Boneh, Shay Golan, Shay Mozes, and Oren Weimann

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Dynamic Time Warping (DTW) distance is the optimal cost of matching two strings when extending runs of letters is for free. Therefore, it is natural to measure the time complexity of DTW in terms of the number of runs n (rather than the string lengths N). In this paper, we give an Õ(n²) time algorithm for computing the DTW distance. This matches (up to log factors) the known (conditional) lower bound, and should be compared with the previous fastest O(n³) time exact algorithm and the Õ(n²) time approximation algorithm. Our method also immediately implies an Õ(nk) time algorithm when the distance is bounded by k. This should be compared with the previous fastest O(n²k) and O(Nk) time exact algorithms and the Õ(nk) time approximation algorithm.
Cite as

Itai Boneh, Shay Golan, Shay Mozes, and Oren Weimann. Õptimal Dynamic Time Warping on Run-Length Encoded Strings. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boneh_et_al:LIPIcs.ICALP.2024.30,
  author =	{Boneh, Itai and Golan, Shay and Mozes, Shay and Weimann, Oren},
  title =	{{\~{O}ptimal Dynamic Time Warping on Run-Length Encoded Strings}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{30:1--30:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.30},
  URN =		{urn:nbn:de:0030-drops-201730},
  doi =		{10.4230/LIPIcs.ICALP.2024.30},
  annote =	{Keywords: Dynamic time warping, Fr\'{e}chet distance, edit distance, run-length encoding}
}
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)

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@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.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)

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@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)

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@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)

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@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)

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@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)

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@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)

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@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}
}
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