As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for the development of data analysis methods applicable to simplicial complexes, including techniques based on computational topology, higher-order random processes, generalized Cheeger inequalities, isoperimetric inequalities, and spectral methods. In particular, spectral learning methods (e.g. label propagation and clustering) that directly operate on simplicial complexes represent a new direction for analyzing such complex datasets. To apply spectral learning methods to massive datasets modeled as simplicial complexes, we develop a method for sparsifying simplicial complexes that preserves the spectrum of the associated Laplacian matrices. We show that the theory of Spielman and Srivastava for the sparsification of graphs extends to simplicial complexes via the up Laplacian. In particular, we introduce a generalized effective resistance for simplices, provide an algorithm for sparsifying simplicial complexes at a fixed dimension, and give a specific version of the generalized Cheeger inequality for weighted simplicial complexes. Finally, we introduce higher-order generalizations of spectral clustering and label propagation for simplicial complexes and demonstrate via experiments the utility of the proposed spectral sparsification method for these applications.
{"title":"Spectral sparsification of simplicial complexes for clustering and label propagation","authors":"B. Osting, Sourabh Palande, Bei Wang","doi":"10.20382/jocg.v11i1a8","DOIUrl":"https://doi.org/10.20382/jocg.v11i1a8","url":null,"abstract":"As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for the development of data analysis methods applicable to simplicial complexes, including techniques based on computational topology, higher-order random processes, generalized Cheeger inequalities, isoperimetric inequalities, and spectral methods. In particular, spectral learning methods (e.g. label propagation and clustering) that directly operate on simplicial complexes represent a new direction for analyzing such complex datasets. \u0000To apply spectral learning methods to massive datasets modeled as simplicial complexes, we develop a method for sparsifying simplicial complexes that preserves the spectrum of the associated Laplacian matrices. We show that the theory of Spielman and Srivastava for the sparsification of graphs extends to simplicial complexes via the up Laplacian. In particular, we introduce a generalized effective resistance for simplices, provide an algorithm for sparsifying simplicial complexes at a fixed dimension, and give a specific version of the generalized Cheeger inequality for weighted simplicial complexes. Finally, we introduce higher-order generalizations of spectral clustering and label propagation for simplicial complexes and demonstrate via experiments the utility of the proposed spectral sparsification method for these applications.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"25 1","pages":"176-211"},"PeriodicalIF":0.0,"publicationDate":"2017-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77829882","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}
Bahareh Banyassady, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, Yannik Stein
Let $P$ be a planar set of $n$ sites in general position. For $kin{1,dots,n-1}$, the Voronoi diagram of order $k$ for $P$ is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest $k$ neighbors in $P$. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of $k=1$ and $k=n-1$, respectively. For any given $Kin{1,dots,n-1}$, the family of all higher-order Voronoi diagrams of order $k=1,dots,K$ for $P$ can be computed in total time $O(nK^2+ nlog n)$ using $O(K^2(n-K))$ space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for $P$ can be computed in $O(nlog n)$ time using $O(n)$ space [Preparata, Shamos, Springer'85]. For $sin{1,dots,n}$, 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 or modified afterwards. We describe a deterministic $s$-workspace algorithm for computing NVD and FVD for $P$ that runs in $O((n^2/s)log s)$ time. Moreover, we generalize our $s$-workspace algorithm so that for any given $Kin O(sqrt{s})$, we compute the family of all higher-order Voronoi diagrams of order $k=1,dots,K$ for $P$ in total expected time $O (frac{n^2 K^5}{s}(log s+K2^{O(log^* K)}))$ or in total deterministic time $O(frac{n^2 K^5}{s}(log s+Klog K))$. Previously, for Voronoi diagrams, the only known $s$-workspace algorithm runs in expected time $Obigl((n^2/s)log s+nlog slog^* s)$ [Korman et al., WADS'15] and only works for NVD (i.e., $k=1$). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.
