Let S be a set of n points in R d . A Steiner convex partition is a tiling of conv( S ) with empty convex bodies. For every integer d , we show that S admits a Steiner convex partition with at most ⌈( n -1)/ d ⌉ tiles. This bound is the best possible for points in general position in the plane, and it is the best possible apart from constant factors in every fixed dimension d ≥3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/ n ). Here we give a (1-epsilon)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d -dimensional unit box [0,1] d .
{"title":"Minimum Convex Partitions and Maximum Empty Polytopes","authors":"A. Dumitrescu, Sariel Har-Peled, Csaba D. Tóth","doi":"10.20382/jocg.v5i1a5","DOIUrl":"https://doi.org/10.20382/jocg.v5i1a5","url":null,"abstract":"Let S be a set of n points in R d . A Steiner convex partition is a tiling of conv( S ) with empty convex bodies. For every integer d , we show that S admits a Steiner convex partition with at most ⌈( n -1)/ d ⌉ tiles. This bound is the best possible for points in general position in the plane, and it is the best possible apart from constant factors in every fixed dimension d ≥3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/ n ). Here we give a (1-epsilon)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d -dimensional unit box [0,1] d .","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"100 1","pages":"213-224"},"PeriodicalIF":0.3,"publicationDate":"2011-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76066576","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}
Dynamic maps that allow continuous map rotations, e.g., on mobile devices, encounter new issues unseen in static map labeling before. We study the following dynamic map labeling problem: The input is a static, labeled map, i.e., a set P of points in the plane with attached non-overlapping horizontal rectangular labels. The goal is to find a consistent labeling of P under rotation that maximizes the number of visible labels for all rotation angles such that the labels remain horizontal while the map is rotated. A labeling is consistent if a single active interval of angles is selected for each label such that labels neither intersect each other nor occlude points in P at any rotation angle. We first introduce a general model for labeling rotating maps and derive basic geometric properties of consistent solutions. We show NP-completeness of the active interval maximization problem even for unit-square labels. We then present a constant-factor approximation for this problem based on line stabbing, and refine it further into an EPTAS. Finally, we extend the EPTAS to the more general setting of rectangular labels of bounded size and aspect ratio.
{"title":"Consistent labeling of rotating maps","authors":"Andreas Gemsa, M. Nöllenburg, Ignaz Rutter","doi":"10.20382/jocg.v7i1a15","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a15","url":null,"abstract":"Dynamic maps that allow continuous map rotations, e.g., on mobile devices, encounter new issues unseen in static map labeling before. We study the following dynamic map labeling problem: The input is a static, labeled map, i.e., a set P of points in the plane with attached non-overlapping horizontal rectangular labels. The goal is to find a consistent labeling of P under rotation that maximizes the number of visible labels for all rotation angles such that the labels remain horizontal while the map is rotated. A labeling is consistent if a single active interval of angles is selected for each label such that labels neither intersect each other nor occlude points in P at any rotation angle. \u0000 \u0000We first introduce a general model for labeling rotating maps and derive basic geometric properties of consistent solutions. We show NP-completeness of the active interval maximization problem even for unit-square labels. We then present a constant-factor approximation for this problem based on line stabbing, and refine it further into an EPTAS. Finally, we extend the EPTAS to the more general setting of rectangular labels of bounded size and aspect ratio.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"13 1","pages":"451-462"},"PeriodicalIF":0.3,"publicationDate":"2011-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80579454","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}
Let S be a set of n points in ℝ d . A graph G = (S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0 < θ < π/3, a θ-angle-constrained t-spanner can be computed in O(n logn) time, where t depends only on θ.
{"title":"An optimal algorithm for computing angle-constrained spanners","authors":"Paz Carmi, M. Smid","doi":"10.20382/jocg.v3i1a10","DOIUrl":"https://doi.org/10.20382/jocg.v3i1a10","url":null,"abstract":"Let S be a set of n points in ℝ d . A graph G = (S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0 < θ < π/3, a θ-angle-constrained t-spanner can be computed in O(n logn) time, where t depends only on θ.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"47 1","pages":"316-327"},"PeriodicalIF":0.3,"publicationDate":"2010-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82229895","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 k-means algorithm is the method of choice for clustering large-scale data sets and it performs exceedingly well in practice. Most of the theoretical work is restricted to the case that squared Euclidean distances are used as similarity measure. In many applications, however, data is to be clustered with respect to other measures like, e.g., relative entropy, which is commonly used to cluster web pages. In this paper, we analyze the running-time of the k-means method for Bregman divergences, a very general class of similarity measures including squared Euclidean distances and relative entropy. We show that the exponential lower bound known for the Euclidean case carries over to almost every Bregman divergence. To narrow the gap between theory and practice, we also study k-means in the semi-random input model of smoothed analysis. For the case that n data points in ? d are perturbed by noise with standard deviation ?, we show that for almost arbitrary Bregman divergences the expected running-time is bounded by ${rm poly}(n^{sqrt k}, 1/sigma)$ and k kd ·poly(n, 1/?).
