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

Polynomial partitioning techniques have recently led to improved geometric data structures for a variety of fundamental problems related to semialgebraic range searching and intersection searching in 3D and higher dimensions (e.g., see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021; Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to improved algorithms for offline versions of semialgebraic range searching in 2D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that these techniques can yield new data structures for a number of other 2D problems even for online queries:
1) Semialgebraic range stabbing. We present a data structure for n semialgebraic ranges in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of ranges containing a query point in O(n^{1/4+ε}) time, for an arbitrarily small constant ε > 0. (The query time bound is likely close to tight for this space bound.)
2) Ray shooting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in O(n^{1/4+ε}) time. (The query bound is again likely close to tight for this space bound, and they improve a result by Ezra and Sharir with near n^{3/2} space and near √n query time.)
3) Intersection counting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in O(n^{1/2+ε}) time. In particular, this implies an O(n^{3/2+ε})-time algorithm for counting intersections between two sets of n algebraic arcs in 2D. (This generalizes a classical O(n^{3/2+ε})-time algorithm for circular arcs by Agarwal and Sharir from SoCG 1991.)

Timothy M. Chan, Pingan Cheng, and Da Wei Zheng. Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2024.33, author = {Chan, Timothy M. and Cheng, Pingan and Zheng, Da Wei}, title = {{Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {33:1--33:15}, 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.2024.33}, URN = {urn:nbn:de:0030-drops-199785}, doi = {10.4230/LIPIcs.SoCG.2024.33}, annote = {Keywords: Computational geometry, range searching, intersection searching, semialgebraic sets, data structures, polynomial partitioning} }

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

We revisit a standard polygon containment problem: given a convex k-gon P and a convex n-gon Q in the plane, find a placement of P inside Q under translation and rotation (if it exists), or more generally, find the largest copy of P inside Q under translation, rotation, and scaling.
Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required Ω(n²) time, even in the simplest k = 3 case. We present a significantly faster new algorithm for k = 3 achieving O(n polylog n) running time. Moreover, we extend the result for general k, achieving O(k^O(1/ε) n^{1+ε}) running time for any ε > 0.
Along the way, we also prove a new O(k^O(1) n polylog n) bound on the number of similar copies of P inside Q that have 4 vertices of P in contact with the boundary of Q (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998).

Timothy M. Chan and Isaac M. Hair. Convex Polygon Containment: Improving Quadratic to Near Linear Time. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2024.34, author = {Chan, Timothy M. and Hair, Isaac M.}, title = {{Convex Polygon Containment: Improving Quadratic to Near Linear Time}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {34:1--34:15}, 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.2024.34}, URN = {urn:nbn:de:0030-drops-199795}, doi = {10.4230/LIPIcs.SoCG.2024.34}, annote = {Keywords: Polygon containment, convex polygons, translations, rotations} }

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

Let X be a set of points in ℝ² and 𝒪 be a set of geometric objects in ℝ², where |X| + |𝒪| = n. We study the problem of computing a minimum subset 𝒪^* ⊆ 𝒪 that encloses all points in X. Here a point x ∈ X is enclosed by 𝒪^* if it lies in a bounded connected component of ℝ²∖(⋃_{O ∈ 𝒪^*} O). We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in O(1)-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an O(α(n)log n)-approximation algorithm for segments, where α(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks.

Timothy M. Chan, Qizheng He, and Jie Xue. Enclosing Points with Geometric Objects. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2024.35, author = {Chan, Timothy M. and He, Qizheng and Xue, Jie}, title = {{Enclosing Points with Geometric Objects}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {35:1--35:15}, 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.2024.35}, URN = {urn:nbn:de:0030-drops-199802}, doi = {10.4230/LIPIcs.SoCG.2024.35}, annote = {Keywords: obstacle placement, geometric optimization, approximation algorithms} }

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

We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for many classes of geometric objects in 2D . Our data structures can answer connectivity queries between two objects, as well as "global" connectivity queries (e.g., deciding whether the entire graph is connected). Previously, the data structure by Afshani and Chan (ESA'06) achieved such bounds only in the special case of axis-aligned line segments or rectangles but did not work for arbitrary line segments or disks, whereas the data structures by Chan, Pătraşcu, and Roditty (FOCS'08) worked for more general classes of geometric objects but required n^{Ω(1)} query time and could not handle global connectivity queries.
Specifically, we obtain new data structures with O(1) query time and amortized update time near n^{4/5}, n^{7/8}, and n^{20/21} for axis-aligned line segments, disks, and arbitrary line segments respectively. Besides greatly reducing the query time, our data structures also improve the previous update times for axis-aligned line segments by Afshani and Chan (from near n^{10/11} to n^{4/5}) and for disks by Chan, Pătraşcu, and Roditty (from near n^{20/21} to n^{7/8}).

Timothy M. Chan and Zhengcheng Huang. Dynamic Geometric Connectivity in the Plane with Constant Query Time. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2024.36, author = {Chan, Timothy M. and Huang, Zhengcheng}, title = {{Dynamic Geometric Connectivity in the Plane with Constant Query Time}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {36:1--36:13}, 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.2024.36}, URN = {urn:nbn:de:0030-drops-199819}, doi = {10.4230/LIPIcs.SoCG.2024.36}, annote = {Keywords: Connectivity, dynamic data structures, geometric intersection graphs} }

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Track A: Algorithms, Complexity and Games

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

We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results:
- We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time.
- We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs.
- Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}).
- We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2.
- We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2.
- We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound.

