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Enumeration and visibility problems in integer lattices (extended abstract) 整数格的枚举和可见性问题(扩展抽象)
SCG '90 Pub Date : 1990-05-01 DOI: 10.1145/98524.98582
E. Kranakis, M. Pocchiola
{"title":"Enumeration and visibility problems in integer lattices (extended abstract)","authors":"E. Kranakis, M. Pocchiola","doi":"10.1145/98524.98582","DOIUrl":"https://doi.org/10.1145/98524.98582","url":null,"abstract":"We study enumeration and visibility problems in the <italic>d</italic>-dimensional integer lattice <italic>L</italic><supscrpt>d</supscrpt><subscrpt>n</subscrpt> of <italic>d</italic>-tuples of integers ≤ <italic>n</italic>. In the first part of the paper we give several useful enumeration principles and use them to study the asymptotic behavior of the number of straight lines traversing a certain fixed number of lattice vertices of <italic>L</italic><supscrpt>d</supscrpt><subscrpt>n</subscrpt>, the line incidence problem and the edge visibility region. In the second part of the paper we consider an art gallery problem for point obstacles. More specifically we study the camera placement problem for the infinite lattice <italic>L</italic><supscrpt>d</supscrpt>. A lattice point is visible from a camera <italic>C</italic> (positioned at a vertex of <italic>L</italic><supscrpt>d</supscrpt>) if the line segment joining <italic>A</italic> and <italic>C</italic> crosses no other lattice vertex. For any given number <italic>s</italic> ≤ 3<supscrpt>d</supscrpt> of cameras we determine the position they must occupy in the lattice <italic>L</italic><supscrpt>d</supscrpt> in order to maximize their visibility.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"95 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133400321","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: 4
Maximin location of convex objects in a polygon and related dynamic Voronoi diagrams 多边形和相关动态Voronoi图中凸对象的最大位置
SCG '90 Pub Date : 1990-05-01 DOI: 10.1145/98524.98575
Hiromi Aonuma, H. Imai, K. Imai, T. Tokuyama
{"title":"Maximin location of convex objects in a polygon and related dynamic Voronoi diagrams","authors":"Hiromi Aonuma, H. Imai, K. Imai, T. Tokuyama","doi":"10.1145/98524.98575","DOIUrl":"https://doi.org/10.1145/98524.98575","url":null,"abstract":"This paper considers the maximin placement of a convex polygon <italic>P</italic> inside a polygon <italic>Q</italic>, and introduce several new static and dynamic Voronoi diagrams to solve the problem. It is shown that <italic>P</italic> can be placed inside <italic>Q</italic>, using translation and rotation, so that the minimum Euclidean distance between any point on <italic>P</italic> and any point on <italic>Q</italic> is maximized in <italic>&Ogr;</italic>(<italic>m</italic><supscrpt>4</supscrpt><italic>n λ</italic><subscrpt>16</subscrpt>(<italic>mn</italic>) log <italic>mn</italic>) time, where <italic>m</italic> and <italic>n</italic> are the numbers of edges of <italic>P</italic> and <italic>Q</italic>, respectively, and <italic>λ</italic><subscrpt>16</subscrpt>(<italic>N</italic>) is the maximum length of Davenport-Schinzel sequences on <italic>N</italic> alphabets of order 16. If only translation is allowed, the problem can be solved in <italic>&Ogr;</italic>(<italic>mn</italic> log <italic>mn</italic>) time. The problem of placing multiple translates of <italic>P</italic> inside <italic>Q</italic> in a maximum manner is also considered, and in connection with this problem the dynamic Voronoi diagram of <italic>&kgr;</italic> rigidly moving sets of <italic>n</italic> points is investigated. The combinatorial complexity of this canonical dynamic diagram for <italic>&kgr;n</italic> points is shown to be <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt>) and <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>3</supscrpt><italic>&kgr;</italic><supscrpt>4</supscrpt> log<supscrpt>*</supscrpt> <italic>&kgr;</italic>) for <italic>&kgr;</italic> = 2, 3 and <italic>&kgr;</italic> ≥ 4, respectively. Several related problems are also treated in a unified way.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116532829","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: 32
How to net a lot with little: small ε-nets for disks and halfspaces 如何用小的ε-网(用于磁盘和半空间)网到很多东西
SCG '90 Pub Date : 1990-05-01 DOI: 10.1145/98524.98530
J. Matoušek, R. Seidel, E. Welzl
{"title":"How to net a lot with little: small ε-nets for disks and halfspaces","authors":"J. Matoušek, R. Seidel, E. Welzl","doi":"10.1145/98524.98530","DOIUrl":"https://doi.org/10.1145/98524.98530","url":null,"abstract":"It is known that in general range spaces of VC-dimension <italic>d</italic> > 1 require <italic>ε</italic>-nets to be of size at least &OHgr;(<italic>d</italic>/<italic>ε</italic> log 1/<italic>ε</italic>). We investigate the question whether this general lower bound is valid for the special range spaces that typically arise in computational geometry. We show that disks and pseudo-disks in the plane as well as halfspaces in R<supscrpt>3</supscrpt> allow <italic>ε</italic>-nets of size only <italic>&Ogr;</italic>(1/<italic>ε</italic>), which is best possible up to a multiplicative constant. The analogous questions for higher-dimensional spaces remain open.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124488438","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: 127
Tiling the plane with one tile 用一块瓷砖铺平面
SCG '90 Pub Date : 1990-05-01 DOI: 10.1145/98524.98553
D. Beauquier, M. Nivat
{"title":"Tiling the plane with one tile","authors":"D. Beauquier, M. Nivat","doi":"10.1145/98524.98553","DOIUrl":"https://doi.org/10.1145/98524.98553","url":null,"abstract":"","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125530201","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: 51
Linear programming and convex hulls made easy 线性规划和凸包变得很容易
SCG '90 Pub Date : 1990-05-01 DOI: 10.1145/98524.98570
R. Seidel
