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

A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions.
First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ℝ^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an Ω((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete.
Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ℝ^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ≥ d+1 finite point sets in ℝ^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d.

Helena Bergold, Daniel Bertschinger, Nicolas Grelier, Wolfgang Mulzer, and Patrick Schnider. Well-Separation and Hyperplane Transversals in High Dimensions. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bergold_et_al:LIPIcs.SWAT.2022.16, author = {Bergold, Helena and Bertschinger, Daniel and Grelier, Nicolas and Mulzer, Wolfgang and Schnider, Patrick}, title = {{Well-Separation and Hyperplane Transversals in High Dimensions}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.16}, URN = {urn:nbn:de:0030-drops-161766}, doi = {10.4230/LIPIcs.SWAT.2022.16}, annote = {Keywords: hyperplane transversal, high-dimension, hardness} }

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

Let 𝒟 be a set of straight-line segments in the plane, potentially crossing, and let c be a positive integer. We denote by P the union of the endpoints of the straight-line segments of 𝒟 and of the intersection points between pairs of segments. We say that 𝒟 has a nearest-neighbor decomposition into c parts if we can partition P into c point sets P₁, … , P_c such that 𝒟 is the union of the nearest neighbor graphs on P₁, … , P_c. We show that it is NP-complete to decide whether 𝒟 can be drawn as the union of c ≥ 3 nearest-neighbor graphs, even when no two segments cross. We show that for c = 2, it is NP-complete in the general setting and polynomial-time solvable when no two segments cross. We show the existence of an O(log n)-approximation algorithm running in subexponential time for partitioning 𝒟 into a minimum number of nearest-neighbor graphs.
As a main tool in our analysis, we establish the notion of the conflict graph for a drawing 𝒟. The vertices of the conflict graph are the connected components of 𝒟, with the assumption that each connected component is the nearest neighbor graph of its vertices, and there is an edge between two components U and V if and only if the nearest neighbor graph of U ∪ V contains an edge between a vertex in U and a vertex in V. We show that string graphs are conflict graphs of certain planar drawings. For planar graphs and complete k-partite graphs, we give additional, more efficient constructions. We furthermore show that there are subdivisions of non-planar graphs that are not conflict graphs. Lastly, we show a separator lemma for conflict graphs.

Jonas Cleve, Nicolas Grelier, Kristin Knorr, Maarten Löffler, Wolfgang Mulzer, and Daniel Perz. Nearest-Neighbor Decompositions of Drawings. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 21:1-21:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cleve_et_al:LIPIcs.SWAT.2022.21, author = {Cleve, Jonas and Grelier, Nicolas and Knorr, Kristin and L\"{o}ffler, Maarten and Mulzer, Wolfgang and Perz, Daniel}, title = {{Nearest-Neighbor Decompositions of Drawings}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.21}, URN = {urn:nbn:de:0030-drops-161812}, doi = {10.4230/LIPIcs.SWAT.2022.21}, annote = {Keywords: nearest-neighbors, decompositions, drawing} }

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

In the longest plane spanning tree problem, we are given a finite planar point set 𝒫, and our task is to find a plane (i.e., noncrossing) spanning tree T_OPT for 𝒫 with maximum total Euclidean edge length |T_OPT|. Despite more than two decades of research, it remains open if this problem is NP-hard. Thus, previous efforts have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates |T_OPT|. The approximate trees in these algorithms all have small unweighted diameter, typically three or four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms.
We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree T_ALG with diameter at most four and |T_ALG| ≥ 0.546 ⋅ |T_OPT|. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter d ≥ 3, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most d (compared to a longest plane tree without constraints).

Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec. Long Plane Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 23:1-23:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2022.23, author = {Cabello, Sergio and Hoffmann, Michael and Klost, Katharina and Mulzer, Wolfgang and Tkadlec, Josef}, title = {{Long Plane Trees}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.23}, URN = {urn:nbn:de:0030-drops-160311}, doi = {10.4230/LIPIcs.SoCG.2022.23}, annote = {Keywords: geometric network design, spanning trees, plane straight-line graphs, approximation algorithms} }