{"title":"Improved Time-Space Trade-offs for Computing Voronoi Diagrams","authors":"Bahareh Banyassady, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, Yannik Stein","doi":"10.20382/jocg.v9i1a6","DOIUrl":"https://doi.org/10.20382/jocg.v9i1a6","url":null,"abstract":"Let $P$ be a planar set of $n$ sites in general position. For $kin{1,dots,n-1}$, the Voronoi diagram of order $k$ for $P$ is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest $k$ neighbors in $P$. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of $k=1$ and $k=n-1$, respectively. For any given $Kin{1,dots,n-1}$, the family of all higher-order Voronoi diagrams of order $k=1,dots,K$ for $P$ can be computed in total time $O(nK^2+ nlog n)$ using $O(K^2(n-K))$ space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for $P$ can be computed in $O(nlog n)$ time using $O(n)$ space [Preparata, Shamos, Springer'85]. \u0000For $sin{1,dots,n}$, 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 or modified afterwards. \u0000We describe a deterministic $s$-workspace algorithm for computing NVD and FVD for $P$ that runs in $O((n^2/s)log s)$ time. Moreover, we generalize our $s$-workspace algorithm so that for any given $Kin O(sqrt{s})$, we compute the family of all higher-order Voronoi diagrams of order $k=1,dots,K$ for $P$ in total expected time $O (frac{n^2 K^5}{s}(log s+K2^{O(log^* K)}))$ or in total deterministic time $O(frac{n^2 K^5}{s}(log s+Klog K))$. Previously, for Voronoi diagrams, the only known $s$-workspace algorithm runs in expected time $Obigl((n^2/s)log s+nlog slog^* s)$ [Korman et al., WADS'15] and only works for NVD (i.e., $k=1$). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"13 1","pages":"191-212"},"PeriodicalIF":0.0,"publicationDate":"2017-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87491978","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}
Venn diagrams are used to display all relations between a finite number of sets. Recent researches in this domain concern the mathematical aspects of these constructions, but are not directed towards the readability of the diagram. This article presents a new way to draw easy-to-read Venn diagrams, in which each region tends to be drawn with the same size when the number of sets grows, and tends to draw a grid. Finally, using linear algebra, we prove that this construction gives a simple Venn diagram for any number of sets.
{"title":"A new drawing for simple Venn diagrams based on algebraic construction","authors":"Arnaud Bannier, N. Bodin","doi":"10.20382/jocg.v8i1a8","DOIUrl":"https://doi.org/10.20382/jocg.v8i1a8","url":null,"abstract":"Venn diagrams are used to display all relations between a finite number of sets. Recent researches in this domain concern the mathematical aspects of these constructions, but are not directed towards the readability of the diagram. This article presents a new way to draw easy-to-read Venn diagrams, in which each region tends to be drawn with the same size when the number of sets grows, and tends to draw a grid. Finally, using linear algebra, we prove that this construction gives a simple Venn diagram for any number of sets.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"3 1","pages":"153-173"},"PeriodicalIF":0.0,"publicationDate":"2017-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74644477","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 projection median of a set $P$ of $n$ points in $mathbb{R}^d$ is a robust geometric generalization of the notion of univariate median to higher dimensions. In its original definition, the projection median is expressed as a normalized integral of the medians of the projections of $P$ onto all lines through the origin. We introduce a new definition in which the projection median is expressed as a weighted mean of $P$, and show the equivalence of the two definitions. In addition to providing a definition whose form is more consistent with those of traditional statistical estimators of location, this new definition for the projection median allows many of its geometric properties to be established more easily, as well as enabling new randomized algorithms that compute approximations of the projection median with increased accuracy and efficiency, reducing computation time from $O(n^{d+epsilon})$ to $O(mnd)$, where $m$ denotes the number of random projections sampled. Selecting $m in Theta(epsilon^{-2} d^2 log n)$ or $m in Theta(min ( d + epsilon^{-2} log n, epsilon^{-2} n))$, suffices for our algorithms to return a point within relative distance $epsilon$ of the true projection median with high probability, resulting in running times $O(d^3 n log n)$ and $O(min(d^2 n, d n^2))$ respectively, for any fixed $epsilon$.
