We introduce a new metric framework which is based on an injective embedding of [0,1]2 into Rm, for m ≥ 2, and an additional scaling function for re-scaling the distances. The framework is used to construct a new type of generalized Voronoi diagrams in [0,1]2, which is possibly anisotropic. We present different possible applications of these Voronoi diagrams with several examples of generated diagrams.
{"title":"Voronoi Diagrams from (Possibly Discontinuous) Embeddings","authors":"M. Kapl, F. Aurenhammer, B. Jüttler","doi":"10.1109/ISVD.2013.13","DOIUrl":"https://doi.org/10.1109/ISVD.2013.13","url":null,"abstract":"We introduce a new metric framework which is based on an injective embedding of [0,1]2 into Rm, for m ≥ 2, and an additional scaling function for re-scaling the distances. The framework is used to construct a new type of generalized Voronoi diagrams in [0,1]2, which is possibly anisotropic. We present different possible applications of these Voronoi diagrams with several examples of generated diagrams.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"19 17-19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133193407","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}
We consider the farthest-neighbor Voronoi diagram of a set of line segments in three dimensions. To understand the structure of the diagram, we define the “farthest hull” of the segments and investigate it by its representation in a Gaussian map. We then provide lower and upper bounds on the worst-case complexities of the farthest hull and of the Voronoi diagram.
{"title":"On the Farthest-Neighbor Voronoi Diagram of Segments in Three Dimensions","authors":"G. Barequet, Evanthia Papadopoulou","doi":"10.1109/ISVD.2013.15","DOIUrl":"https://doi.org/10.1109/ISVD.2013.15","url":null,"abstract":"We consider the farthest-neighbor Voronoi diagram of a set of line segments in three dimensions. To understand the structure of the diagram, we define the “farthest hull” of the segments and investigate it by its representation in a Gaussian map. We then provide lower and upper bounds on the worst-case complexities of the farthest hull and of the Voronoi diagram.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131852942","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}
A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon or planar straight-line graph. In this paper, we ask the reverse question: Given the straight skeleton (in form of a planar straight-line graph, with some rays to infinity), can we reconstruct a planar straight-line graph for which this was the straight skeleton? We show how to reduce this problem to the problem of finding a line that intersects a set of convex polygons. We can find these convex polygons and all such lines in $O(nlog n)$ time in the Real RAM computer model, where $n$ denotes the number of edges of the input graph. We also explain how our approach can be used for recognizing Voronoi diagrams of points, thereby completing a partial solution provided by Ash and Bolker in 1985.
{"title":"Recognizing Straight Skeletons and Voronoi Diagrams and Reconstructing Their Input","authors":"T. Biedl, M. Held, S. Huber","doi":"10.1109/ISVD.2013.11","DOIUrl":"https://doi.org/10.1109/ISVD.2013.11","url":null,"abstract":"A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon or planar straight-line graph. In this paper, we ask the reverse question: Given the straight skeleton (in form of a planar straight-line graph, with some rays to infinity), can we reconstruct a planar straight-line graph for which this was the straight skeleton? We show how to reduce this problem to the problem of finding a line that intersects a set of convex polygons. We can find these convex polygons and all such lines in $O(nlog n)$ time in the Real RAM computer model, where $n$ denotes the number of edges of the input graph. We also explain how our approach can be used for recognizing Voronoi diagrams of points, thereby completing a partial solution provided by Ash and Bolker in 1985.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115834829","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}
We present an algorithm to construct order-k Voronoi diagrams with a sweepline technique. The sites can be points or line segments. The algorithm has O(nk2 log n) time complexity and O(nk) space complexity.
{"title":"A Sweepline Algorithm for Higher Order Voronoi Diagrams","authors":"Maksym Zavershynskyi, Evanthia Papadopoulou","doi":"10.1109/ISVD.2013.17","DOIUrl":"https://doi.org/10.1109/ISVD.2013.17","url":null,"abstract":"We present an algorithm to construct order-k Voronoi diagrams with a sweepline technique. The sites can be points or line segments. The algorithm has O(nk2 log n) time complexity and O(nk) space complexity.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126217698","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}
Summary form only given. Molecular structure determines molecular function and the geometry is one of the most fundamental aspects of the molecular structure regardless it is organic or inorganic. Despite of its importance, the theory for understanding the geometry of molecules has not been sufficiently developed. In this talk, we will present a unified theory of molecular geometry (MG) as a new discipline and demonstrate how the theory can be used for "accurately", "efficiently", and "conveniently" solving all molecular problems related on structure.The MG theory is based on the beta-complex which is a derived structure from the Voronoi diagram of atoms and its dual structure called the quasi-triangulation. Voronoi diagrams are everywhere in nature and are useful for understanding the spatial structure among generators. Unlike the well-known ordinary Voronoi diagram of points, the Voronoi diagram of spherical atoms has been known to be difficult to compute and to possess a few anomaly cases. Once computed, however, it nicely defines the proximity among the atoms in molecules.This talk will discuss the quasi-triangulation, the dual structure of the Voronoi diagram of atoms, and the beta-complex in the three-dimensional space. It turns out that the beta-complex, together with the Voronoi diagram and quasi-triangulation, can be used to accurately, efficiently, and conveniently solve seemingly unrelated geometry and topology problems for molecules within a single theoretical and computational framework. Among many application areas which will be explained, structural molecular biology and noble material design are the most immediate application area. In this talk, we will also demonstrate our molecular modeling and analysis software, BetaMol in 3D and BetaConcept in 2D, which are entirely based on the beta-complex and the Voronoi diagram. Programs are freely available at the Voronoi Diagram Research Center (VDRC, http://voronoi.hanyang.ac.kr/).