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

Let S ⊆ ℝ² be a set of n planar sites, such that each s ∈ S has an associated radius r_s > 0. Let 𝒟(S) be the disk intersection graph for S. It has vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii r_s, r_t intersect. Our goal is to design data structures that maintain the connectivity structure of 𝒟(S) as sites are inserted and/or deleted.
First, we consider unit disk graphs, i.e., r_s = 1, for all s ∈ S. We describe a data structure that has O(log² n) amortized update and O(log n/log log n) amortized query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ r_s ≤ Ψ, for a Ψ ≥ 1 known in advance. In the fully dynamic case, we achieve amortized update time O(Ψ λ₆(log n) log⁷ n) and query time O(log n/log log n), where λ_s(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. In the incremental case, where only insertions are allowed, we get logarithmic dependency on Ψ, with O(α(n)) query time and O(logΨ λ₆(log n) log⁷ n) update time. For the decremental setting, where only deletions are allowed, we first develop an efficient disk revealing structure: given two sets R and B of disks, we can delete disks from R, and upon each deletion, we receive a list of all disks in B that no longer intersect the union of R. Using this, we get decremental data structures with amortized query time O(log n/log log n) that support m deletions in O((nlog⁵ n + m log⁷ n) λ₆(log n) + nlog Ψ log⁴n) overall time for bounded radius ratio Ψ and O((nlog⁶ n + m log⁸n) λ₆(log n)) for arbitrary radii.

Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic Connectivity in Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 49:1-49:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kaplan_et_al:LIPIcs.SoCG.2022.49, author = {Kaplan, Haim and Kauer, Alexander and Klost, Katharina and Knorr, Kristin and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul}, title = {{Dynamic Connectivity in Disk Graphs}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {49:1--49:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.49}, URN = {urn:nbn:de:0030-drops-160572}, doi = {10.4230/LIPIcs.SoCG.2022.49}, annote = {Keywords: Disk Graphs, Connectivity, Lower Envelopes} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Let V ⊂ ℝ² be a set of n sites in the plane. The unit disk graph DG(V) of V is the graph with vertex set V where two sites v and w are adjacent if and only if their Euclidean distance is at most 1.
We develop a compact routing scheme ℛ for DG(V). The routing scheme ℛ preprocesses DG(V) by assigning a label 𝓁(v) to every site v in V. After that, for any two sites s and t, the scheme ℛ must be able to route a packet from s to t as follows: given the label of a current vertex r (initially, r = s), the label of the target vertex t, and additional information in the header of the packet, the scheme determines a neighbor r' of r. Then, the packet is forwarded to r', and the process continues until the packet reaches its desired target t. The resulting path between the source s and the target t is called the routing path of s and t. The stretch of the routing scheme is the maximum ratio of the total Euclidean length of the routing path and of the shortest path in DG(V), between any two sites s, t ∈ V.
We show that for any given ε > 0, we can construct a routing scheme for DG(V) with diameter D that achieves stretch 1+ε, has label size (1/ε)^{O(ε^(-2))} log Dlog³n/log log n, and the header has at most O(log²n/log log n) bits. In the past, several routing schemes for unit disk graphs have been proposed. Our scheme achieves poly-logarithmic label and header size, small stretch and does not use any neighborhood oracles.

Wolfgang Mulzer and Max Willert. Compact Routing in Unit Disk Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mulzer_et_al:LIPIcs.ISAAC.2020.16, author = {Mulzer, Wolfgang and Willert, Max}, title = {{Compact Routing in Unit Disk Graphs}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.16}, URN = {urn:nbn:de:0030-drops-133602}, doi = {10.4230/LIPIcs.ISAAC.2020.16}, annote = {Keywords: routing scheme, unit disk graph, separator} }

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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)

The classic Ham-Sandwich theorem states that for any d measurable sets in ℝ^d, there is a hyperplane that bisects them simultaneously. An extension by Bárány, Hubard, and Jerónimo [DCG 2008] states that if the sets are convex and well-separated, then for any given α₁, … , α_d ∈ [0, 1], there is a unique oriented hyperplane that cuts off a respective fraction α₁, … , α_d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the α-Ham-Sandwich theorem. They gave an algorithm to find the hyperplane in time O(n (log n)^{d-3}), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]).
Recently, Fearnley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called Unique End-of-Potential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the α-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P-matrix linear complementarity problem.

