Pub Date : 2023-11-07DOI: 10.1142/s0218195923500036
Sergey Korotov, Jon Eivind Vatne
Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms.
{"title":"On Dihedral Angle Sums of Prisms and Hexahedra","authors":"Sergey Korotov, Jon Eivind Vatne","doi":"10.1142/s0218195923500036","DOIUrl":"https://doi.org/10.1142/s0218195923500036","url":null,"abstract":"Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135545464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-02DOI: 10.1142/s0218195923500024
Abderrazzak Benroummane
We give some properties of semi-symmetric pseudo-Riemannian manifolds as an indecomposable irreducible Ricci pseudo-Riemannian manifold (i.e. the minimal polynomial of its Ricci operator is irreducible) is semi symmetric if and only if it is locally symmetric. We also show that any semi-symmetric pseudo-Riemannian manifold will be foliated. Moreover, if the metric is Lorentzian, the Ricci operator has only real eigenvalues and more precisely, on each leaf, it is diagonalizable with at most a single non zero eigenvalue or isotropic.
{"title":"Some Results on Semi-Symmetric Spaces","authors":"Abderrazzak Benroummane","doi":"10.1142/s0218195923500024","DOIUrl":"https://doi.org/10.1142/s0218195923500024","url":null,"abstract":"We give some properties of semi-symmetric pseudo-Riemannian manifolds as an indecomposable irreducible Ricci pseudo-Riemannian manifold (i.e. the minimal polynomial of its Ricci operator is irreducible) is semi symmetric if and only if it is locally symmetric. We also show that any semi-symmetric pseudo-Riemannian manifold will be foliated. Moreover, if the metric is Lorentzian, the Ricci operator has only real eigenvalues and more precisely, on each leaf, it is diagonalizable with at most a single non zero eigenvalue or isotropic.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"29 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135973335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-01DOI: 10.1142/s0218195922410035
Stephane Durocher, J. Mark Keil, Saeed Mehrabi, Debajyoti Mondal
Chvátal and Klincsek (1980) gave an [Formula: see text]-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set [Formula: see text] of [Formula: see text] points in the plane. This paper examines a generalization of the problem, the Bottleneck Convex Subsets problem: given a set [Formula: see text] of [Formula: see text] points in the plane and a positive integer [Formula: see text], select [Formula: see text] pairwise disjoint convex subsets of [Formula: see text] such that the cardinality of the smallest subset is maximized. Equivalently, a solution maximizes the cardinality of [Formula: see text] mutually disjoint convex subsets of [Formula: see text] of equal cardinality. We give an algorithm that solves the problem exactly, with running time polynomial in [Formula: see text] when [Formula: see text] is fixed. We then show the problem to be NP-hard when [Formula: see text] is an arbitrary input parameter, even for points in general position. Finally, we give a fixed-parameter tractable algorithm parameterized in terms of the number of points strictly interior to the convex hull.
Chvátal和Klincsek(1980)给出了一种[公式:见文]时间算法，用于在平面上的[公式:见文]点的任意给定集合[公式:见文]中找到一个最大基数凸子集的问题。本文研究瓶颈凸子集问题的一个推广:给定平面上[公式:见文]点的[公式:见文]集和一个正整数[公式:见文]，选择[公式:见文]的[公式:见文]的[公式:见文]对不相交的[公式:见文]凸子集，使最小子集的基数最大化。同样地，一个解最大化相等基数的[公式:见文]的[公式:见文]互不相交的凸子集的基数。我们给出了一种算法，当[Formula: see text]固定时，算法的运行时间多项式为[Formula: see text]。然后，当[公式:见文本]是任意输入参数时，即使对于一般位置的点，我们也会证明问题是np困难的。最后，给出了一种以凸包内严格点数为参数的定参数易处理算法。
