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Computing Geometric Minimum-Dilation Graphs is NP-Hard 几何最小膨胀图的计算是np困难的
International Journal of Computational Geometry and Applications Pub Date : 2006-09-18 DOI: 10.1142/S0218195910003244
P. Giannopoulos, R. Klein, Christian Knauer, Martin Kutz, D. Marx
{"title":"Computing Geometric Minimum-Dilation Graphs is NP-Hard","authors":"P. Giannopoulos, R. Klein, Christian Knauer, Martin Kutz, D. Marx","doi":"10.1142/S0218195910003244","DOIUrl":"https://doi.org/10.1142/S0218195910003244","url":null,"abstract":"Consider a geometric graph G, drawn with straight lines in the plane. For every pair a, b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual Euclidean distance in the plane. The worst-case ratio of these two values, for all pairs of vertices, is called the vertex-to-vertex dilation of G. \u0000 \u0000We prove that computing a minimum-dilation graph that connects a given n-point set in the plane, using not more than a given number m of edges, is an NP-hard problem, no matter if edge crossings are allowed or forbidden. In addition, we show that the minimum dilation tree over a given point set may in fact contain edge crossings.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130525838","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: 37
Counting d-Dimensional Polycubes and nonrectangular Planar polyominoes 计算d维多边形和非矩形平面多边形
International Journal of Computational Geometry and Applications Pub Date : 2006-08-15 DOI: 10.1142/S0218195909002927
Gadi Aleksandrowicz, G. Barequet
{"title":"Counting d-Dimensional Polycubes and nonrectangular Planar polyominoes","authors":"Gadi Aleksandrowicz, G. Barequet","doi":"10.1142/S0218195909002927","DOIUrl":"https://doi.org/10.1142/S0218195909002927","url":null,"abstract":"A planar polyomino of size n is an edge-connected set of n squares on a rectangular 2-D lattice. Similarly, a d-dimensional polycube (for d ≥2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d–1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. In this paper we describe a generalization of Redelmeier’s algorithm for counting two-dimensional rectangular polyominoes [Re81], which counts all the above types of polyominoes. For example, our program computed the number of distinct 3-D polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114756993","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: 31
On Unfolding Lattice Polygons/Trees and Diameter-4 Trees 关于展开格子多边形/树和直径-4树
International Journal of Computational Geometry and Applications Pub Date : 2006-08-15 DOI: 10.1007/11809678_21
{"title":"On Unfolding Lattice Polygons/Trees and Diameter-4 Trees","authors":"","doi":"10.1007/11809678_21","DOIUrl":"https://doi.org/10.1007/11809678_21","url":null,"abstract":"","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"35 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116643195","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: 7
Embedding Point Sets into Plane Graphs of Small Dilation 在小膨胀平面图中嵌入点集
International Journal of Computational Geometry and Applications Pub Date : 2005-12-19 DOI: 10.1142/S0218195907002318
Annette Ebbers-Baumann, Ansgar Grüne, R. Klein, Marek Karpinski, Christian Knauer, A. Lingas
{"title":"Embedding Point Sets into Plane Graphs of Small Dilation","authors":"Annette Ebbers-Baumann, Ansgar Grüne, R. Klein, Marek Karpinski, Christian Knauer, A. Lingas","doi":"10.1142/S0218195907002318","DOIUrl":"https://doi.org/10.1142/S0218195907002318","url":null,"abstract":"Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound >1. In this paper we provide the first upper and lower bounds for the embedding problem. \u0000 \u0000Each finite point set can be embedded into the vertex set of a finite triangulation of dilation ≤ 1.1247. \u0000 \u0000Each embedding of a closed convex curve has dilation ≥ 1.00157. \u0000 \u0000Let P be the plane graph that results from intersecting n infinite families of equidistant, parallel lines in general position. Then the vertex set of P has dilation $geq 2/sqrt{3} approx 1.1547$.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133430279","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: 13
Guarding Art Galleries by Guarding Witnesses 护卫证人护卫艺术画廊
International Journal of Computational Geometry and Applications Pub Date : 2004-12-20 DOI: 10.1142/S0218195906002002
Kyung-Yong Chwa, Byung-Cheol Jo, Christian Knauer, Esther Moet, R. V. Oostrum, C. Shin
{"title":"Guarding Art Galleries by Guarding Witnesses","authors":"Kyung-Yong Chwa, Byung-Cheol Jo, Christian Knauer, Esther Moet, R. V. Oostrum, C. Shin","doi":"10.1142/S0218195906002002","DOIUrl":"https://doi.org/10.1142/S0218195906002002","url":null,"abstract":"Let P be a simple polygon We define a witness setW to be a set of points such that if any (prospective) guard set G guards W, then it is guaranteed that G guards P Not all polygons admit a finite witness set If a finite minimal witness set exists, then it cannot contain any witness in the interior of P; all witnesses must lie on the boundary of P, and there can be at most one witness in the interior of every edge We give an algorithm to compute a minimum witness set for P in O(n2log n) time, if such a set exists, or to report the non-existence within the same time bounds.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116938436","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: 18
Cutting out Polygons with Lines and Rays 切割多边形与线和射线