Man-Kwun Chiu, Aruni Choudhary, and Wolfgang Mulzer. Computational Complexity of the α-Ham-Sandwich Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 31:1-31:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chiu_et_al:LIPIcs.ICALP.2020.31, author = {Chiu, Man-Kwun and Choudhary, Aruni and Mulzer, Wolfgang}, title = {{Computational Complexity of the \alpha-Ham-Sandwich Problem}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {31:1--31:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.31}, URN = {urn:nbn:de:0030-drops-124382}, doi = {10.4230/LIPIcs.ICALP.2020.31}, annote = {Keywords: Ham-Sandwich Theorem, Computational Complexity, Continuous Local Search} }

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

Tverberg’s theorem states that for any k ≥ 2 and any set P ⊂ ℝ^d of at least (d + 1)(k - 1) + 1 points, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem lies in the complexity class PPAD ∩ PLS, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa [SODA 2019] gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm.
We present a deterministic algorithm that finds for any n-point set P ⊂ ℝ^d and any k ∈ {2, … , n} in O(nd ⌈log k⌉) time a k-partition of P such that there is a ball of radius O((k/√n)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, and that there are no approximation algorithms that are truly polynomial in any dimension, our result provides a remarkably efficient and simple new notion of approximation.
Our main contribution is to generalize Sarkaria’s method [Israel Journal Math., 1992] to reduce the Tverberg problem to the Colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.

Aruni Choudhary and Wolfgang Mulzer. No-Dimensional Tverberg Theorems and Algorithms. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 31:1-31:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{choudhary_et_al:LIPIcs.SoCG.2020.31, author = {Choudhary, Aruni and Mulzer, Wolfgang}, title = {{No-Dimensional Tverberg Theorems and Algorithms}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {31:1--31:17}, 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.2020.31}, URN = {urn:nbn:de:0030-drops-121893}, doi = {10.4230/LIPIcs.SoCG.2020.31}, annote = {Keywords: Tverberg’s theorem, Colorful Carath\'{e}odory Theorem, Tensor lifting} }

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

Let P be a set of 2n points in convex position, such that n points are colored red and n points are colored blue. A non-crossing alternating path on P of length 𝓁 is a sequence p₁, … , p_𝓁 of 𝓁 points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors, for i ≠ j.
We show that there is an absolute constant ε > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + ε)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + ε)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3+o(n).

Wolfgang Mulzer and Pavel Valtr. Long Alternating Paths Exist. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 57:1-57:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mulzer_et_al:LIPIcs.SoCG.2020.57, author = {Mulzer, Wolfgang and Valtr, Pavel}, title = {{Long Alternating Paths Exist}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {57:1--57:16}, 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.2020.57}, URN = {urn:nbn:de:0030-drops-122152}, doi = {10.4230/LIPIcs.SoCG.2020.57}, annote = {Keywords: Non-crossing path, bichromatic point sets} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ^{3ω/2}n^{ω/2}) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n × n matrices can be multiplied in O(n^ω) time.
The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators.
We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(n^{ω/2}) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1, Ψ] can be found in O(Ψ⁶log^11 n + Ψ^{12 ω} n^{ω/2}) time with high probability.

Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum Matchings in Geometric Intersection Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 31:1-31:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bonnet_et_al:LIPIcs.STACS.2020.31, author = {Bonnet, \'{E}douard and Cabello, Sergio and Mulzer, Wolfgang}, title = {{Maximum Matchings in Geometric Intersection Graphs}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {31:1--31:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.31}, URN = {urn:nbn:de:0030-drops-118926}, doi = {10.4230/LIPIcs.STACS.2020.31}, annote = {Keywords: computational geometry, geometric intersection graph, maximum matching, disk graph, unit-disk graph} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s.
We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.

Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Triangles and Girth in Disk Graphs and Transmission Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 64:1-64:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kaplan_et_al:LIPIcs.ESA.2019.64, author = {Kaplan, Haim and Klost, Katharina and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha}, title = {{Triangles and Girth in Disk Graphs and Transmission Graphs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {64:1--64:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.64}, URN = {urn:nbn:de:0030-drops-111859}, doi = {10.4230/LIPIcs.ESA.2019.64}, annote = {Keywords: disk graph, transmission graph, triangle, girth} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain.
We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space.

Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth. On the Stretch Factor of Polygonal Chains. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.MFCS.2019.56, author = {Chen, Ke and Dumitrescu, Adrian and Mulzer, Wolfgang and T\'{o}th, Csaba D.}, title = {{On the Stretch Factor of Polygonal Chains}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {56:1--56:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.56}, URN = {urn:nbn:de:0030-drops-110005}, doi = {10.4230/LIPIcs.MFCS.2019.56}, annote = {Keywords: polygonal chain, vertex dilation, Koch curve, recursive construction} }

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

We present efficient data structures for problems on unit discs and arcs of their boundary in the plane. (i) We give an output-sensitive algorithm for the dynamic maintenance of the union of n unit discs under insertions in O(k log^2 n) update time and O(n) space, where k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. (ii) As part of the solution of (i) we devise a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines, which we believe is of independent interest. The structure has O(log^2 n) update time and O(log n) vertical ray shooting query time. To achieve this performance, we devise a new algorithm for finding the intersection between two lower envelopes of pseudo-lines in O(log n) time, using tentative binary search; the lower envelopes are special in that at x=-infty any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. (iii) We also present a dynamic range searching structure for a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), where the ranges are unit discs, with O(n log n) preprocessing time, O(n^{1/2+epsilon} + l) query time and O(log^2 n) amortized update time, where l is the size of the output and for any epsilon>0. The structure requires O(n) storage space.