{"title":"Molecular Geometry: A New Challenge and Opportunity for Geometers","authors":"Deok-Soo Kim","doi":"10.1109/ISVD.2013.16","DOIUrl":"https://doi.org/10.1109/ISVD.2013.16","url":null,"abstract":"Summary form only given. Molecular structure determines molecular function and the geometry is one of the most fundamental aspects of the molecular structure regardless it is organic or inorganic. Despite of its importance, the theory for understanding the geometry of molecules has not been sufficiently developed. In this talk, we will present a unified theory of molecular geometry (MG) as a new discipline and demonstrate how the theory can be used for \"accurately\", \"efficiently\", and \"conveniently\" solving all molecular problems related on structure.The MG theory is based on the beta-complex which is a derived structure from the Voronoi diagram of atoms and its dual structure called the quasi-triangulation. Voronoi diagrams are everywhere in nature and are useful for understanding the spatial structure among generators. Unlike the well-known ordinary Voronoi diagram of points, the Voronoi diagram of spherical atoms has been known to be difficult to compute and to possess a few anomaly cases. Once computed, however, it nicely defines the proximity among the atoms in molecules.This talk will discuss the quasi-triangulation, the dual structure of the Voronoi diagram of atoms, and the beta-complex in the three-dimensional space. It turns out that the beta-complex, together with the Voronoi diagram and quasi-triangulation, can be used to accurately, efficiently, and conveniently solve seemingly unrelated geometry and topology problems for molecules within a single theoretical and computational framework. Among many application areas which will be explained, structural molecular biology and noble material design are the most immediate application area. In this talk, we will also demonstrate our molecular modeling and analysis software, BetaMol in 3D and BetaConcept in 2D, which are entirely based on the beta-complex and the Voronoi diagram. Programs are freely available at the Voronoi Diagram Research Center (VDRC, http://voronoi.hanyang.ac.kr/).","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128588781","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}
Fast and accurate calculation of the volume of additively weighted Voronoi region (Johnson-Mehl nucleus, or Voronoi S-region) is needed in various physical applications. The problem will be solved if an analytical expression could be derived for volume calculation of the pyramid which base is a piece of the hyperboloid surface.
在各种物理应用中需要快速准确地计算加加权Voronoi区域(Johnson-Mehl核或Voronoi s -区域)的体积。若能导出以双曲面为底面的锥体体积计算的解析表达式，则该问题将得到解决。
{"title":"Open Problem: A Formula for Calculation of the Voronoi S-region Volume","authors":"N. N. Medvedev, V. Voloshin","doi":"10.1109/ISVD.2013.14","DOIUrl":"https://doi.org/10.1109/ISVD.2013.14","url":null,"abstract":"Fast and accurate calculation of the volume of additively weighted Voronoi region (Johnson-Mehl nucleus, or Voronoi S-region) is needed in various physical applications. The problem will be solved if an analytical expression could be derived for volume calculation of the pyramid which base is a piece of the hyperboloid surface.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115286537","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}
We investigated the use of weighted straight skeletons in the computer-aided creation of pop-up cards. A pop-up card is a sheet of paper that can be folded to a flat plane but, when opened, produces a meaningful three-dimensional structure. Weighted straight skeletons are a special type of Voronoi diagrams, and they are closely related to the foldability of a sheet of paper along specified lines. We characterize the weights that make a paper foldable and apply them to an interactive system for the design of pop-up cards.
{"title":"Design of Pop-Up Cards Based on Weighted Straight Skeletons","authors":"K. Sugihara","doi":"10.1109/ISVD.2013.9","DOIUrl":"https://doi.org/10.1109/ISVD.2013.9","url":null,"abstract":"We investigated the use of weighted straight skeletons in the computer-aided creation of pop-up cards. A pop-up card is a sheet of paper that can be folded to a flat plane but, when opened, produces a meaningful three-dimensional structure. Weighted straight skeletons are a special type of Voronoi diagrams, and they are closely related to the foldability of a sheet of paper along specified lines. We characterize the weights that make a paper foldable and apply them to an interactive system for the design of pop-up cards.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133953352","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}
Wenjie Liu, Lin Lu, B. Lévy, Chenglei Yang, Xiangxu Meng
Centroidal Voronoi tessellation (CVT) and its extensions have a wide spectrum of applications including computational geometry, image processing, cellular biology and scientific visualization etc. In this paper, we propose the concept of the complete streamline and the CVT of streamlines, and then formulate the computation of CVT of complete streamlines as a continuous variational problem. To reduce the computing complexity, we present a simple, approximation method for solving this problem. Given a flow field and a number of complete streamlines, our method can optimize the placement of the streamlines so that the streamlines best approximate the geometric characteristics of the flow field. Experimental results show the effectiveness of our method for flow visualization, especially in terms of continuity and uniformity.