{"title":"Bottleneck Convex Subsets: Finding k Large Convex Sets in a Point Set","authors":"Stephane Durocher, J. Mark Keil, Saeed Mehrabi, Debajyoti Mondal","doi":"10.1142/s0218195922410035","DOIUrl":"https://doi.org/10.1142/s0218195922410035","url":null,"abstract":"Chvátal and Klincsek (1980) gave an [Formula: see text]-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set [Formula: see text] of [Formula: see text] points in the plane. This paper examines a generalization of the problem, the Bottleneck Convex Subsets problem: given a set [Formula: see text] of [Formula: see text] points in the plane and a positive integer [Formula: see text], select [Formula: see text] pairwise disjoint convex subsets of [Formula: see text] such that the cardinality of the smallest subset is maximized. Equivalently, a solution maximizes the cardinality of [Formula: see text] mutually disjoint convex subsets of [Formula: see text] of equal cardinality. We give an algorithm that solves the problem exactly, with running time polynomial in [Formula: see text] when [Formula: see text] is fixed. We then show the problem to be NP-hard when [Formula: see text] is an arbitrary input parameter, even for points in general position. Finally, we give a fixed-parameter tractable algorithm parameterized in terms of the number of points strictly interior to the convex hull.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136019487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-12-09DOI: 10.1142/S0218195917600020
Siu-Wing Cheng, Man-Kwun Chiu, Jiongxin Jin, A. Vigneron
We propose an algorithm for finding a ((1+varepsilon ))-approximate shortest path through a weighted 3D simplicial complex (mathcal T). The weights are integers from the range [1, W] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in (mathcal T). Let (rho ) be some arbitrary constant. Let (kappa ) be the size of the largest connected component of tetrahedra whose aspect ratios exceed (rho ). There exists a constant C dependent on (rho ) but independent of (mathcal T) such that if (kappa le frac{1}{C}log log n + O(1)), the running time of our algorithm is polynomial in n, (1/varepsilon ) and (log (NW)). If (kappa = O(1)), the running time reduces to (O(n varepsilon ^{-O(1)}(log (NW))^{O(1)})).
{"title":"Navigating Weighted Regions with Scattered Skinny Tetrahedra","authors":"Siu-Wing Cheng, Man-Kwun Chiu, Jiongxin Jin, A. Vigneron","doi":"10.1142/S0218195917600020","DOIUrl":"https://doi.org/10.1142/S0218195917600020","url":null,"abstract":"We propose an algorithm for finding a ((1+varepsilon ))-approximate shortest path through a weighted 3D simplicial complex (mathcal T). The weights are integers from the range [1, W] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in (mathcal T). Let (rho ) be some arbitrary constant. Let (kappa ) be the size of the largest connected component of tetrahedra whose aspect ratios exceed (rho ). There exists a constant C dependent on (rho ) but independent of (mathcal T) such that if (kappa le frac{1}{C}log log n + O(1)), the running time of our algorithm is polynomial in n, (1/varepsilon ) and (log (NW)). If (kappa = O(1)), the running time reduces to (O(n varepsilon ^{-O(1)}(log (NW))^{O(1)})).","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114326403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-12-15DOI: 10.1142/S0218195917600044
O. Aichholzer, Vincent Kusters, Wolfgang Mulzer, Alexander Pilz, Manuel Wettstein
Let $P$ be a set of $n$ labeled points in the plane. The radial system of $P$ describes, for each $pin P$, the order in which a ray that rotates around $p$ encounters the points in $P setminus {p}$. This notion is related to the order type of $P$, which describes the orientation (clockwise or counterclockwise) of every ordered triple in $P$. Given only the order type, the radial system is uniquely determined and can easily be obtained. The converse, however, is not true. Indeed, let $R$ be the radial system of $P$, and let $T(R)$ be the set of all order types with radial system $R$ (we define $T(R) = emptyset$ for the case that $R$ is not a valid radial system). Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in ISAAC 2014) show that $T(R)$ may contain up to $n-1$ order types. They also provide polynomial-time algorithms to compute $T(R)$ when only $R$ is given. We describe a new algorithm for finding $T(R)$. The algorithm constructs the convex hulls of all possible point sets with the radial system $R$. After that, orientation queries on point triples can be answered in constant time. A representation of this set of convex hulls can be found in $O(n)$ queries to the radial system, using $O(n)$ additional processing time. This is optimal. Our results also generalize to abstract order types.
