We present the first algorithm to approximate the medial axis $M_{gamma}$ of a smooth, closed curve $gamma subset mathbb{R}^3$ in near-linear time. Our algorithm works on a sufficiently dense eps-sample and comes with a convergence guarantee for the non-discrete, but continuous approximation object. As our approach also works correctly for a set of curves, we discuss the following application of the medial axis: The medial axis of two curves $gamma_1$ and $gamma_2$ can be applied to compute piecewise-linear simplifications of $gamma_1$ and $gamma_2$. In particular, a controllable tradeoff between the degree of simplification and the degree of falsification of the summed Fr'{e}chet distance between $gamma_1$ and $gamma_2$ is obtained. Finally, we show that for simplifying $gamma_1$ and $gamma_2$, our approximation, instead of $M_{gamma}$, can be applied while guaranteeing the same result.
{"title":"Near-linear time medial axis approximation of smooth curves in $mathbb{R}^3$","authors":"Christian Scheffer","doi":"10.20382/jocg.v7i1a17","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a17","url":null,"abstract":"We present the first algorithm to approximate the medial axis $M_{gamma}$ of a smooth, closed curve $gamma subset mathbb{R}^3$ in near-linear time. Our algorithm works on a sufficiently dense eps-sample and comes with a convergence guarantee for the non-discrete, but continuous approximation object. As our approach also works correctly for a set of curves, we discuss the following application of the medial axis: The medial axis of two curves $gamma_1$ and $gamma_2$ can be applied to compute piecewise-linear simplifications of $gamma_1$ and $gamma_2$. In particular, a controllable tradeoff between the degree of simplification and the degree of falsification of the summed Fr'{e}chet distance between $gamma_1$ and $gamma_2$ is obtained. Finally, we show that for simplifying $gamma_1$ and $gamma_2$, our approximation, instead of $M_{gamma}$, can be applied while guaranteeing the same result.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"81 4 1","pages":"360-429"},"PeriodicalIF":0.0,"publicationDate":"2016-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86532459","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}
Given $n$ intervals on a line $ell$, we consider the problem of moving these intervals on $ell$ such that no two intervals overlap and the maximum moving distance of the intervals is minimized. The difficulty for solving the problem lies in determining the order of the intervals in an optimal solution. By interesting observations, we show that it is sufficient to consider at most $n$ "candidate" lists of ordered intervals. Further, although explicitly maintaining these lists takes $Omega(n^2)$ time and space, by more observations and a pruning technique, we present an algorithm that can compute an optimal solution in $O(nlog n)$ time and $O(n)$ space. We also prove an $Omega(nlog n)$ time lower bound for solving the problem, which implies the optimality of our algorithm.
{"title":"Separating Overlapped Intervals on a Line","authors":"Shimin Li, Haitao Wang","doi":"10.20382/jocg.v10i1a11","DOIUrl":"https://doi.org/10.20382/jocg.v10i1a11","url":null,"abstract":"Given $n$ intervals on a line $ell$, we consider the problem of moving these intervals on $ell$ such that no two intervals overlap and the maximum moving distance of the intervals is minimized. The difficulty for solving the problem lies in determining the order of the intervals in an optimal solution. By interesting observations, we show that it is sufficient to consider at most $n$ \"candidate\" lists of ordered intervals. Further, although explicitly maintaining these lists takes $Omega(n^2)$ time and space, by more observations and a pruning technique, we present an algorithm that can compute an optimal solution in $O(nlog n)$ time and $O(n)$ space. We also prove an $Omega(nlog n)$ time lower bound for solving the problem, which implies the optimality of our algorithm.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"3 1","pages":"281-321"},"PeriodicalIF":0.0,"publicationDate":"2016-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77299733","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 show that the memoryless routing algorithms Greedy Walk, Compass Walk, and all variants of visibility walk based on orientation predicates are asymptotically optimal in the average case on the Delaunay triangulation. More specifically, we consider the Delaunay triangulation of an unbounded Poisson point process of unit rate and demonstrate that, for any pair of vertices $(s,t)$ inside $[0,n]^2$, the ratio between the longest and shortest visibility walks between $s$ and $t$ is bounded by a constant with probability converging to one (as long as the vertices are sufficiently far apart). As a corollary, it follows that the worst-case path has $O(sqrt{n},)$ steps in the limiting case, under the same conditions. Our results have applications in routing in mobile networks and also settle a long-standing conjecture in point location using walking algorithms. Our proofs use techniques from percolation theory and stochastic geometry.