{"title":"Worst-Case and Smoothed Analysis of k-Means Clustering with Bregman Divergences","authors":"B. Manthey, Heiko Röglin","doi":"10.20382/jocg.v4i1a5","DOIUrl":"https://doi.org/10.20382/jocg.v4i1a5","url":null,"abstract":"The k-means algorithm is the method of choice for clustering large-scale data sets and it performs exceedingly well in practice. Most of the theoretical work is restricted to the case that squared Euclidean distances are used as similarity measure. In many applications, however, data is to be clustered with respect to other measures like, e.g., relative entropy, which is commonly used to cluster web pages. In this paper, we analyze the running-time of the k-means method for Bregman divergences, a very general class of similarity measures including squared Euclidean distances and relative entropy. We show that the exponential lower bound known for the Euclidean case carries over to almost every Bregman divergence. To narrow the gap between theory and practice, we also study k-means in the semi-random input model of smoothed analysis. For the case that n data points in ? d are perturbed by noise with standard deviation ?, we show that for almost arbitrary Bregman divergences the expected running-time is bounded by ${rm poly}(n^{sqrt k}, 1/sigma)$ and k kd ·poly(n, 1/?).","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"16 1","pages":"1024-1033"},"PeriodicalIF":0.3,"publicationDate":"2009-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79781341","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 : 2009-02-05DOI: 10.1007/978-3-642-00219-9_30
M. Löffler, E. Mumford
{"title":"Connected Rectilinear Graphs on Point Sets","authors":"M. Löffler, E. Mumford","doi":"10.1007/978-3-642-00219-9_30","DOIUrl":"https://doi.org/10.1007/978-3-642-00219-9_30","url":null,"abstract":"","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"369 1","pages":"313-318"},"PeriodicalIF":0.3,"publicationDate":"2009-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84920439","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}
N. Atienza, N. D. Castro, C. Cortés, Maria Angeles Garrido, C. Grima, G. Hernández, A. Márquez, A. Moreno-González, M. Nöllenburg, J. Portillo, Pedro Reyes, Jesus Valenzuela, M. Villar, A. Wolff
We study problems that arise in the context of covering certain geometric objects called seeds (e.g., points or disks) by a set of other geometric objects called cover (e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, respectively, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in three types of tasks, both in the general case and in the special case of seeds on a line: (a) deciding whether a given seed set has a connected CCG, (b) deciding whether a given graph has a realization as a CCG on a given seed set, and (c) bounding the sizes of certain classes of CCG's. Concerning (a) we give efficient algorithms for the case that seeds are points and show that the problem becomes hard if seeds and covers are disks. Concerning (b) we show that this problem is hard even for point seeds and disk covers (given a fixed correspondence between graph vertices and seeds). Concerning (c) we obtain upper and lower bounds on the number of CCG's for point seeds.
{"title":"Cover Contact Graphs","authors":"N. Atienza, N. D. Castro, C. Cortés, Maria Angeles Garrido, C. Grima, G. Hernández, A. Márquez, A. Moreno-González, M. Nöllenburg, J. Portillo, Pedro Reyes, Jesus Valenzuela, M. Villar, A. Wolff","doi":"10.20382/jocg.v3i1a6","DOIUrl":"https://doi.org/10.20382/jocg.v3i1a6","url":null,"abstract":"We study problems that arise in the context of covering certain geometric objects called seeds (e.g., points or disks) by a set of other geometric objects called cover (e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, respectively, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in three types of tasks, both in the general case and in the special case of seeds on a line: (a) deciding whether a given seed set has a connected CCG, (b) deciding whether a given graph has a realization as a CCG on a given seed set, and (c) bounding the sizes of certain classes of CCG's. Concerning (a) we give efficient algorithms for the case that seeds are points and show that the problem becomes hard if seeds and covers are disks. Concerning (b) we show that this problem is hard even for point seeds and disk covers (given a fixed correspondence between graph vertices and seeds). Concerning (c) we obtain upper and lower bounds on the number of CCG's for point seeds.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"4 1","pages":"171-182"},"PeriodicalIF":0.3,"publicationDate":"2007-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87097764","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：