{"title":"Centroidal Voronoi Tessellation of Streamlines for Flow Visualization","authors":"Wenjie Liu, Lin Lu, B. Lévy, Chenglei Yang, Xiangxu Meng","doi":"10.1109/ISVD.2013.8","DOIUrl":"https://doi.org/10.1109/ISVD.2013.8","url":null,"abstract":"Centroidal Voronoi tessellation (CVT) and its extensions have a wide spectrum of applications including computational geometry, image processing, cellular biology and scientific visualization etc. In this paper, we propose the concept of the complete streamline and the CVT of streamlines, and then formulate the computation of CVT of complete streamlines as a continuous variational problem. To reduce the computing complexity, we present a simple, approximation method for solving this problem. Given a flow field and a number of complete streamlines, our method can optimize the placement of the streamlines so that the streamlines best approximate the geometric characteristics of the flow field. Experimental results show the effectiveness of our method for flow visualization, especially in terms of continuity and uniformity.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134515022","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}
The Delaunay triangulation of n points in the plane can be constructed in o(n log n) time when the coordinates of the points are integers from a restricted range. However, algorithms that are known to achieve such running times had not been implemented so far. We explore ways to obtain a practical algorithm for Delaunay triangulations in the plane that runs in linear time for small integers. For this, we first implement and evaluate two variants of BrioDC, an algorithm that is known to achieve this bound. We implement the first O(n)-time algorithm for constructing Delaunay triangulations and found that our implementations are practical. While we do not improve upon fast existing algorithms (with non-optimal worst-case running time) for realistic data sets, our BrioDC implementations do give us insight into the optimal time needed for point location. Secondly, we implement and evaluate variants of BRIO, an algorithm which has an O(n log n) worst-case running time on small integers but runs faster for many distributions. Our variants aim to avoid bad worst-case behavior, which is due to high point location time. Our BrioDC implementation shows that point location time can be reduced by 25% and our squarified space-filling curve orders show the first improvement by reducing this by 3%.
{"title":"Delaunay Triangulations on the Word RAM: Towards a Practical Worst-Case Optimal Algorithm","authors":"Okke Schrijvers, F.F.J.M. van Bommel, K. Buchin","doi":"10.1109/ISVD.2013.10","DOIUrl":"https://doi.org/10.1109/ISVD.2013.10","url":null,"abstract":"The Delaunay triangulation of n points in the plane can be constructed in o(n log n) time when the coordinates of the points are integers from a restricted range. However, algorithms that are known to achieve such running times had not been implemented so far. We explore ways to obtain a practical algorithm for Delaunay triangulations in the plane that runs in linear time for small integers. For this, we first implement and evaluate two variants of BrioDC, an algorithm that is known to achieve this bound. We implement the first O(n)-time algorithm for constructing Delaunay triangulations and found that our implementations are practical. While we do not improve upon fast existing algorithms (with non-optimal worst-case running time) for realistic data sets, our BrioDC implementations do give us insight into the optimal time needed for point location. Secondly, we implement and evaluate variants of BRIO, an algorithm which has an O(n log n) worst-case running time on small integers but runs faster for many distributions. Our variants aim to avoid bad worst-case behavior, which is due to high point location time. Our BrioDC implementation shows that point location time can be reduced by 25% and our squarified space-filling curve orders show the first improvement by reducing this by 3%.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123923246","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}
In this paper, we propose and solve several queries called common influence region queries. They are related to the simultaneous influence, i.e. capacity of attracting customers, of two sets of facilities of different type. In them a facility of the first type competes with the other facilities of the first type and cooperates with several facilities of the second type. The studied queries find applications, for example, in decision making support systems. We present GPU parallel algorithms, designed under CUDA architecture, for approximately solving the studied queries and provide and discuss experimental results showing the efficiency and scalability of our approach.
{"title":"Common Influence Region Queries","authors":"Marta Fort, J. A. Sellarès","doi":"10.1109/ISVD.2013.12","DOIUrl":"https://doi.org/10.1109/ISVD.2013.12","url":null,"abstract":"In this paper, we propose and solve several queries called common influence region queries. They are related to the simultaneous influence, i.e. capacity of attracting customers, of two sets of facilities of different type. In them a facility of the first type competes with the other facilities of the first type and cooperates with several facilities of the second type. The studied queries find applications, for example, in decision making support systems. We present GPU parallel algorithms, designed under CUDA architecture, for approximately solving the studied queries and provide and discuss experimental results showing the efficiency and scalability of our approach.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"16 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132275539","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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