Timothy M. Chan, Qizheng He, and Yuancheng Yu. On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2023.34, author = {Chan, Timothy M. and He, Qizheng and Yu, Yuancheng}, title = {{On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {34:1--34:19}, 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.2023.34}, URN = {urn:nbn:de:0030-drops-180868}, doi = {10.4230/LIPIcs.ICALP.2023.34}, annote = {Keywords: Geometric set cover, discrete k-center, conditional lower bounds} }

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

In SoCG 2022, Conroy and Tóth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an O(nlog n)-size 3-hop spanner for n disks (or fat convex objects) in the plane, and an O(nlog² n)-size 3-hop spanner for n axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs?
We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an O(1)-hop spanner for arbitrary string graphs with O(nα_k(n)) size for any constant k, where α_k(n) denotes the k-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an O(1)-hop spanner for intersection graphs of d-dimensional fat objects with O(nα_k(n)) size for any constant k and d.
We also improve on some of Conroy and Tóth’s specific previous results, in either the number of hops or the size: we describe an O(nlog n)-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an O(nlog n)-size 3-hop spanner for axis-aligned rectangles in the plane.
Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.

Timothy M. Chan and Zhengcheng Huang. Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2023.23, author = {Chan, Timothy M. and Huang, Zhengcheng}, title = {{Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {23:1--23:16}, 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.2023.23}, URN = {urn:nbn:de:0030-drops-178738}, doi = {10.4230/LIPIcs.SoCG.2023.23}, annote = {Keywords: Hop spanners, geometric intersection graphs, string graphs, fat objects, separators, shallow cuttings} }

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

We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ Hausdorff distance between two sets of n points under translations in any constant dimension d. This substantially improves the best previous time bound near O(n^{5d/4}) by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan’s algorithm [FOCS'13] for Klee’s measure problem.
To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis.

Timothy M. Chan. Minimum L_∞ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee’s Measure Problem. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chan:LIPIcs.SoCG.2023.24, author = {Chan, Timothy M.}, title = {{Minimum L\underline∞ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee’s Measure Problem}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {24:1--24:13}, 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.2023.24}, URN = {urn:nbn:de:0030-drops-178741}, doi = {10.4230/LIPIcs.SoCG.2023.24}, annote = {Keywords: Hausdorff distance, geometric optimization, Klee’s measure problem, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Given a graph with n vertices and real edge weights in [0,1], we investigate an approximate version of the standard all-pairs shortest paths (APSP) problem where distances are estimated with additive error at most ε. Yuster (2012) introduced this natural variant of approximate APSP, and presented an algorithm for directed graphs running in Õ(n^{(3+ω)/2}) ≤ O(n^{2.687}) time for an arbitrarily small constant ε > 0, where ω denotes the matrix multiplication exponent. We give a faster algorithm for undirected graphs running in Õ(n^{(3+ω²)/(ω+1)}) ≤ O(n^{2.559}) time for any constant ε > 0. If ω = 2, the time bound is Õ(n^{7/3}), matching a previous result for undirected graphs by Dor, Halperin, and Zwick (2000) which only guaranteed additive error at most 2.

Timothy M. Chan. All-Pairs Shortest Paths for Real-Weighted Undirected Graphs with Small Additive Error. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 27:1-27:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chan:LIPIcs.ESA.2021.27, author = {Chan, Timothy M.}, title = {{All-Pairs Shortest Paths for Real-Weighted Undirected Graphs with Small Additive Error}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {27:1--27:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.27}, URN = {urn:nbn:de:0030-drops-146086}, doi = {10.4230/LIPIcs.ESA.2021.27}, annote = {Keywords: Shortest paths, approximation, matrix multiplication} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

In the colored orthogonal range reporting problem, we want a data structure for storing n colored points so that given a query axis-aligned rectangle, we can report the distinct colors among the points inside the rectangle. This natural problem has been studied in a series of papers, but most prior work focused on the static case. In this paper, we give a dynamic data structure in the 2D case which can answer queries in O(log^{1+o(1)} n + klog^{1/2+o(1)}n) time, where k denotes the output size (the number of distinct colors in the query range), and which can support insertions and deletions in O(log^{2+o(1)}n) time (amortized) in the standard RAM model. This is the first fully dynamic structure with polylogarithmic update time whose query cost per color reported is sublogarithmic (near √{log n}). We also give an alternative data structure with O(log^{1+o(1)} n + klog^{3/4+o(1)}n) query time and O(log^{3/2+o(1)}n) update time (amortized). We also mention extensions to higher constant dimensions.

Timothy M. Chan and Zhengcheng Huang. Dynamic Colored Orthogonal Range Searching. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chan_et_al:LIPIcs.ESA.2021.28, author = {Chan, Timothy M. and Huang, Zhengcheng}, title = {{Dynamic Colored Orthogonal Range Searching}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.28}, URN = {urn:nbn:de:0030-drops-146090}, doi = {10.4230/LIPIcs.ESA.2021.28}, annote = {Keywords: Range searching, dynamic data structures, word RAM} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

