International Journal of Computational Geometry & Applications最新文献_第4页

An Improved Cost Function for Hierarchical Cluster Trees 一种改进的层次聚类树代价函数

International Journal of Computational Geometry & Applications Pub Date : 2018-12-06 DOI: 10.20382/jocg.v11i1a11

Dingkang Wang, Yusu Wang

{"title":"An Improved Cost Function for Hierarchical Cluster Trees","authors":"Dingkang Wang, Yusu Wang","doi":"10.20382/jocg.v11i1a11","DOIUrl":"https://doi.org/10.20382/jocg.v11i1a11","url":null,"abstract":"Hierarchical clustering has been a popular method in various data analysis applications. It partitions a data set into a hierarchical collection of clusters, and can provide a global view of (cluster) structure behind data across different granularity levels. A hierarchical clustering (HC) of a data set can be naturally represented by a tree, called a HC-tree, where leaves correspond to input data and subtrees rooted at internal nodes correspond to clusters. Many hierarchical clustering algorithms used in practice are developed in a procedure manner. Dasgupta proposed to study the hierarchical clustering problem from an optimization point of view, and introduced an intuitive cost function for similarity-based hierarchical clustering with nice properties as well as natural approximation algorithms. \u0000We observe that while Dasgupta's cost function is effective at differentiating a good HC-tree from a bad one for a fixed graph, the value of this cost function does not reflect how well an input similarity graph is consistent to a hierarchical structure. In this paper, we present a new cost function, which is developed based on Dasgupta's cost function, to address this issue. The optimal tree under the new cost function remains the same as the one under Dasgupta's cost function. However, the value of our cost function is more meaningful. The new way of formulating the cost function also leads to a polynomial time algorithm to compute the optimal cluster tree when the input graph has a perfect HC-structure, or an approximation algorithm when the input graph 'almost' has a perfect HC-structure. Finally, we provide further understanding of the new cost function by studying its behavior for random graphs sampled from an edge probability matrix.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"40 1","pages":"283-331"},"PeriodicalIF":0.0,"publicationDate":"2018-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89788009","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}

引用次数: 15

Polyline Simplification has Cubic Complexity 折线化简具有三次复杂度

International Journal of Computational Geometry & Applications Pub Date : 2018-10-01 DOI: 10.4230/LIPIcs.SoCG.2019.18

K. Bringmann, B. Chaudhury

{"title":"Polyline Simplification has Cubic Complexity","authors":"K. Bringmann, B. Chaudhury","doi":"10.4230/LIPIcs.SoCG.2019.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2019.18","url":null,"abstract":"In the classic polyline simplification problem we want to replace a given polygonal curve $P$, consisting of $n$ vertices, by a subsequence $P'$ of $k$ vertices from $P$ such that the polygonal curves $P$ and $P'$ are as close as possible. Closeness is usually measured using the Hausdorff or Fr'echet distance. These distance measures can be applied \"globally\", i.e., to the whole curves $P$ and $P'$, or \"locally\", i.e., to each simplified subcurve and the line segment that it was replaced with separately (and then taking the maximum). This gives rise to four problem variants: Global-Hausdorff (known to be NP-hard), Local-Hausdorff (in time $O(n^3)$), Global-Fr'echet (in time $O(k n^5)$), and Local-Fr'echet (in time $O(n^3)$). \u0000Our contribution is as follows. \u0000- Cubic time for all variants: For Global-Fr'echet we design an algorithm running in time $O(n^3)$. This shows that all three problems (Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet) can be solved in cubic time. All these algorithms work over a general metric space such as $(mathbb{R}^d,L_p)$, but the hidden constant depends on $p$ and (linearly) on $d$. \u0000- Cubic conditional lower bound: We provide evidence that in high dimensions cubic time is essentially optimal for all three problems (Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet). Specifically, improving the cubic time to $O(n^{3-epsilon} textrm{poly}(d))$ for polyline simplification over $(mathbb{R}^d,L_p)$ for $p = 1$ would violate plausible conjectures. We obtain similar results for all $p in [1,infty), p ne 2$. \u0000In total, in high dimensions and over general $L_p$-norms we resolve the complexity of polyline simplification with respect to Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet, by providing new algorithms and conditional lower bounds.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"62 1","pages":"94-130"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87700816","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}

引用次数: 17

On the tree-width of knot diagrams 在结图的树宽度上

International Journal of Computational Geometry & Applications Pub Date : 2018-09-06 DOI: 10.20382/jocg.v10i1a6