International Journal of Computational Geometry and Applications Pub Date : 2004-12-01 DOI: 10.1142/S0218195906002014
O. Daescu, Jun Luo
{"title":"Cutting out Polygons with Lines and Rays","authors":"O. Daescu, Jun Luo","doi":"10.1142/S0218195906002014","DOIUrl":"https://doi.org/10.1142/S0218195906002014","url":null,"abstract":"We present approximation algorithms for cutting out polygons with line cuts and ray cuts Our results answer a number of open problems and are either the first solutions or significantly improve over previously known solutions For the line cutting version, we prove a key property that leads to a simple, constant factor approximation algorithm For the ray cutting version, we prove it is possible to compute in almost linear time a cutting sequence that is an O(log2n)-factor approximation No algorithms were previously known for the ray cutting version.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"124 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124652427","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: 11
Inner Rectangular Drawings of Plane Graphs 平面图形的内矩形图
International Journal of Computational Geometry and Applications Pub Date : 2004-10-14 DOI: 10.1142/S0218195906002026
Kazuyuki Miura, Hiroki Haga, Takao Nishizeki
{"title":"Inner Rectangular Drawings of Plane Graphs","authors":"Kazuyuki Miura, Hiroki Haga, Takao Nishizeki","doi":"10.1142/S0218195906002026","DOIUrl":"https://doi.org/10.1142/S0218195906002026","url":null,"abstract":"A drawing of a plane graph is called an inner rectangular drawing if every edge is drawn as a horizontal or vertical line segment so that every inner face is a rectangle In this paper we show that a plane graph G has an inner rectangular drawing D if and only if a new bipartite graph constructed from G has a perfect matching We also show that D can be found in time O(n1.5/log n) if G has n vertices and a sketch of the outer face is prescribed, that is, all the convex outer vertices and concave ones are prescribed.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127821828","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: 16
Grid Vertex-Unfolding Orthostacks 网格顶点展开正交
International Journal of Computational Geometry and Applications Pub Date : 2004-10-08 DOI: 10.1142/S0218195910003281
E. Demaine, J. Iacono, S. Langerman
{"title":"Grid Vertex-Unfolding Orthostacks","authors":"E. Demaine, J. Iacono, S. Langerman","doi":"10.1142/S0218195910003281","DOIUrl":"https://doi.org/10.1142/S0218195910003281","url":null,"abstract":"An algorithm was presented in [BDD+98] for unfolding orthostacks into one piece without overlap by using arbitrary cuts along the surface. It was conjectured that orthostacks could be unfolded using cuts that lie in a plane orthogonal to a coordinate axis and containing a vertex of the orthostack. We prove the existence of a vertex-unfolding using only such cuts.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124441323","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: 11
Maximizing the Area of Overlap of Two Unions of Disks under Rigid Motion 在刚性运动条件下，最大化两盘并集的重叠面积
International Journal of Computational Geometry and Applications Pub Date : 2004-07-08 DOI: 10.1142/S0218195909003118
Sergio Cabello, M. D. Berg, P. Giannopoulos, Christian Knauer, R. V. Oostrum, R. Veltkamp
{"title":"Maximizing the Area of Overlap of Two Unions of Disks under Rigid Motion","authors":"Sergio Cabello, M. D. Berg, P. Giannopoulos, Christian Knauer, R. V. Oostrum, R. Veltkamp","doi":"10.1142/S0218195909003118","DOIUrl":"https://doi.org/10.1142/S0218195909003118","url":null,"abstract":"Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a trans- lation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 − � )- approximation algorithms for translations and for rigid motions, which run in O((nm/� 2 ) log(m/� )) and O((n 2 m 2 /� 3 ) log m)) time, respectively. For rigid motions, we can also compute a (1−� )-approximation in O((m 2 n 4/3 ∆ 1/3 /� 3 ) log n log m) time, where ∆ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1−� )-approximation al- gorithm for rigid motions that runs in O((m 2 /� 4 ) log(m/� ) log 2 m) time. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126673604","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: 11
Proximity Structures for Geometric Graphs 几何图的邻近结构
International Journal of Computational Geometry and Applications Pub Date : 2003-07-30 DOI: 10.1142/S0218195910003360
S. Kapoor, Xiangyang Li
{"title":"Proximity Structures for Geometric Graphs","authors":"S. Kapoor, Xiangyang Li","doi":"10.1142/S0218195910003360","DOIUrl":"https://doi.org/10.1142/S0218195910003360","url":null,"abstract":"In this paper we study proximity structures like Delauney triangulations based on geometric graphs, i.e. graphs which are subgraphs of the complete geometric graph. Given an arbitrary geometric graph G, we define several restricted Voronoi diagrams, restricted Delaunay triangulations, relative neighborhood graphs, Gabriel graphs and then study their complexities when G is a general geometric graph or G is some special graph derived from the application area of wireless networks. Besides being of fundamental interest these structures have applications in topology control for wireless networks.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129108433","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: 24
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