All-Pairs Shortest Paths (APSP) is one of the most well studied problems in graph algorithms. This paper studies several variants of APSP in unweighted graphs or graphs with small integer weights.
APSP with small integer weights in undirected graphs [Seidel'95, Galil and Margalit'97] has an Õ(n^ω) time algorithm, where ω < 2.373 is the matrix multiplication exponent. APSP in directed graphs with small weights however, has a much slower running time that would be Ω(n^{2.5}) even if ω = 2 [Zwick'02]. To understand this n^{2.5} bottleneck, we build a web of reductions around directed unweighted APSP . We show that it is fine-grained equivalent to computing a rectangular Min-Plus product for matrices with integer entries; the dimensions and entry size of the matrices depend on the value of ω. As a consequence, we establish an equivalence between APSP in directed unweighted graphs, APSP in directed graphs with small (Õ(1)) integer weights, All-Pairs Longest Paths in DAGs with small weights, cRed-APSP in undirected graphs with small weights, for any c ≥ 2 (computing all-pairs shortest path distances among paths that use at most c red edges), #_{≤ c}APSP in directed graphs with small weights (counting the number of shortest paths for each vertex pair, up to c), and approximate APSP with additive error c in directed graphs with small weights, for c ≤ Õ(1).
We also provide fine-grained reductions from directed unweighted APSP to All-Pairs Shortest Lightest Paths (APSLP) in undirected graphs with {0,1} weights and #_{mod c}APSP in directed unweighted graphs (computing counts mod c), thus showing that unless the current algorithms for APSP in directed unweighted graphs can be improved substantially, these problems need at least Ω(n^{2.528}) time.
We complement our hardness results with new algorithms. We improve the known algorithms for APSLP in directed graphs with small integer weights (previously studied by Zwick [STOC'99]) and for approximate APSP with sublinear additive error in directed unweighted graphs (previously studied by Roditty and Shapira [ICALP'08]). Our algorithm for approximate APSP with sublinear additive error is optimal, when viewed as a reduction to Min-Plus product. We also give new algorithms for variants of #APSP (such as #_{≤ U}APSP and #_{mod U}APSP for U ≤ n^{Õ(1)}) in unweighted graphs, as well as a near-optimal Õ(n³)-time algorithm for the original #APSP problem in unweighted graphs (when counts may be exponentially large). This also implies an Õ(n³)-time algorithm for Betweenness Centrality, improving on the previous Õ(n⁴) running time for the problem. Our techniques also lead to a simpler alternative to Shoshan and Zwick’s algorithm [FOCS'99] for the original APSP problem in undirected graphs with small integer weights.

Timothy M. Chan, Virginia Vassilevska Williams, and Yinzhan Xu. Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 47:1-47:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2021.47, author = {Chan, Timothy M. and Vassilevska Williams, Virginia and Xu, Yinzhan}, title = {{Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {47:1--47:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.47}, URN = {urn:nbn:de:0030-drops-141166}, doi = {10.4230/LIPIcs.ICALP.2021.47}, annote = {Keywords: All-Pairs Shortest Paths, Fine-Grained Complexity, Graph Algorithm} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst n given points in d dimensions. Previously, the best algorithms known have running time O(nlog²n) for d = 2 (by Aggarwal and Suri [SoCG'87]) and near n^d for d ≥ 3. We describe faster algorithms with running time
- O(n2^{O(log^*n)}log n) for d = 2,
- O(n^{2.5+o(1)}) time for d = 3, and
- Õ(n^{(5d+2)/6}) time for any constant d ≥ 4.
To obtain the higher-dimensional result, we adapt and extend previous techniques for Klee’s measure problem to optimize certain objective functions over the complement of a union of orthants.

Timothy M. Chan. Faster Algorithms for Largest Empty Rectangles and Boxes. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chan:LIPIcs.SoCG.2021.24, author = {Chan, Timothy M.}, title = {{Faster Algorithms for Largest Empty Rectangles and Boxes}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.24}, URN = {urn:nbn:de:0030-drops-138231}, doi = {10.4230/LIPIcs.SoCG.2021.24}, annote = {Keywords: Largest empty rectangle, largest empty box, Klee’s measure problem} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an O(1)-approximation in sublinear update time for set cover for axis-aligned squares in 2D . More precisely, we obtain randomized update time O(n^{2/3+δ}) for an arbitrarily small constant δ > 0. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D . As a byproduct, our techniques for dynamic set cover also yield an optimal randomized O(nlog n)-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier O(nlog n(log log n)^{O(1)}) result [SoCG 2020].

Timothy M. Chan and Qizheng He. More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2021.25, author = {Chan, Timothy M. and He, Qizheng}, title = {{More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {25:1--25:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.25}, URN = {urn:nbn:de:0030-drops-138244}, doi = {10.4230/LIPIcs.SoCG.2021.25}, annote = {Keywords: Geometric set cover, approximation algorithms, dynamic data structures, sublinear algorithms, random sampling} }

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

In this paper, we present simple randomized multi-pass streaming algorithms for fundamental computational geometry problems of finding the skyline (maximal) points and the extreme points of the convex hull. For the skyline problem, one of our algorithm occupies O(h) space and performs O(log n) passes, where h is the number of skyline points. This improves the space bound of the currently best known result by Das Sarma, Lall, Nanongkai, and Xu [VLDB'09] by a logarithmic factor. For the extreme points problem, we present the first non-trivial result for any constant dimension greater than two: an O(h log^{O(1)}n) space and O(log^dn) pass algorithm, where h is the number of extreme points. Finally, we argue why randomization seems unavoidable for these problems, by proving lower bounds on the performance of deterministic algorithms for a related problem of finding maximal elements in a poset.