Pankaj K. Agarwal, Ravid Cohen, Dan Halperin, and Wolfgang Mulzer. Maintaining the Union of Unit Discs Under Insertions with Near-Optimal Overhead. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 26:1-26:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2019.26, author = {Agarwal, Pankaj K. and Cohen, Ravid and Halperin, Dan and Mulzer, Wolfgang}, title = {{Maintaining the Union of Unit Discs Under Insertions with Near-Optimal Overhead}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {26:1--26:15}, 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.2019.26}, URN = {urn:nbn:de:0030-drops-104307}, doi = {10.4230/LIPIcs.SoCG.2019.26}, annote = {Keywords: lower envelopes, pseudo-lines, unit discs, range search, dynamic algorithms, tentative binary search} }

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**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

We consider asymmetric convex intersection testing (ACIT).
Let P subset R^d be a set of n points and H a set of n halfspaces in d dimensions. We denote by {ch(P)} the polytope obtained by taking the convex hull of P, and by {fh(H)} the polytope obtained by taking the intersection of the halfspaces in H. Our goal is to decide whether the intersection of H and the convex hull of P are disjoint. Even though ACIT is a natural variant of classic LP-type problems that have been studied at length in the literature, and despite its applications in the analysis of high-dimensional data sets, it appears that the problem has not been studied before.
We discuss how known approaches can be used to attack the ACIT problem, and we provide a very simple strategy that leads to a deterministic algorithm, linear on n and m, whose running time depends reasonably on the dimension d.

Luis Barba and Wolfgang Mulzer. Asymmetric Convex Intersection Testing. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barba_et_al:OASIcs.SOSA.2019.9, author = {Barba, Luis and Mulzer, Wolfgang}, title = {{Asymmetric Convex Intersection Testing}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {9:1--9:14}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.9}, URN = {urn:nbn:de:0030-drops-100358}, doi = {10.4230/OASIcs.SOSA.2019.9}, annote = {Keywords: polytope intersection, LP-type problem, randomized algorithm} }

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

Our goal is to compare two planar point sets by finding subsets of a given size such that a minimum-weight matching between them has the smallest weight. This can be done by a translation of one set that minimizes the weight of the matching. We give efficient algorithms (a) for finding approximately optimal matchings, when the cost of a matching is the L_p-norm of the tuple of the Euclidean distances between the pairs of matched points, for any p in [1,infty], and (b) for constructing small-size approximate minimization (or matching) diagrams: partitions of the translation space into regions, together with an approximate optimal matching for each region.

Pankaj K. Agarwal, Haim Kaplan, Geva Kipper, Wolfgang Mulzer, Günter Rote, Micha Sharir, and Allen Xiao. Approximate Minimum-Weight Matching with Outliers Under Translation. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 26:1-26:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{agarwal_et_al:LIPIcs.ISAAC.2018.26, author = {Agarwal, Pankaj K. and Kaplan, Haim and Kipper, Geva and Mulzer, Wolfgang and Rote, G\"{u}nter and Sharir, Micha and Xiao, Allen}, title = {{Approximate Minimum-Weight Matching with Outliers Under Translation}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {26:1--26:13}, 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.2018.26}, URN = {urn:nbn:de:0030-drops-99747}, doi = {10.4230/LIPIcs.ISAAC.2018.26}, annote = {Keywords: Minimum-weight partial matching, Pattern matching, Approximation} }

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

Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points.
This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.

Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert. Stabbing Pairwise Intersecting Disks by Five Points. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 50:1-50:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{harpeled_et_al:LIPIcs.ISAAC.2018.50, author = {Har-Peled, Sariel and Kaplan, Haim and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha and Willert, Max}, title = {{Stabbing Pairwise Intersecting Disks by Five Points}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {50:1--50:12}, 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.2018.50}, URN = {urn:nbn:de:0030-drops-99989}, doi = {10.4230/LIPIcs.ISAAC.2018.50}, annote = {Keywords: Disk graph, piercing set, LP-type problem} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Motivated by map labeling, we study the problem in which we
are given a collection of n disks in the
plane that grow at possibly different speeds. Whenever two
disks meet, the one with the higher index disappears. This
problem was introduced by Funke, Krumpe, and Storandt[IWOCA 2016].
We provide the first general subquadratic algorithm for computing
the times and the order of disappearance.
Our algorithm also works for other shapes (such as rectangles)
and in any fixed dimension.
Using quadtrees, we provide an alternative
algorithm that runs in near linear time, although
this second algorithm has a logarithmic dependence
on either the ratio of the fastest speed to the slowest speed of disks
or the spread of the disk centers
(the ratio of the maximum to the minimum distance between them).
Our result improves the running times of previous algorithms by
Funke, Krumpe, and
Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and
Funke and Storandt [EWCG 2017].
Finally, we give an \Omega(n\log n) lower bound on the
problem, showing that our quadtree algorithms are almost tight.

Hee-Kap Ahn, Sang Won Bae, Jongmin Choi, Matias Korman, Wolfgang Mulzer, Eunjin Oh, Ji-won Park, André van Renssen, and Antoine Vigneron. Faster Algorithms for Growing Prioritized Disks and Rectangles. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 3:1-3:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ahn_et_al:LIPIcs.ISAAC.2017.3, author = {Ahn, Hee-Kap and Bae, Sang Won and Choi, Jongmin and Korman, Matias and Mulzer, Wolfgang and Oh, Eunjin and Park, Ji-won and van Renssen, Andr\'{e} and Vigneron, Antoine}, title = {{Faster Algorithms for Growing Prioritized Disks and Rectangles}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {3:1--3:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.3}, URN = {urn:nbn:de:0030-drops-82199}, doi = {10.4230/LIPIcs.ISAAC.2017.3}, annote = {Keywords: map labeling, growing disks, elimination order} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex. Then, we must be able to route a data packet between any two vertices p and q of P , where each step must use only the label of the target node q and the routing table of the current node.
For any fixed eps > 0, we pre ent a routing scheme that always achieves a routing path that exceeds the shortest path by a factor of at most 1 + eps. The labels have O(log n) bits, and the routing tables are of size O((eps^{-1} + h) log n). The preprocessing time is O(n^2 log n + hn^2 + eps^{-1}hn). It can be improved to O(n 2 + eps^{-1}n) for simple polygons.

Bahareh Banyassady, Man-Kwun Chiu, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, Yannik Stein, Birgit Vogtenhuber, and Max Willert. Routing in Polygonal Domains. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 10:1-10:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{banyassady_et_al:LIPIcs.ISAAC.2017.10, author = {Banyassady, Bahareh and Chiu, Man-Kwun and Korman, Matias and Mulzer, Wolfgang and van Renssen, Andr\'{e} and Roeloffzen, Marcel and Seiferth, Paul and Stein, Yannik and Vogtenhuber, Birgit and Willert, Max}, title = {{Routing in Polygonal Domains}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {10:1--10:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.10}, URN = {urn:nbn:de:0030-drops-82379}, doi = {10.4230/LIPIcs.ISAAC.2017.10}, annote = {Keywords: polygonal domains, routing scheme, small stretch,Yao graph} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space.
For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards.
We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.

Bahareh Banyassady, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, and Yannik Stein. Improved Time-Space Trade-Offs for Computing Voronoi Diagrams. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{banyassady_et_al:LIPIcs.STACS.2017.9, author = {Banyassady, Bahareh and Korman, Matias and Mulzer, Wolfgang and van Renssen, Andr\'{e} and Roeloffzen, Marcel and Seiferth, Paul and Stein, Yannik}, title = {{Improved Time-Space Trade-Offs for Computing Voronoi Diagrams}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.9}, URN = {urn:nbn:de:0030-drops-70249}, doi = {10.4230/LIPIcs.STACS.2017.9}, annote = {Keywords: memory-constrained model, Voronoi diagram, time-space trade-off} }

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

Let P_1,...,P_{d+1} be d-dimensional point sets such that the convex hull of each P_i contains the origin. We call the sets P_i color classes, and we think of the points in P_i as having color i. A colorful choice is a set with at most one point of each color. The colorful Caratheodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the computational complexity of finding such a colorful choice is unknown.
We approach this problem from two directions. First, we consider approximation algorithms: an m-colorful choice is a set that contains at most m points from each color class. We show that for any fixed epsilon > 0, an (epsilon d)-colorful choice containing the origin in its convex hull can be found in polynomial time. This notion of approximation has not been studied before, and it is motivated through the applications of the colorful Caratheodory theorem in the literature. In the second part, we present a natural generalization of the colorful Caratheodory problem: in the Nearest Colorful Polytope problem (NCP), we are given d-dimensional point sets P_1,...,P_n that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing local optima for the NCP problem is PLS-complete, while computing a global optimum is NP-hard.