{"title":"Linear programming and convex hulls made easy","authors":"R. Seidel","doi":"10.1145/98524.98570","DOIUrl":"https://doi.org/10.1145/98524.98570","url":null,"abstract":"We present two randomized algorithms. One solves linear programs involving <italic>m</italic> constraints in <italic>d</italic> variables in expected time <italic>&Ogr;</italic>(<italic>m</italic>). The other constructs convex hulls of <italic>n</italic> points in R<italic><supscrpt>d</supscrpt>, d</italic> > 3, in expected time <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>⌈<italic>d</italic>/2⌉</supscrpt>). In both bounds <italic>d</italic> is considered to be a constant. In the linear programming algorithm the dependence of the time bound on <italic>d</italic> is of the form <italic>d</italic>!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115003527","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: 206
Polygon triangulation in O(n log log n) time with simple data-structures 用简单的数据结构在O(n log log n)时间内进行多边形三角剖分
SCG '90 Pub Date : 1990-05-01 DOI: 10.1145/98524.98533
D. Kirkpatrick, M. Klawe, R. Tarjan
{"title":"Polygon triangulation in O(n log log n) time with simple data-structures","authors":"D. Kirkpatrick, M. Klawe, R. Tarjan","doi":"10.1145/98524.98533","DOIUrl":"https://doi.org/10.1145/98524.98533","url":null,"abstract":"We give a new <italic>&Ogr;</italic>(<italic>n</italic> log log <italic>n</italic>)-time deterministic linear-time algorithm for triangulating simple <italic>n</italic>-vertex polygons, which avoids the use of complicated data-structures. In addition, for polygons whose vertices have integer coordinates of polynomially bounded size, the algorithm can be modified to run in <italic>&Ogr;</italic>(<italic>n</italic> log<supscrpt>*</supscrpt> <italic>n</italic>) time. The major new techniques employed are the efficient location of horizontal visibility edges which partition the interior of the polygon into regions of approximately equal size, and a linear-time algorithm for obtaining the horizontal visibility partition of a subchain of a polygonal chain, from the horizontal visibility partition of the entire chain. This latter technique has other interesting applications, including a linear-time algorithm to convert a Steiner triangulation of a polygon into a true triangulation.\u0000This research was partially supported by DIMACS and the following grants: NSERC 583584, NSERC 580485, NSF-STC88-09648, ONR-N00014-87-0467.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123764854","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: 49
Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open chain manipulators 可证明的笛卡尔机器人和开链机械臂最优动力学规划的良好逼近算法
SCG '90 Pub Date : 1990-05-01 DOI: 10.1145/98524.98594
B. Donald, P. Xavier
{"title":"Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open chain manipulators","authors":"B. Donald, P. Xavier","doi":"10.1145/98524.98594","DOIUrl":"https://doi.org/10.1145/98524.98594","url":null,"abstract":"We consider the following problem: given a robot system, find a minimal-time trajectory from a start state to a goal state, while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [CDRX] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds. This algorithm differed from previous work in three ways: it is possible (1) to bound the goodness of the approximation by an error term ε (2) to polynomially bound the running time (complexity) of our algorithm; and (3) to express the complexity as a polynomial function of the error term.\u0000We extend these results to <italic>d</italic>-link, revolute-joint 3D robots will full rigid body dynamics. Specifically, we first prove a generalized trajectory-tracking lemma for robots with coupled dynamics bounds. Using this result we describe polynomial-time approximation algorithms for Cartesian robots obeying <italic>L</italic><subscrpt>2</subscrpt> dynamics bounds and open kinematic chain manipulators with revolute and prismatic joints; the latter class includes most industrial manipulators. We obtain a general <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt> (log <italic>n</italic>)(1/ε<supscrpt>6<italic>d</italic>-1</supscrpt>) algorithm, where <italic>n</italic> is the geometric complexity. The algorithm is simple, but the new game-theoretic proof techniques we introduce are subtle, and employ tools from disparate parts of computational geometry, robotics, and dynamical systems.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130950182","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: 39
Structured visibility profiles with applications to problems in simple polygons (extended abstract) 简单多边形问题应用的结构化可见性概要文件(扩展摘要)
SCG '90 Pub Date : 1989-12-01 DOI: 10.1145/98524.98536
P. Heffernan, Joseph S. B. Mitchell
{"title":"Structured visibility profiles with applications to problems in simple polygons (extended abstract)","authors":"P. Heffernan, Joseph S. B. Mitchell","doi":"10.1145/98524.98536","DOIUrl":"https://doi.org/10.1145/98524.98536","url":null,"abstract":"A number of problems in computational geometry involving simple polygons can be solved in linear time once the polygon has been triangulated. Since the worst-case time bound for triangulating a general simple polygon is currently super-linear, these algorithms are not linear time in the worst case. In this paper we define the structured visibility profile of a polygonal path and show how to compute it in linear time. We apply this result to solve many problems in linear time that previously required triangulation. Our list of problems includes: translation separability of two simple polygons, computing the weak visibility region for a segment within a simple polygon, finding shortest monotone paths in a simple polygon, ray shooting from an edge, and the convex rope problem.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128777805","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
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