Timothy M. Chan and Saladi Rahul. Simple Multi-Pass Streaming Algorithms for Skyline Points and Extreme Points. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chan_et_al:LIPIcs.STACS.2021.22, author = {Chan, Timothy M. and Rahul, Saladi}, title = {{Simple Multi-Pass Streaming Algorithms for Skyline Points and Extreme Points}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.22}, URN = {urn:nbn:de:0030-drops-136674}, doi = {10.4230/LIPIcs.STACS.2021.22}, annote = {Keywords: multi-pass streaming algorithms, skyline, convex hull, extreme points, randomized algorithms} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Given a set of n integer-valued coin types and a target value t, the well-known change-making problem asks for the minimum number of coins that sum to t, assuming an unlimited number of coins in each type. In the more general all-targets version of the problem, we want the minimum number of coins summing to j, for every j = 0,…,t. For example, the textbook dynamic programming algorithms can solve the all-targets problem in O(nt) time. Recently, Chan and He (SOSA'20) described a number of O(t polylog t)-time algorithms for the original (single-target) version of the change-making problem, but not the all-targets version.
In this paper, we obtain a number of new results on change-making and related problems:
- We present a new algorithm for the all-targets change-making problem with running time Õ(t^{4/3}), improving a previous Õ(t^{3/2})-time algorithm.
- We present a very simple Õ(u²+t)-time algorithm for the all-targets change-making problem, where u denotes the maximum coin value. The analysis of the algorithm uses a theorem of Erdős and Graham (1972) on the Frobenius problem. This algorithm can be extended to solve the all-capacities version of the unbounded knapsack problem (for integer item weights bounded by u).
- For the original (single-target) coin changing problem, we describe a simple modification of one of Chan and He’s algorithms that runs in Õ(u) time (instead of Õ(t)).
- For the original (single-capacity) unbounded knapsack problem, we describe a simple algorithm that runs in Õ(nu) time, improving previous near-u²-time algorithms.
- We also observe how one of our ideas implies a new result on the minimum word break problem, an optimization version of a string problem studied by Bringmann et al. (FOCS'17), generalizing change-making (which corresponds to the unary special case).

Timothy M. Chan and Qizheng He. More on Change-Making and Related Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.ESA.2020.29, author = {Chan, Timothy M. and He, Qizheng}, title = {{More on Change-Making and Related Problems}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.29}, URN = {urn:nbn:de:0030-drops-128958}, doi = {10.4230/LIPIcs.ESA.2020.29}, annote = {Keywords: Coin changing, knapsack, dynamic programming, Frobenius problem, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We improve the running times of O(1)-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized O(n log⁴n)-time, O(1)-approximation algorithm, by using variants of the multiplicative weight update (MWU) method combined with geometric data structures. We simplify the data structure requirement in one of their methods and obtain a deterministic O(n log³n log log n)-time algorithm. With further new ideas, we obtain a still faster randomized O(n log n(log log n)^O(1))-time algorithm.
For the weighted problem, we also give a randomized O(n log⁴n log log n)-time, O(1)-approximation algorithm, by simple modifications to the MWU method and the quasi-uniform sampling technique.

Timothy M. Chan and Qizheng He. Faster Approximation Algorithms for Geometric Set Cover. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2020.27, author = {Chan, Timothy M. and He, Qizheng}, title = {{Faster Approximation Algorithms for Geometric Set Cover}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.27}, URN = {urn:nbn:de:0030-drops-121856}, doi = {10.4230/LIPIcs.SoCG.2020.27}, annote = {Keywords: Set cover, approximation algorithms, multiplicate weight update method, random sampling, shallow cuttings} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We present a number of new results about range searching for colored (or "categorical") data:
1) For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1,…,n}. Previous data structures require O((log n)/(log log n) + k) query time. Our result also implies improvements in higher constant dimensions.
2) Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log²n) query time.
3) For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored "type-2" range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O((log n)/(log log n) + k log log n), where k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k (log n)/(log log n)) time.
Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.

Timothy M. Chan, Qizheng He, and Yakov Nekrich. Further Results on Colored Range Searching. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2020.28, author = {Chan, Timothy M. and He, Qizheng and Nekrich, Yakov}, title = {{Further Results on Colored Range Searching}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.28}, URN = {urn:nbn:de:0030-drops-121868}, doi = {10.4230/LIPIcs.SoCG.2020.28}, annote = {Keywords: Range searching, geometric data structures, randomized incremental construction, random sampling, word RAM} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box.
For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods.
We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points.

Sergio Cabello and Timothy M. Chan. Computing Shapley Values in the Plane. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2019.20, author = {Cabello, Sergio and Chan, Timothy M.}, title = {{Computing Shapley Values in the Plane}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {20:1--20:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.20}, URN = {urn:nbn:de:0030-drops-104244}, doi = {10.4230/LIPIcs.SoCG.2019.20}, annote = {Keywords: Shapley values, stochastic computational geometry, convex hull, minimum enclosing disk, bounding box, arrangements, convolutions, airport problem} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Given a set of n points in the plane, and a parameter k, we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing k points. We present the first near quadratic time algorithm for this problem, improving over the previous near-O(n^{5/2})-time algorithm by Kaplan et al. [Haim Kaplan et al., 2017]. We provide an almost matching conditional lower bound, under the assumption that (min,+)-convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for either perimeter or area) that can make the time bound sensitive to k, giving near O(n k) time. We also present a near linear time (1+epsilon)-approximation algorithm to the minimum area of the optimal rectangle containing k points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.

Timothy M. Chan and Sariel Har-Peled. Smallest k-Enclosing Rectangle Revisited. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2019.23, author = {Chan, Timothy M. and Har-Peled, Sariel}, title = {{Smallest k-Enclosing Rectangle Revisited}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.23}, URN = {urn:nbn:de:0030-drops-104276}, doi = {10.4230/LIPIcs.SoCG.2019.23}, annote = {Keywords: Geometric optimization, outliers, approximation algorithms, conditional lower bounds} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We present new results on a number of fundamental problems about dynamic geometric data structures:
1) We describe the first fully dynamic data structures with sublinear amortized update time for maintaining (i) the number of vertices or the volume of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the discrete 1-center of a 2D point set, (v) the number of maximal (i.e., skyline) points in a 3D point set. The update times are near n^{11/12} for (i) and (ii), n^{7/8} for (iii) and (iv), and n^{2/3} for (v). Previously, sublinear bounds were known only for restricted "semi-online" settings [Chan, SODA 2002].
2) We slightly improve previous fully dynamic data structures for answering extreme point queries for the convex hull of a 3D point set and nearest neighbor search for a 2D point set. The query time is O(log^2n), and the amortized update time is O(log^4n) instead of O(log^5n) [Chan, SODA 2006; Kaplan et al., SODA 2017].
3) We also improve previous fully dynamic data structures for maintaining the bichromatic closest pair between two 2D point sets and the diameter of a 2D point set. The amortized update time is O(log^4n) instead of O(log^7n) [Eppstein 1995; Chan, SODA 2006; Kaplan et al., SODA 2017].