Wolfgang Mulzer and Yannik Stein. Computational Aspects of the Colorful Carathéodory Theorem. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 44-58, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{mulzer_et_al:LIPIcs.SOCG.2015.44, author = {Mulzer, Wolfgang and Stein, Yannik}, title = {{Computational Aspects of the Colorful Carath\'{e}odory Theorem}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {44--58}, 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.2015.44}, URN = {urn:nbn:de:0030-drops-51019}, doi = {10.4230/LIPIcs.SOCG.2015.44}, annote = {Keywords: colorful Carath\'{e}odory theorem, high-dimensional approximation, PLS} }

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

Let P be a set of n points in d dimensions, each with an associated radius r_p > 0. The transmission graph G for P has vertex set P and an edge from p to q if and only if q lies in the ball with radius r_p around p. Let t > 1. A t-spanner H for G is a sparse subgraph of G such that for any two vertices p, q connected by a path of length l in G, there is a p-q-path of length at most tl in H. We show how to compute a t-spanner for G if d=2. The running time is O(n (log n + log Psi)), where Psi is the ratio of the largest and smallest radius of two points in P. We extend this construction to be independent of Psi at the expense of a polylogarithmic overhead in the running time. As a first application, we prove a property of the t-spanner that allows us to find a BFS tree in G for any given start vertex s of P in the same time.
After that, we deal with reachability oracles for G. These are data structures that answer reachability queries: given two vertices, is there a directed path between them? The quality of a reachability oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. For d=1, we show how to compute an oracle with Q(n) = O(1) and S(n) = O(n) in time O(n log n). For d=2, the radius ratio Psi again turns out to be an important measure for the complexity of the problem. We present three different data structures whose quality depends on Psi: (i) if Psi < sqrt(3), we achieve Q(n) = O(1) with S(n) = O(n) and preproccesing time O(n log n); (ii) if Psi >= sqrt(3), we get Q(n) = O(Psi^3 sqrt(n)) and S(n) = O(Psi^5 n^(3/2)); and (iii) if Psi is polynomially bounded in n, we use probabilistic methods to obtain an oracle with Q(n) = O(n^(2/3)log n) and S(n) = O(n^(5/3) log n) that answers queries correctly with high probability. We employ our t-spanner to achieve a fast preproccesing time of O(Psi^5 n^(3/2)) and O(n^(5/3) log^2 n) in case (ii) and (iii), respectively.

Haim Kaplan, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Spanners and Reachability Oracles for Directed Transmission Graphs. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 156-170, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kaplan_et_al:LIPIcs.SOCG.2015.156, author = {Kaplan, Haim and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul}, title = {{Spanners and Reachability Oracles for Directed Transmission Graphs}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {156--170}, 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.2015.156}, URN = {urn:nbn:de:0030-drops-51062}, doi = {10.4230/LIPIcs.SOCG.2015.156}, annote = {Keywords: Transmission Graphs, Reachability Oracles, Spanner, Intersection Graph} }

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

The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.
In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.
This raises the question of how well we can approximate the Fréchet distance (of two given d-dimensional point sequences of length n) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be 2^Theta(n). Moreover, we design an alpha-approximation algorithm that runs in time O(n log n + n^2 / alpha), for any alpha in [1, n]. Hence, an n^epsilon-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any epsilon > 0.

Karl Bringmann and Wolfgang Mulzer. Approximability of the Discrete Fréchet Distance. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 739-753, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bringmann_et_al:LIPIcs.SOCG.2015.739, author = {Bringmann, Karl and Mulzer, Wolfgang}, title = {{Approximability of the Discrete Fr\'{e}chet Distance}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {739--753}, 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.2015.739}, URN = {urn:nbn:de:0030-drops-51072}, doi = {10.4230/LIPIcs.SOCG.2015.739}, annote = {Keywords: Fr\'{e}chet distance, approximation, lower bounds, Strong Exponential Time Hypothesis} }

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