S. Schleimer, A. D. Mesmay, J. Purcell, E. Sedgwick

引用次数: 11

A stability theorem on cube tessellations 关于立方体镶嵌的一个稳定性定理

International Journal of Computational Geometry & Applications Pub Date : 2018-07-13 DOI: 10.20382/jocg.v9i1a13

J. Pach, P. Frankl

引用次数: 0

Orthogonal Point Location and Rectangle Stabbing Queries in 3-d 三维正交点定位和矩形刺入查询

International Journal of Computational Geometry & Applications Pub Date : 2018-05-19 DOI: 10.4230/LIPIcs.ICALP.2018.31

Timothy M. Chan, Yakov Nekrich, S. Rahul, Konstantinos Tsakalidis

{"title":"Orthogonal Point Location and Rectangle Stabbing Queries in 3-d","authors":"Timothy M. Chan, Yakov Nekrich, S. Rahul, Konstantinos Tsakalidis","doi":"10.4230/LIPIcs.ICALP.2018.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.ICALP.2018.31","url":null,"abstract":"In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing. \u0000-We give the first linear-space data structure that supports 3-d point location queries on $n$ disjoint axis-aligned boxes with optimal $Oleft( log nright)$ query time in the (arithmetic) pointer machine model. This improves the previous $Oleft( log^{3/2} n right)$ bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time. \u0000-We give the first linear-space data structure that supports 3-d $4$-sided and $5$-sided rectangle stabbing queries in optimal $O(log_wn+k)$ time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-$k$ rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing. \u0000For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79509961","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}

引用次数: 15

An Efficient Line Clipping Algorithm for Circular Windows Using Vector Calculus and Parallelization 一种基于向量微积分和并行化的圆形窗口线裁剪算法

International Journal of Computational Geometry & Applications Pub Date : 2018-04-30 DOI: 10.5121/IJCGA.2018.8201

P. Kumar, Fenil Patel, R. Kanna

引用次数: 4

Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms 稳定匹配Voronoi图:组合复杂性和算法

International Journal of Computational Geometry & Applications Pub Date : 2018-04-25 DOI: 10.4230/LIPIcs.ICALP.2018.89

G. Barequet, D. Eppstein, M. Goodrich, Nil Mamano

{"title":"Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms","authors":"G. Barequet, D. Eppstein, M. Goodrich, Nil Mamano","doi":"10.4230/LIPIcs.ICALP.2018.89","DOIUrl":"https://doi.org/10.4230/LIPIcs.ICALP.2018.89","url":null,"abstract":"We study algorithms and combinatorial complexity bounds for emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $mathbb{R}^2$ and the sites in $S$ such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota or \"appetite\" indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the well-known post office problem with the added (realistic) constraint that each post office has a limit on the size of its jurisdiction. Previous work on the stable-matching Voronoi diagram provided existence and uniqueness proofs, but did not analyze its combinatorial or algorithmic complexity. In this paper, we show that a stable-matching Voronoi diagram of $n$ point sites has $O(n^{2+varepsilon})$ faces and edges, for any $varepsilon>0$, and show that this bound is almost tight by giving a family of diagrams with $Theta(n^2)$ faces and edges. We also provide a discrete algorithm for constructing it in $O(n^3log n+n^2f(n))$ time in the real-RAM model of computation, where $f(n)$ is the runtime of a geometric primitive (which we define) that can be approximated numerically, but cannot, in general, be performed exactly in an algebraic model of computation. We show, however, how to compute the geometric primitive exactly for polygonal convex distance functions.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"33 1","pages":"26-59"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89825362","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}

引用次数: 1

Extending Drawings of Graphs to Arrangements of Pseudolines 将图的绘制扩展到伪线的排列

International Journal of Computational Geometry & Applications Pub Date : 2018-04-25 DOI: 10.4230/LIPICS.SOCG.2020.9

Alan Arroyo, Julien Bensmail, R. Richter

引用次数: 8

Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension 任意大维不可分解持久模块的实现

International Journal of Computational Geometry & Applications Pub Date : 2018-03-15 DOI: 10.4230/LIPIcs.SoCG.2018.15

M. Buchet, Emerson G. Escolar

引用次数: 23

Routing on the Visibility Graph 可见性图上的路由

International Journal of Computational Geometry & Applications Pub Date : 2018-03-08 DOI: 10.20382/jocg.v9i1a15

P. Bose, Matias Korman, André van Renssen, S. Verdonschot

引用次数: 3