Timothy M. Chan. Dynamic Geometric Data Structures via Shallow Cuttings. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chan:LIPIcs.SoCG.2019.24, author = {Chan, Timothy M.}, title = {{Dynamic Geometric Data Structures via Shallow Cuttings}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.24}, URN = {urn:nbn:de:0030-drops-104288}, doi = {10.4230/LIPIcs.SoCG.2019.24}, annote = {Keywords: dynamic data structures, convex hulls, nearest neighbor search, closest pair, shallow cuttings} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

For any constant d and parameter epsilon > 0, we show the existence of (roughly) 1/epsilon^d orderings on the unit cube [0,1)^d, such that any two points p, q in [0,1)^d that are close together under the Euclidean metric are "close together" in one of these linear orderings in the following sense: the only points that could lie between p and q in the ordering are points with Euclidean distance at most epsilon | p - q | from p or q. These orderings are extensions of the Z-order, and they can be efficiently computed.
Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.

Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones. On Locality-Sensitive Orderings and Their Applications. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chan_et_al:LIPIcs.ITCS.2019.21, author = {Chan, Timothy M. and Har-Peled, Sariel and Jones, Mitchell}, title = {{On Locality-Sensitive Orderings and Their Applications}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {21:1--21:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.21}, URN = {urn:nbn:de:0030-drops-101140}, doi = {10.4230/LIPIcs.ITCS.2019.21}, annote = {Keywords: Approximation algorithms, Data structures, Computational geometry} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far.
Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. While for general set cover the best possible approximation ratio is Theta(log n), it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity.
Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.

Timothy M. Chan, Thomas C. van Dijk, Krzysztof Fleszar, Joachim Spoerhase, and Alexander Wolff. Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan_et_al:LIPIcs.ISAAC.2018.61, author = {Chan, Timothy M. and van Dijk, Thomas C. and Fleszar, Krzysztof and Spoerhase, Joachim and Wolff, Alexander}, title = {{Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {61:1--61:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.61}, URN = {urn:nbn:de:0030-drops-100094}, doi = {10.4230/LIPIcs.ISAAC.2018.61}, annote = {Keywords: Geometric optimization, NP-hard, approximation, shallow-cell complexity, line stabbing} }

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

In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing.
- Orthogonal point location. We give the first linear-space data structure that supports 3-d point location queries on n disjoint axis-aligned boxes with optimal O(log n) query time in the (arithmetic) pointer machine model. This improves the previous O(log^{3/2} n) bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time.
- Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(log_wn+k) time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-k rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing.
For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.

Timothy M. Chan, Yakov Nekrich, Saladi Rahul, and Konstantinos Tsakalidis. Orthogonal Point Location and Rectangle Stabbing Queries in 3-d. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2018.31, author = {Chan, Timothy M. and Nekrich, Yakov and Rahul, Saladi and Tsakalidis, Konstantinos}, title = {{Orthogonal Point Location and Rectangle Stabbing Queries in 3-d}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.31}, URN = {urn:nbn:de:0030-drops-90352}, doi = {10.4230/LIPIcs.ICALP.2018.31}, annote = {Keywords: geometric data structures, orthogonal point location, rectangle stabbing, pointer machines, I/O model, word RAM model} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.

Jean Cardinal, Timothy M. Chan, John Iacono, Stefan Langerman, and Aurélien Ooms. Subquadratic Encodings for Point Configurations. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cardinal_et_al:LIPIcs.SoCG.2018.20, author = {Cardinal, Jean and Chan, Timothy M. and Iacono, John and Langerman, Stefan and Ooms, Aur\'{e}lien}, title = {{Subquadratic Encodings for Point Configurations}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.20}, URN = {urn:nbn:de:0030-drops-87337}, doi = {10.4230/LIPIcs.SoCG.2018.20}, annote = {Keywords: point configuration, order type, chirotope, succinct data structure} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that
1) every tree of size n (with arbitrarily large degree) has a straight-line drawing with area n2^{O(sqrt{log log n log log log n})}, improving the longstanding O(n log n) bound;
2) every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area n sqrt{log n}(log log n)^{O(1)}, improving the longstanding O(n log n) bound;
3) every binary tree of size n has a straight-line orthogonal drawing with area n2^{O(log^*n)}, improving the previous O(n log log n) bound by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996);
4) every binary tree of size n has a straight-line order-preserving drawing with area n2^{O(log^*n)}, improving the previous O(n log log n) bound by Garg and Rusu (2003);
5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area n2^{O(sqrt{log n})}, improving the O(n^{3/2}) previous bound by Frati (2007).

Timothy M. Chan. Tree Drawings Revisited. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan:LIPIcs.SoCG.2018.23, author = {Chan, Timothy M.}, title = {{Tree Drawings Revisited}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.23}, URN = {urn:nbn:de:0030-drops-87364}, doi = {10.4230/LIPIcs.SoCG.2018.23}, annote = {Keywords: graph drawing, trees, recursion} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We present the first near-linear-time (1 + epsilon)-approximation algorithm for the diameter of a weighted unit-disk graph of n vertices, running in O(n log^2 n) time, for any constant epsilon>0, improving the near-O(n^{3/2})-time algorithm of Gao and Zhang [STOC 2003]. Using similar ideas, we can construct a (1+epsilon)-approximate distance oracle for weighted unit-disk graphs with O(1) query time, with a similar improvement in the preprocessing time, from near O(n^{3/2}) to O(n log^3 n). We also obtain new results for a number of other related problems in the weighted unit-disk graph metric, such as the radius and bichromatic closest pair.
As a further application, we use our new distance oracle, along with additional ideas, to solve the (1 + epsilon)-approximate all-pairs bounded-leg shortest paths problem for a set of n planar points, with near O(n^{2.579}) preprocessing time, O(n^2 log n) space, and O(log{log n}) query time, improving thus the near-cubic preprocessing bound by Roditty and Segal [SODA 2007].