设P是平面上n个标记点的集合。$P$的径向系统描述了，对于$P$中的每个$P$，围绕$P$旋转的射线遇到$P setminus {P }$中的点的顺序。这个概念与$P$的顺序类型有关，它描述了$P$中每个有序三元组的方向(顺时针或逆时针)。仅给定阶型，径向系统是唯一确定的，可以很容易地得到。然而，反之则不成立。确实，设$R$是$P$的径向系统，设$T(R)$是具有径向系统$R$的所有阶类型的集合(我们定义$T(R) = emptyset$，因为$R$不是一个有效的径向系统)。Aichholzer等人(Reconstructing Point Set Order Types from Radial Orderings, in ISAAC 2014)表明$T(R)$可能包含多达$n-1$阶类型。他们还提供了多项式时间算法来计算只有R给定的T(R)$。我们描述了一种求T(R)的新算法。该算法用径向系统$R$构造所有可能点集的凸包。之后，对点三元组的方向查询可以在常数时间内得到回答。这组凸包的表示可以在对径向系统的$O(n)$查询中找到，使用$O(n)$额外的处理时间。这是最优的。我们的结果也推广到抽象顺序类型。
{"title":"An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings","authors":"O. Aichholzer, Vincent Kusters, Wolfgang Mulzer, Alexander Pilz, Manuel Wettstein","doi":"10.1142/S0218195917600044","DOIUrl":"https://doi.org/10.1142/S0218195917600044","url":null,"abstract":"Let $P$ be a set of $n$ labeled points in the plane. The radial system of $P$ describes, for each $pin P$, the order in which a ray that rotates around $p$ encounters the points in $P setminus {p}$. This notion is related to the order type of $P$, which describes the orientation (clockwise or counterclockwise) of every ordered triple in $P$. Given only the order type, the radial system is uniquely determined and can easily be obtained. The converse, however, is not true. Indeed, let $R$ be the radial system of $P$, and let $T(R)$ be the set of all order types with radial system $R$ (we define $T(R) = emptyset$ for the case that $R$ is not a valid radial system). Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in ISAAC 2014) show that $T(R)$ may contain up to $n-1$ order types. They also provide polynomial-time algorithms to compute $T(R)$ when only $R$ is given. \u0000We describe a new algorithm for finding $T(R)$. The algorithm constructs the convex hulls of all possible point sets with the radial system $R$. After that, orientation queries on point triples can be answered in constant time. A representation of this set of convex hulls can be found in $O(n)$ queries to the radial system, using $O(n)$ additional processing time. This is optimal. Our results also generalize to abstract order types.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"155 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116633218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-12-15DOI: 10.1142/S0218195916600025
S. Suri, Kevin Verbeek
We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple ((s_i, pi _i)) consisting of a point (s_i) in a (d)-dimensional space and a probability (pi _i) determining its existence. The problem is interesting and non-trivial even in (1)-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity (Omega (n^2)). We then show that under more natural and less adversarial conditions, the size of the (1)-dimensional LVD is significantly smaller: (1) (Theta (k n)) if the input has only (k) distinct probability values, (2) (O(n log n)) on average, and (3) (O(n sqrt{n})) under smoothed analysis. We also present an alternative approach to the most likely nearest neighbor (LNN) search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models, as well as worst-case with a bounded number of distinct probabilities. Using the Pareto-set approach, we can also reduce the multi-dimensional LNN search to a sequence of nearest neighbor and spherical range queries.
{"title":"On the Most Likely Voronoi Diagram and Nearest Neighbor Searching","authors":"S. Suri, Kevin Verbeek","doi":"10.1142/S0218195916600025","DOIUrl":"https://doi.org/10.1142/S0218195916600025","url":null,"abstract":"We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple ((s_i, pi _i)) consisting of a point (s_i) in a (d)-dimensional space and a probability (pi _i) determining its existence. The problem is interesting and non-trivial even in (1)-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity (Omega (n^2)). We then show that under more natural and less adversarial conditions, the size of the (1)-dimensional LVD is significantly smaller: (1) (Theta (k n)) if the input has only (k) distinct probability values, (2) (O(n log n)) on average, and (3) (O(n sqrt{n})) under smoothed analysis. We also present an alternative approach to the most likely nearest neighbor (LNN) search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models, as well as worst-case with a bounded number of distinct probabilities. Using the Pareto-set approach, we can also reduce the multi-dimensional LNN search to a sequence of nearest neighbor and spherical range queries.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"302 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115439808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-12-01DOI: 10.1142/S0218195914600127
K. Buchin, Dirk H. P. Gerrits
An important but strongly NP-hard problem in automated cartography is how to best place textual labels for point features on a static map. We examine the complexity of various generalizations of this problem for dynamic and/or interactive maps. Specifically, we show that it is strongly PSPACE/complete to decide whether there is a smooth dynamic labeling (function from time to static labelings) when the points move, when points are added and removed, or when the user pans, rotates, and/or zooms their view of the points.