$mathbb{R}^d$中$n$点的集合$P$的投影中值是单变量中值概念到高维的鲁棒几何推广。在其原始定义中，投影中位数表示为$P$在经过原点的所有直线上的投影中位数的归一化积分。我们引入了一个新的定义，其中投影中值表示为$P$的加权平均值，并证明了这两个定义的等价性。除了提供一个与传统位置统计估计更一致的定义形式外，这个投影中值的新定义允许更容易地建立其许多几何属性，以及启用新的随机算法，以更高的精度和效率计算投影中值的近似值，减少计算时间从$O(n^{d+epsilon})$到$O(mnd)$。其中$m$为随机抽样的投影数。选择$m in Theta(epsilon^{-2} d^2 log n)$或$m in Theta(min ( d + epsilon^{-2} log n, epsilon^{-2} n))$，足以使我们的算法以高概率返回与真实投影中值相对距离$epsilon$内的点，从而对任何固定$epsilon$分别产生运行时间$O(d^3 n log n)$和$O(min(d^2 n, d n^2))$。
{"title":"The projection median as a weighted average","authors":"Stephane Durocher, Alexandre Leblanc, M. Skala","doi":"10.20382/jocg.v8i1a5","DOIUrl":"https://doi.org/10.20382/jocg.v8i1a5","url":null,"abstract":"The projection median of a set $P$ of $n$ points in $mathbb{R}^d$ is a robust geometric generalization of the notion of univariate median to higher dimensions. In its original definition, the projection median is expressed as a normalized integral of the medians of the projections of $P$ onto all lines through the origin. We introduce a new definition in which the projection median is expressed as a weighted mean of $P$, and show the equivalence of the two definitions. In addition to providing a definition whose form is more consistent with those of traditional statistical estimators of location, this new definition for the projection median allows many of its geometric properties to be established more easily, as well as enabling new randomized algorithms that compute approximations of the projection median with increased accuracy and efficiency, reducing computation time from $O(n^{d+epsilon})$ to $O(mnd)$, where $m$ denotes the number of random projections sampled. Selecting $m in Theta(epsilon^{-2} d^2 log n)$ or $m in Theta(min ( d + epsilon^{-2} log n, epsilon^{-2} n))$, suffices for our algorithms to return a point within relative distance $epsilon$ of the true projection median with high probability, resulting in running times $O(d^3 n log n)$ and $O(min(d^2 n, d n^2))$ respectively, for any fixed $epsilon$.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"19 1","pages":"78-104"},"PeriodicalIF":0.0,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81166743","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 PLant ANImation (PLANI) framework allows a designer’s ideas and decisions about virtual plants to be guided through a structured process that results in an animation of a plant. The process proceeds by selecting relevant objects with properties from four logically grouped domains to simplify implementation. The resulting grouped objects are used as the baseline parameters for the coding process to create the virtual plant. PLANI’s construction is based on more than a thousand years of biological research, fifty years of functional-structural plant modelling, and ten years of ontology development, instantiated into an animation environment. PLANI ensures that, when designing virtual plants, a selection of objects derived from an appropriate ontology are considered, and that this selection depends on the purpose of the animation, e.g., whether it is for gaming animation, biological simulation, or film animation. The use of PLANI provides the developer with a framework that is flexible, covers a wide variety of structural, functional, and animation objects for plants, and provides classification of current computer algorithms by their applications to designing virtual plants.
{"title":"The Plani Plant Animation Framework","authors":"Tina L. M. Derzaph, Howard J. Hamilton","doi":"10.5121/IJCGA.2017.7201","DOIUrl":"https://doi.org/10.5121/IJCGA.2017.7201","url":null,"abstract":"The PLant ANImation (PLANI) framework allows a designer’s ideas and decisions about virtual plants to be guided through a structured process that results in an animation of a plant. The process proceeds by selecting relevant objects with properties from four logically grouped domains to simplify implementation. The resulting grouped objects are used as the baseline parameters for the coding process to create the virtual plant. PLANI’s construction is based on more than a thousand years of biological research, fifty years of functional-structural plant modelling, and ten years of ontology development, instantiated into an animation environment. PLANI ensures that, when designing virtual plants, a selection of objects derived from an appropriate ontology are considered, and that this selection depends on the purpose of the animation, e.g., whether it is for gaming animation, biological simulation, or film animation. The use of PLANI provides the developer with a framework that is flexible, covers a wide variety of structural, functional, and animation objects for plants, and provides classification of current computer algorithms by their applications to designing virtual plants.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"7 1","pages":"1-20"},"PeriodicalIF":0.0,"publicationDate":"2017-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.5121/IJCGA.2017.7201","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49088057","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 : 2017-01-01DOI: 10.4230/LIPIcs.SoCG.2017.18