Timothy M. Chan and Dimitrios Skrepetos. Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2018.24, author = {Chan, Timothy M. and Skrepetos, Dimitrios}, title = {{Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.24}, URN = {urn:nbn:de:0030-drops-87375}, doi = {10.4230/LIPIcs.SoCG.2018.24}, annote = {Keywords: shortest paths, distance oracles, unit-disk graphs, planar graphs} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We study a longstanding problem in computational geometry: dynamic 2-d orthogonal point location, i.e., vertical ray shooting among n horizontal line segments. We present a data structure achieving O(log n / log log n) optimal expected query time and O(log^{1/2+epsilon} n) update time (amortized) in the word-RAM model for any constant epsilon>0, under the assumption that the x-coordinates are integers bounded polynomially in n. This substantially improves previous results of Giyora and Kaplan [SODA 2007] and Blelloch [SODA 2008] with O(log n) query and update time, and of Nekrich (2010) with O(log n / log log n) query time and O(log^{1+epsilon} n) update time. Our result matches the best known upper bound for simpler problems such as dynamic 2-d dominance range searching.
We also obtain similar bounds for orthogonal line segment intersection reporting queries, vertical ray stabbing, and vertical stabbing-max, improving previous bounds, respectively, of Blelloch [SODA 2008] and Mortensen [SODA 2003], of Tao (2014), and of Agarwal, Arge, and Yi [SODA 2005] and Nekrich [ISAAC 2011].

Timothy M. Chan and Konstantinos Tsakalidis. Dynamic Planar Orthogonal Point Location in Sublogarithmic Time. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2018.25, author = {Chan, Timothy M. and Tsakalidis, Konstantinos}, title = {{Dynamic Planar Orthogonal Point Location in Sublogarithmic Time}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {25:1--25:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.25}, URN = {urn:nbn:de:0030-drops-87382}, doi = {10.4230/LIPIcs.SoCG.2018.25}, annote = {Keywords: dynamic data structures, point location, word RAM} }

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**Published in:** OASIcs, Volume 61, 1st Symposium on Simplicity in Algorithms (SOSA 2018)

We revisit the standard 0-1 knapsack problem. The latest polynomial-time approximation scheme by Rhee (2015) with approximation factor 1+eps has running time near O(n+(1/eps)^{5/2}) (ignoring polylogarithmic factors), and is randomized. We present a simpler algorithm which achieves the same result and is deterministic.
With more effort, our ideas can actually lead to an improved time bound near O(n + (1/eps)^{12/5}), and still further improvements for small n.

Timothy M. Chan. Approximation Schemes for 0-1 Knapsack. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan:OASIcs.SOSA.2018.5, author = {Chan, Timothy M.}, title = {{Approximation Schemes for 0-1 Knapsack}}, booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)}, pages = {5:1--5:12}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-064-4}, ISSN = {2190-6807}, year = {2018}, volume = {61}, editor = {Seidel, Raimund}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.5}, URN = {urn:nbn:de:0030-drops-82994}, doi = {10.4230/OASIcs.SOSA.2018.5}, annote = {Keywords: knapsack problem, approximation algorithms, optimization, (min,+)-convolution} }

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

We present an algorithm that computes a (1+varepsilon)-approximation of the diameter of a weighted, undirected planar graph of n vertices with non-negative edge lengths in O(nlog n(log n + (1/varepsilon)^5)) expected time, improving upon the O(n((1/varepsilon)^4 log^4(n) + 2^{O(1/varepsilon)}))-time algorithm of Weimann and Yuster [ICALP 2013]. Our algorithm makes two improvements over that result: first and foremost, it replaces the exponential dependency on 1/varepsilon with a polynomial one, by adapting and specializing Cabello's recent abstract-Voronoi-diagram-based technique [SODA 2017] for approximation purposes; second, it shaves off two logarithmic factors by choosing a better sequence of error parameters during recursion.
Moreover, using similar techniques, we improve the (1+varepsilon)-approximate distance oracle of Gu and Xu [ISAAC 2015] by first replacing the exponential dependency on 1/varepsilon on the preprocessing time and space with a polynomial one and second removing a logarithmic factor from the preprocessing time.

Timothy M. Chan and Dimitrios Skrepetos. Faster Approximate Diameter and Distance Oracles in Planar Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chan_et_al:LIPIcs.ESA.2017.25, author = {Chan, Timothy M. and Skrepetos, Dimitrios}, title = {{Faster Approximate Diameter and Distance Oracles in Planar Graphs}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {25:1--25:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.25}, URN = {urn:nbn:de:0030-drops-78382}, doi = {10.4230/LIPIcs.ESA.2017.25}, annote = {Keywords: planar graphs, diameter, abstract Voronoi diagrams} }

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

We apply the polynomial method - specifically, Chebyshev polynomials - to obtain a number of new results on geometric approximation algorithms in low constant dimensions. For example, we give an algorithm for constructing epsilon-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/epsilon)^{(d-1)/2}), up to a small near-(1/epsilon)^{3/2} factor, for any d-dimensional n-point set. We obtain an improved data structure for Euclidean *approximate nearest neighbor search* with close to O(n log n + (1/epsilon)^{d/4} n) preprocessing time and O((1/epsilon)^{d/4} log n) query time. We obtain improved approximation algorithms for discrete Voronoi diagrams, diameter, and bichromatic closest pair in the L_s-metric for any even integer constant s >= 2. The techniques are general and may have further applications.