{"title":"Dynamic Point Labeling is Strongly PSPACE-Complete","authors":"K. Buchin, Dirk H. P. Gerrits","doi":"10.1142/S0218195914600127","DOIUrl":"https://doi.org/10.1142/S0218195914600127","url":null,"abstract":"An important but strongly NP-hard problem in automated cartography is how to best place textual labels for point features on a static map. We examine the complexity of various generalizations of this problem for dynamic and/or interactive maps. Specifically, we show that it is strongly PSPACE/complete to decide whether there is a smooth dynamic labeling (function from time to static labelings) when the points move, when points are added and removed, or when the user pans, rotates, and/or zooms their view of the points.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124975914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-09-16DOI: 10.1142/S0218195914600097
O. Aichholzer, T. Hackl, Matias Korman, Alexander Pilz, B. Vogtenhuber
Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon (mathcal{P}) by a polygon (mathcal{P}') such that (1) (mathcal{P}') contains (mathcal{P}), (2) (mathcal{P}') has its reflex vertices at the same positions as (mathcal{P}), and (3) the number of vertices of (mathcal{P}') is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both (mathcal{P}) and (mathcal{P}'), our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of (mathcal{P}).
{"title":"Geodesic-Preserving Polygon Simplification","authors":"O. Aichholzer, T. Hackl, Matias Korman, Alexander Pilz, B. Vogtenhuber","doi":"10.1142/S0218195914600097","DOIUrl":"https://doi.org/10.1142/S0218195914600097","url":null,"abstract":"Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon (mathcal{P}) by a polygon (mathcal{P}') such that (1) (mathcal{P}') contains (mathcal{P}), (2) (mathcal{P}') has its reflex vertices at the same positions as (mathcal{P}), and (3) the number of vertices of (mathcal{P}') is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both (mathcal{P}) and (mathcal{P}'), our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of (mathcal{P}).","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127498266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-03-07DOI: 10.1142/S0218195914600103
Patrizio Angelini, Thomas Bläsius, Ignaz Rutter
We introduce and study the problem Mutual Planar Duality, which asks for planar graphs G 1 and G 2 whether G 1 can be embedded such that its dual is isomorphic to G 2. We show NP-completeness for general graphs and give a linear-time algorithm for biconnected graphs.
{"title":"Testing Mutual duality of Planar graphs","authors":"Patrizio Angelini, Thomas Bläsius, Ignaz Rutter","doi":"10.1142/S0218195914600103","DOIUrl":"https://doi.org/10.1142/S0218195914600103","url":null,"abstract":"We introduce and study the problem Mutual Planar Duality, which asks for planar graphs G 1 and G 2 whether G 1 can be embedded such that its dual is isomorphic to G 2. We show NP-completeness for general graphs and give a linear-time algorithm for biconnected graphs.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117043808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-12-19DOI: 10.1142/S0218195913600121
Evanthia Papadopoulou, S. Dey
The farthest line-segment Voronoi diagram shows properties surprisingly different from the farthest point Voronoi diagram: Voronoi regions may be disconnected and they are not characterized by convex-hull properties. In this paper we introduce the farthest line-segment hull and its Gaussian map, a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram similarly to the way an ordinary convex hull characterizes the regions of the farthest-point Voronoi diagram. We also derive tighter bounds on the (linear) size of the farthest line-segment Voronoi diagram. With the purpose of unifying construction algorithms for farthest-point and farthest line-segment Voronoi diagrams, we adapt standard techniques for the construction of a convex hull to compute the farthest line-segment hull in O(n logn) or output-sensitive O(n logh) time, where n is the number of segments and h is the size of the hull (number of Voronoi faces). As a result, the farthest line-segment Voronoi diagram can be constructed in output sensitive O(n logh) time.
{"title":"On the Farthest Line-Segment Voronoi Diagram","authors":"Evanthia Papadopoulou, S. Dey","doi":"10.1142/S0218195913600121","DOIUrl":"https://doi.org/10.1142/S0218195913600121","url":null,"abstract":"The farthest line-segment Voronoi diagram shows properties surprisingly different from the farthest point Voronoi diagram: Voronoi regions may be disconnected and they are not characterized by convex-hull properties. In this paper we introduce the farthest line-segment hull and its Gaussian map, a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram similarly to the way an ordinary convex hull characterizes the regions of the farthest-point Voronoi diagram. We also derive tighter bounds on the (linear) size of the farthest line-segment Voronoi diagram. With the purpose of unifying construction algorithms for farthest-point and farthest line-segment Voronoi diagrams, we adapt standard techniques for the construction of a convex hull to compute the farthest line-segment hull in O(n logn) or output-sensitive O(n logh) time, where n is the number of segments and h is the size of the hull (number of Voronoi faces). As a result, the farthest line-segment Voronoi diagram can be constructed in output sensitive O(n logh) time.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130437402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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