C. Biró, Édouard Bonnet, D. Marx, Tillmann Miltzow, Paweł Rzaͅżewski
On planar graphs, many classic algorithmic problems enjoy a certain ``square root phenomenon'' and can be solved significantly faster than what is known to be possible on general graphs: for example, Independent Set , 3-Coloring , Hamiltonian Cycle , Dominating Set can be solved in time $2^{O(sqrt{n})}$ on an $n$-vertex planar graph, while no $2^{o(n)}$ algorithms exist for general graphs, assuming the Exponential Time Hypothesis (ETH). The square root in the exponent seems to be best possible for planar graphs: assuming the ETH, the running time for these problems cannot be improved to $2^{o(sqrt{n})}$. In some cases, a similar speedup can be obtained for 2-dimensional geometric problems, for example, there are $2^{O(sqrt{n}log n)}$ time algorithms for Independent Set on unit disk graphs or for TSP on 2-dimensional point sets. In this paper, we explore whether such a speedup is possible for geometric coloring problems. On the one hand, geometric objects can behave similarly to planar graphs: 3-Coloring can be solved in time $2^{O(sqrt{n})}$ on the intersection graph of $n$ disks in the plane and, assuming the ETH, there is no such algorithm with running time $2^{o(sqrt{n})}$. On the other hand, if the number $ell$ of colors is part of the input, then no such speedup is possible: Coloring the intersection graph of $n$ unit disks with $ell$ colors cannot be solved in time $2^{o(n)}$, assuming the ETH. More precisely, we exhibit a smooth increase of complexity as the number $ell$ of colors increases: If we restrict the number of colors to $ell=Theta(n^{alpha})$ for some $0le alphale 1$, then the problem of coloring the intersection graph of $n$ disks with $ell$ colors can be solved in time $exp left( O(n^{frac{1+alpha}{2}}log n) right)=exp left( O(sqrt{nell}log n) right)$, and cannot be solved in time $exp left ( o(n^{frac{1+alpha}{2}})right )=exp left( o(sqrt{nell}) right)$, even on unit disks, unless the ETH fails. More generally, we consider the problem of coloring $d$-dimensional balls in the Euclidean space and obtain analogous results showing that the problem can be solved in time $exp left( O(n^{frac{d-1+alpha}{d}}log n) right)$ $=exp left( O(n^{1-1/d}ell^{1/d}log n) right)$, and cannot be solved in time $exp left(O(n^{frac{d-1+alpha}{d}-epsilon})right)= exp left(O(n^{1-1/d-epsilon}ell^{1/d})right)$ for any $epsilon>0$, even for unit balls, unless the ETH fails. Finally, we prove that fatness is crucial to obtain subexponential algorithms for coloring. We show that existence of an algorithm coloring an intersection graph of segments using a constant number of colors in time $2^{o(n)}$ already refutes the ETH.
{"title":"Fine-Grained Complexity of Coloring Unit Disks and Balls","authors":"C. Biró, Édouard Bonnet, D. Marx, Tillmann Miltzow, Paweł Rzaͅżewski","doi":"10.4230/LIPIcs.SoCG.2017.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2017.18","url":null,"abstract":"On planar graphs, many classic algorithmic problems enjoy a certain ``square root phenomenon'' and can be solved significantly faster than what is known to be possible on general graphs: for example, Independent Set , 3-Coloring , Hamiltonian Cycle , Dominating Set can be solved in time $2^{O(sqrt{n})}$ on an $n$-vertex planar graph, while no $2^{o(n)}$ algorithms exist for general graphs, assuming the Exponential Time Hypothesis (ETH). The square root in the exponent seems to be best possible for planar graphs: assuming the ETH, the running time for these problems cannot be improved to $2^{o(sqrt{n})}$. In some cases, a similar speedup can be obtained for 2-dimensional geometric problems, for example, there are $2^{O(sqrt{n}log n)}$ time algorithms for Independent Set on unit disk graphs or for TSP on 2-dimensional point sets. In this paper, we explore whether such a speedup is possible for geometric coloring problems. On the one hand, geometric objects can behave similarly to planar graphs: 3-Coloring can be solved in time $2^{O(sqrt{n})}$ on the intersection graph of $n$ disks in the plane and, assuming the ETH, there is no such algorithm with running time $2^{o(sqrt{n})}$. On the other hand, if the number $ell$ of colors is part of the input, then no such speedup is possible: Coloring the intersection graph of $n$ unit disks with $ell$ colors cannot be solved in time $2^{o(n)}$, assuming the ETH. More precisely, we exhibit a smooth increase of complexity as the number $ell$ of colors increases: If we restrict the number of colors to $ell=Theta(n^{alpha})$ for some $0le alphale 1$, then the problem of coloring the intersection graph of $n$ disks with $ell$ colors can be solved in time $exp left( O(n^{frac{1+alpha}{2}}log n) right)=exp left( O(sqrt{nell}log n) right)$, and cannot be solved in time $exp left ( o(n^{frac{1+alpha}{2}})right )=exp left( o(sqrt{nell}) right)$, even on unit disks, unless the ETH fails. More generally, we consider the problem of coloring $d$-dimensional balls in the Euclidean space and obtain analogous results showing that the problem can be solved in time $exp left( O(n^{frac{d-1+alpha}{d}}log n) right)$ $=exp left( O(n^{1-1/d}ell^{1/d}log n) right)$, and cannot be solved in time $exp left(O(n^{frac{d-1+alpha}{d}-epsilon})right)= exp left(O(n^{1-1/d-epsilon}ell^{1/d})right)$ for any $epsilon>0$, even for unit balls, unless the ETH fails. Finally, we prove that fatness is crucial to obtain subexponential algorithms for coloring. We show that existence of an algorithm coloring an intersection graph of segments using a constant number of colors in time $2^{o(n)}$ already refutes the ETH.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"28 1","pages":"47-80"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78961457","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}
Matt Gibson, G. Kanade, Rainer Penninger, Kasturi R. Varadarajan, Ivo Vigan
Given a set of points in the plane and a set of disks (that we think of as wireless sensors) which separate the points, we consider the problem of selecting a minimum subset of the disks such that any path between any pair of points is intersected by at least one of the selected disks. We present a $(9 + epsilon)$-approximation algorithm for this problem and show that it is NP-complete even if all disks have unit radius and no disk contains any points. Using a similar hardness reduction, we further show that the Multiterminal Cut problem remains NP-complete on unit disk graphs. Lastly, we prove that removing a minimum subset of a collection of unit disks, such that the plane minus the arrangement of the remaining disks consists of a single connected region is also NP-complete.
{"title":"On isolating points using unit disks","authors":"Matt Gibson, G. Kanade, Rainer Penninger, Kasturi R. Varadarajan, Ivo Vigan","doi":"10.20382/jocg.v7i1a22","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a22","url":null,"abstract":"Given a set of points in the plane and a set of disks (that we think of as wireless sensors) which separate the points, we consider the problem of selecting a minimum subset of the disks such that any path between any pair of points is intersected by at least one of the selected disks. We present a $(9 + epsilon)$-approximation algorithm for this problem and show that it is NP-complete even if all disks have unit radius and no disk contains any points. Using a similar hardness reduction, we further show that the Multiterminal Cut problem remains NP-complete on unit disk graphs. Lastly, we prove that removing a minimum subset of a collection of unit disks, such that the plane minus the arrangement of the remaining disks consists of a single connected region is also NP-complete.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"18 11","pages":"540-557"},"PeriodicalIF":0.0,"publicationDate":"2016-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72618579","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}
Ahmad Biniaz, M. Amani, A. Maheshwari, M. Smid, P. Bose, J. Carufel
Let S be a set of n points in the plane that is in convex position. For a real number t>1, we say that a point p in S is t-good if for every point q of S, the shortest-path distance between p and q along the boundary of the convex hull of S is at most t times the Euclidean distance between p and q. We prove that any point that is part of (an approximation to) the diameter of S is 1.88-good. Using this, we show how to compute a plane 1.88-spanner of S in O(n) time, assuming that the points of S are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was 1.998 (which, in fact, holds for any point set, i.e., even if it is not in convex position).
{"title":"A plane 1.88-spanner for points in convex position","authors":"Ahmad Biniaz, M. Amani, A. Maheshwari, M. Smid, P. Bose, J. Carufel","doi":"10.20382/jocg.v7i1a21","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a21","url":null,"abstract":"Let S be a set of n points in the plane that is in convex position. For a real number t>1, we say that a point p in S is t-good if for every point q of S, the shortest-path distance between p and q along the boundary of the convex hull of S is at most t times the Euclidean distance between p and q. We prove that any point that is part of (an approximation to) the diameter of S is 1.88-good. Using this, we show how to compute a plane 1.88-spanner of S in O(n) time, assuming that the points of S are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was 1.998 (which, in fact, holds for any point set, i.e., even if it is not in convex position).","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"16 1","pages":"520-539"},"PeriodicalIF":0.0,"publicationDate":"2016-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87540077","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：