Timothy M. Chan. Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chan:LIPIcs.SoCG.2017.26, author = {Chan, Timothy M.}, title = {{Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {26:1--26:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.26}, URN = {urn:nbn:de:0030-drops-72279}, doi = {10.4230/LIPIcs.SoCG.2017.26}, annote = {Keywords: diameter, coresets, approximate nearest neighbor search, the polynomial method, streaming} }

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

We revisit the orthogonal range searching problem and the exact l_infinity nearest neighbor searching problem for a static set of n points when the dimension d is moderately large. We give the first data structure with near linear space that achieves truly sublinear query time when the dimension is any constant multiple of log n. Specifically, the preprocessing time and space are O(n^{1+delta}) for any constant delta>0, and the expected query time is n^{1-1/O(c log c)} for d = c log n. The data structure is simple and is based on a new "augmented, randomized, lopsided" variant of k-d trees. It matches (in fact, slightly improves) the performance of previous combinatorial algorithms that work only in the case of offline queries [Impagliazzo, Lovett, Paturi, and Schneider (2014) and Chan (SODA'15)]. It leads to slightly faster combinatorial algorithms for all-pairs shortest paths in general real-weighted graphs and rectangular Boolean matrix multiplication.
In the offline case, we show that the problem can be reduced to the Boolean orthogonal vectors problem and thus admits an n^{2-1/O(log c)}-time non-combinatorial algorithm [Abboud, Williams, and Yu (SODA'15)]. This reduction is also simple and is based on range trees.
Finally, we use a similar approach to obtain a small improvement to Indyk's data structure [FOCS'98] for approximate l_infinity nearest neighbor search when d = c log n.

Timothy M. Chan. Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chan:LIPIcs.SoCG.2017.27, author = {Chan, Timothy M.}, title = {{Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {27:1--27:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.27}, URN = {urn:nbn:de:0030-drops-72262}, doi = {10.4230/LIPIcs.SoCG.2017.27}, annote = {Keywords: computational geometry, data structures, range searching, nearest neighbor searching} }

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

We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range reporting. We present a new data structure achieving O(log n / log log n + k) optimal query time and O(log^{2/3+o(1)}n) update time (amortized) in the word RAM model, where n is the number of data points and k is the output size. This is the first improvement in over 10 years of Mortensen's previous result [SIAM J. Comput., 2006], which has O(log^{7/8+epsilon}n) update time for an arbitrarily small constant epsilon.
In the case of 3-sided queries, our update time reduces to O(log^{1/2+epsilon}n), improving Wilkinson's previous bound [ESA 2014] of O(log^{2/3+epsilon}n).

Timothy M. Chan and Konstantinos Tsakalidis. Dynamic Orthogonal Range Searching on the RAM, Revisited. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2017.28, author = {Chan, Timothy M. and Tsakalidis, Konstantinos}, title = {{Dynamic Orthogonal Range Searching on the RAM, Revisited}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.28}, URN = {urn:nbn:de:0030-drops-72291}, doi = {10.4230/LIPIcs.SoCG.2017.28}, annote = {Keywords: dynamic data structures, range searching, computational geometry} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

In this paper we study the all-pairs shortest paths problem in (unweighted) unit-disk graphs. The previous best solution for this problem required O(n^2 log n) time, by running the O(n log n)-time single-source shortest path algorithm of Cabello and Jejcic [Comput. Geom., 2015] from every source vertex,where n is the number of vertices. We not only manage to eliminate the logarithmic factor, but also obtain the first (slightly) subquadratic algorithm for the problem, running in O(n^2 sqrt{ frac{log log n}{log n} }) time. Our algorithm computes an implicit representation of all the shortest paths, and, in the same amount of time, can also compute the diameter of the graph.

Timothy M. Chan and Dimitrios Skrepetos. All-Pairs Shortest Paths in Unit-Disk Graphs in Slightly Subquadratic Time. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chan_et_al:LIPIcs.ISAAC.2016.24, author = {Chan, Timothy M. and Skrepetos, Dimitrios}, title = {{All-Pairs Shortest Paths in Unit-Disk Graphs in Slightly Subquadratic Time}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.24}, URN = {urn:nbn:de:0030-drops-67948}, doi = {10.4230/LIPIcs.ISAAC.2016.24}, annote = {Keywords: unit-disk graphs, all-pairs shortest paths, computational geometry} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Given a set P of n moving points in fixed dimension d, where the trajectory of each point is a polynomial of degree bounded by some constant, we present a kinetic data structure (KDS) for maintenance of the closest pair on P.
Assuming the closest pair distance is between 1 and Delta over time, our KDS uses O(n log Delta) space and processes O(n^2 beta log Delta log n + n^2 beta log Delta log log Delta)) events, each in worst-case time O(log^2 n + log^2 log Delta). Here, beta is an extremely slow-growing function. The locality of the KDS is O(log n + log log Delta). Our closest pair KDS supports insertions and deletions of points. An insertion or deletion takes worst-case time O(log Delta log^2 n + log Delta log^2 log Delta).
Also, we use a similar approach to provide a KDS for the all epsilon-nearest neighbors in R^d.
The complexities of the previous KDSs, for both closest pair and all epsilon-nearest neighbors, have polylogarithmic factor, where the number of logs depends on dimension d. Assuming Delta is polynomial in n, our KDSs obtain improvements on the previous KDSs.
Our solutions are based on a kinetic clustering on P. Though we use ideas from the previous clustering KDS by Hershberger, we simplify and improve his work.