{"title":"The worst visibility walk in a random Delaunay triangulation is $O(sqrt{n})$","authors":"O. Devillers, Ross Hemsley","doi":"10.20382/jocg.v7i1a16","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a16","url":null,"abstract":"We show that the memoryless routing algorithms Greedy Walk, Compass Walk, and all variants of visibility walk based on orientation predicates are asymptotically optimal in the average case on the Delaunay triangulation. More specifically, we consider the Delaunay triangulation of an unbounded Poisson point process of unit rate and demonstrate that, for any pair of vertices $(s,t)$ inside $[0,n]^2$, the ratio between the longest and shortest visibility walks between $s$ and $t$ is bounded by a constant with probability converging to one (as long as the vertices are sufficiently far apart). As a corollary, it follows that the worst-case path has $O(sqrt{n},)$ steps in the limiting case, under the same conditions. Our results have applications in routing in mobile networks and also settle a long-standing conjecture in point location using walking algorithms. Our proofs use techniques from percolation theory and stochastic geometry.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"30 1","pages":"332-359"},"PeriodicalIF":0.0,"publicationDate":"2016-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87961458","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 : 2016-07-21DOI: 10.4230/LIPIcs.SoCG.2017.39
H. Edelsbrunner, H. Wagner
Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.
{"title":"Topological Data Analysis with Bregman Divergences","authors":"H. Edelsbrunner, H. Wagner","doi":"10.4230/LIPIcs.SoCG.2017.39","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2017.39","url":null,"abstract":"Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. \u0000We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"84 1","pages":"67-86"},"PeriodicalIF":0.0,"publicationDate":"2016-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85929854","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}
Finding nonoverlapping balls with given centers in any metric space, maximizing the sum of radii of the balls, can be expressed as a linear program. Its dual linear program expresses the problem of finding a minimum-weight set of cycles (allowing 2-cycles) covering all vertices in a complete geometric graph. For points in a Euclidean space of any finite dimension~$d$, with any convex distance function on this space, this graph can be replaced by a sparse subgraph obeying a separator theorem. This graph structure leads to an algorithm for finding the optimum set of balls in time $O(n^{2-1/d})$, improving the $O(n^3)$ time of a naive cycle cover algorithm. As a subroutine, we provide an algorithm for weighted bipartite matching in graphs with separators, which speeds up the best previous algorithm for this problem on planar bipartite graphs from $O(n^{3/2}log n)$ to $O(n^{3/2})$ time. We also show how to constrain the balls to all have radius at least a given threshold value, and how to apply our radius-sum optimization algorithms to the problem of embedding a finite metric space into a star metric minimizing the average distance to the hub.
{"title":"Maximizing the Sum of Radii of Disjoint Balls or Disks","authors":"D. Eppstein","doi":"10.20382/jocg.v8i1a12","DOIUrl":"https://doi.org/10.20382/jocg.v8i1a12","url":null,"abstract":"Finding nonoverlapping balls with given centers in any metric space, maximizing the sum of radii of the balls, can be expressed as a linear program. Its dual linear program expresses the problem of finding a minimum-weight set of cycles (allowing 2-cycles) covering all vertices in a complete geometric graph. For points in a Euclidean space of any finite dimension~$d$, with any convex distance function on this space, this graph can be replaced by a sparse subgraph obeying a separator theorem. This graph structure leads to an algorithm for finding the optimum set of balls in time $O(n^{2-1/d})$, improving the $O(n^3)$ time of a naive cycle cover algorithm. As a subroutine, we provide an algorithm for weighted bipartite matching in graphs with separators, which speeds up the best previous algorithm for this problem on planar bipartite graphs from $O(n^{3/2}log n)$ to $O(n^{3/2})$ time. We also show how to constrain the balls to all have radius at least a given threshold value, and how to apply our radius-sum optimization algorithms to the problem of embedding a finite metric space into a star metric minimizing the average distance to the hub.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"89 1","pages":"316-339"},"PeriodicalIF":0.0,"publicationDate":"2016-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83878644","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 : 2016-06-14DOI: 10.4230/LIPIcs.SoCG.2016.33
O. Devillers, M. Karavelas, M. Teillaud
In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in epsilon for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.