Timothy M. Chan and Zahed Rahmati. A Clustering-Based Approach to Kinetic Closest Pair. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chan_et_al:LIPIcs.SWAT.2016.28, author = {Chan, Timothy M. and Rahmati, Zahed}, title = {{A Clustering-Based Approach to Kinetic Closest Pair}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.28}, URN = {urn:nbn:de:0030-drops-60508}, doi = {10.4230/LIPIcs.SWAT.2016.28}, annote = {Keywords: Kinetic Data Structure, Clustering, Closest Pair, All Nearest Neighbors} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Introduced by Agarwal, Har-Peled, and Varadarajan [J. ACM, 2004], an epsilon-kernel of a point set is a coreset that can be used to approximate the width, minimum enclosing cylinder, minimum bounding box, and solve various related geometric optimization problems. Such coresets form one of the most important tools in the design of linear-time approximation algorithms in computational geometry, as well as efficient insertion-only streaming algorithms and dynamic (non-streaming) data structures. In this paper, we continue the theme and explore dynamic streaming algorithms (in the so-called turnstile model).
Andoni and Nguyen [SODA 2012] described a dynamic streaming algorithm for maintaining a (1+epsilon)-approximation of the width using O(polylog U) space and update time for a point set in [U]^d for any constant dimension d and any constant epsilon>0. Their sketch, based on a "polynomial method", does not explicitly maintain an epsilon-kernel. We extend their method to maintain an epsilon-kernel, and at the same time reduce some of logarithmic factors. As an application, we obtain the first randomized dynamic streaming algorithm for the width problem (and related geometric optimization problems) that supports k outliers, using poly(k, log U) space and time.

Timothy M. Chan. Dynamic Streaming Algorithms for Epsilon-Kernels. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 27:1-27:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chan:LIPIcs.SoCG.2016.27, author = {Chan, Timothy M.}, title = {{Dynamic Streaming Algorithms for Epsilon-Kernels}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {27:1--27:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.27}, URN = {urn:nbn:de:0030-drops-59198}, doi = {10.4230/LIPIcs.SoCG.2016.27}, annote = {Keywords: coresets, streaming algorithms, dynamic algorithms, polynomial method, randomization, outliers} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.'s O(n log^2 n) and O(n log n loglog n) solutions respectively.
Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.

Timothy M. Chan and Simon Pratt. Two Approaches to Building Time-Windowed Geometric Data Structures. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2016.28, author = {Chan, Timothy M. and Pratt, Simon}, title = {{Two Approaches to Building Time-Windowed Geometric Data Structures}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.28}, URN = {urn:nbn:de:0030-drops-59201}, doi = {10.4230/LIPIcs.SoCG.2016.28}, annote = {Keywords: time window, geometric data structures, range searching, dynamic convex hull} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (1992) and the optimal but randomized algorithm of Ramos (1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, (<= k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, epsilon-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (1991) and Chazelle (1993).

Timothy M. Chan and Konstantinos Tsakalidis. Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 719-732, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chan_et_al:LIPIcs.SOCG.2015.719, author = {Chan, Timothy M. and Tsakalidis, Konstantinos}, title = {{Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {719--732}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.719}, URN = {urn:nbn:de:0030-drops-51353}, doi = {10.4230/LIPIcs.SOCG.2015.719}, annote = {Keywords: shallow cuttings, derandomization, halfspace range reporting, geometric data structures} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Chazelle [FOCS'89] gave a linear-time algorithm to compute the intersection of two convex polyhedra in three dimensions. We present a simpler algorithm to do the same.

Timothy M. Chan. A Simpler Linear-Time Algorithm for Intersecting Two Convex Polyhedra in Three Dimensions. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 733-738, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chan:LIPIcs.SOCG.2015.733, author = {Chan, Timothy M.}, title = {{A Simpler Linear-Time Algorithm for Intersecting Two Convex Polyhedra in Three Dimensions}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {733--738}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.733}, URN = {urn:nbn:de:0030-drops-51415}, doi = {10.4230/LIPIcs.SOCG.2015.733}, annote = {Keywords: convex polyhedra, intersection, Dobkin–Kirkpatrick hierarchy} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

A mode of a multiset S is an element a in S of maximum multiplicity;
that is, a occurs at least as frequently as any other element in S.
Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i:j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O(sqrt(n) loglog n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O(sqrt(n / log n)) worst-case time. Furthermore, we present strong evidence that a query time significantly below sqrt(n) cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two sqrt(n) by sqrt(n) matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linear-space data structures for orthogonal range mode in higher dimensions (queries in near O(n^(1-1/2d)) time) and for halfspace range mode in higher dimensions (queries in O(n^(1-1/d^2)) time).

Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, and Bryan T. Wilkinson. Linear-Space Data Structures for Range Mode Query in Arrays. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 290-301, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chan_et_al:LIPIcs.STACS.2012.290, author = {Chan, Timothy M. and Durocher, Stephane and Larsen, Kasper Green and Morrison, Jason and Wilkinson, Bryan T.}, title = {{Linear-Space Data Structures for Range Mode Query in Arrays}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {290--301}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.290}, URN = {urn:nbn:de:0030-drops-34254}, doi = {10.4230/LIPIcs.STACS.2012.290}, annote = {Keywords: mode, range query, data structure, linear space, array} }

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Track A: Algorithms, Complexity and Games

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

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry Δ are solvable in time g(d,Δ) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Δ. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth.
We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.

Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková. Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2020.26, author = {Chan, Timothy F. N. and Cooper, Jacob W. and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na}, title = {{Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {26:1--26:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.26}, URN = {urn:nbn:de:0030-drops-124339}, doi = {10.4230/LIPIcs.ICALP.2020.26}, annote = {Keywords: Matroid algorithms, width parameters, integer programming, fixed parameter tractability, branch-width, branch-depth} }

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