{"title":"Qualitative Symbolic Perturbation","authors":"O. Devillers, M. Karavelas, M. Teillaud","doi":"10.4230/LIPIcs.SoCG.2016.33","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2016.33","url":null,"abstract":"In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in epsilon for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"&NA; 1","pages":"282-315"},"PeriodicalIF":0.0,"publicationDate":"2016-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83417812","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}
Animation usually offers the advantage of delivering better representations of dynamic concepts. Compared with the static images and text; animation can present procedural information (e.g. social activities and/or biochemical reaction) more explicitly as they show the steps in an orderly manner. Quite a few empirical studies show promising results of animation reflected on learning (e.g. Trevisan, Oki and Senger, 2009; Hays, 1996). There are, however, there are some limitations to such results. Designing and developing quality animation for teaching and learning can sometimes be challenging (Morrison, Tversky and Betrancourt, 2000). Kesner and Linzey (2005) found no improvement on students' learning in using animation in their study. Thus, researchers may encounter case factors that govern successful use of animation in teaching and learning processes.
动画通常提供了更好地表示动态概念的优势。与静态图文相比较;动画可以更明确地呈现程序信息(例如社会活动和/或生化反应)，因为它们以有序的方式显示步骤。相当多的实证研究表明，动画反映在学习上的结果令人鼓舞(如Trevisan, Oki和Senger, 2009;海斯,1996)。然而，这样的结果也有一些局限性。为教学和学习设计和开发高质量的动画有时是具有挑战性的(Morrison, Tversky和Betrancourt, 2000)。Kesner and Linzey(2005)发现在学习中使用动画对学生的学习没有改善。因此，研究人员可能会遇到在教学过程中成功使用动画的案例因素。
{"title":"BEYOND SEGMENTED INSTRUCTIONAL ANIMATION AND ITS ROLE IN ENRICHMENT OF EDUCATION AND TECHNOLOGY","authors":"N. Z. Amarin","doi":"10.5121/IJCGA.2016.6302","DOIUrl":"https://doi.org/10.5121/IJCGA.2016.6302","url":null,"abstract":"Animation usually offers the advantage of delivering better representations of dynamic concepts. Compared with the static images and text; animation can present procedural information (e.g. social activities and/or biochemical reaction) more explicitly as they show the steps in an orderly manner. Quite a few empirical studies show promising results of animation reflected on learning (e.g. Trevisan, Oki and Senger, 2009; Hays, 1996). There are, however, there are some limitations to such results. Designing and developing quality animation for teaching and learning can sometimes be challenging (Morrison, Tversky and Betrancourt, 2000). Kesner and Linzey (2005) found no improvement on students' learning in using animation in their study. Thus, researchers may encounter case factors that govern successful use of animation in teaching and learning processes.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"48 1","pages":"17-33"},"PeriodicalIF":0.0,"publicationDate":"2016-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.5121/IJCGA.2016.6302","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70617770","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}
Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic $n$–manifolds of finite volume to non-compact strictly convex projective $n$–manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projective surfaces.
{"title":"An algorithm for the Euclidean cell decomposition of a non-compact strictly convex projective surface","authors":"Stephan Tillmann, Sampson Wong","doi":"10.20382/jocg.v7i1a12","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a12","url":null,"abstract":"Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic $n$–manifolds of finite volume to non-compact strictly convex projective $n$–manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projective surfaces.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"95 1","pages":"237-255"},"PeriodicalIF":0.0,"publicationDate":"2016-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79247168","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}
Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is of dimension $d_2$, with $d_{1}
{"title":"Minimax Rates for Estimating the Dimension of a Manifold","authors":"Jisu Kim, A. Rinaldo, L. Wasserman","doi":"10.20382/jocg.v10i1a3","DOIUrl":"https://doi.org/10.20382/jocg.v10i1a3","url":null,"abstract":"Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is of dimension $d_2$, with $d_{1}<d_{2}$. With an i.i.d. sample of size $n$, we provide an upper bound on the probability of choosing the wrong dimension of $Oleft( n^{-left(d_{2}/d_{1}-1-epsilonright)n} right)$, where $epsilon$ is an arbitrarily small positive number. The proof is based on bounding the length of the traveling salesman path through the data points. We also demonstrate a lower bound of $Omega left( n^{-(2d_{2}-2d_{1}+epsilon)n} right)$, by applying Le Cam's lemma with a specific set of $d_{1}$-dimensional probability distributions. We then extend these results to get minimax rates for estimating the dimension of well-behaved manifolds. We obtain an upper bound of order $O left( n^{-(frac{1}{m-1}-epsilon)n} right)$ and a lower bound of order $Omega left( n^{-(2+epsilon)n} right)$, where $m$ is the embedding dimension.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"21 1","pages":"42-95"},"PeriodicalIF":0.0,"publicationDate":"2016-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87229780","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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