Pub Date : 1994-07-06DOI: 10.1142/S021819599700034X
J. Czyzowicz, H. Everett, J. Robert
In this paper, we establish two combinatorial bounds related to the separation problem for sets of n pairwise disjoint translates of convex objects: 1) there exists a line which separates one translate from at least n — co√n translates, for some constant c that depends on the “shape” of the translates and 2) there is a function f such that there exists a line with orientation Θ or f(Θ) which separates one translate from at least ⌈3n⌉/4-4 translates, for any orientation Θ (f is defined only by the “shape” of the translate). We also present an O(n log (n+k)+k) time algorithm for finding a translate which can be separated from the maximum number of translates amongst sets of n pairwise disjoint translates of convex k-gons.
{"title":"Separating Translates in the Plane: Combinatorial Bounds and an Algorithm","authors":"J. Czyzowicz, H. Everett, J. Robert","doi":"10.1142/S021819599700034X","DOIUrl":"https://doi.org/10.1142/S021819599700034X","url":null,"abstract":"In this paper, we establish two combinatorial bounds related to the separation problem for sets of n pairwise disjoint translates of convex objects: 1) there exists a line which separates one translate from at least n — co√n translates, for some constant c that depends on the “shape” of the translates and 2) there is a function f such that there exists a line with orientation Θ or f(Θ) which separates one translate from at least ⌈3n⌉/4-4 translates, for any orientation Θ (f is defined only by the “shape” of the translate). We also present an O(n log (n+k)+k) time algorithm for finding a translate which can be separated from the maximum number of translates amongst sets of n pairwise disjoint translates of convex k-gons.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1994-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124933166","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 : 1994-03-01DOI: 10.1142/S0218195994000069
N. Kanamaru, Takao Nishizeki, T. Asano
This paper first presents an algorithm for enumerating all the integer-grid points in a given convex m-gon in O(K + m + log n) time where K is the number of such grid points and n is the dimension of the m-gon, i.e., the shorter length of the horizontal and vertical sides of an axis-parallel rectangle enclosing the m-gon. The paper next gives a simple algorithm which solves a two-variable integer programming problem with m constraints in O(m log m + log n) time where n is the dimension of a convex polygon corresponding to the feasible solution space. This improves the best known algorithm in complexity and simplicity. The paper finally presents algorithms for counting the number of grid points in a triangle or a simple polygon.
本文首先给出了在O(K + m + log n)时间内枚举给定凸m-gon中所有整数网格点的算法，其中K为整数网格点的个数，n为m-gon的维数，即包围m-gon的轴平行矩形的水平边和垂直边的较短长度。本文给出了在O(m log m + log n)时间内求解具有m个约束条件的两变量整数规划问题的简单算法，其中n为可行解空间对应的凸多边形的维数。这在复杂性和简单性方面改进了最著名的算法。最后给出了计算三角形或简单多边形中网格点数目的算法。
{"title":"Efficient Enumeration of Grid Points in a Polygon and its Application to Integer Programming","authors":"N. Kanamaru, Takao Nishizeki, T. Asano","doi":"10.1142/S0218195994000069","DOIUrl":"https://doi.org/10.1142/S0218195994000069","url":null,"abstract":"This paper first presents an algorithm for enumerating all the integer-grid points in a given convex m-gon in O(K + m + log n) time where K is the number of such grid points and n is the dimension of the m-gon, i.e., the shorter length of the horizontal and vertical sides of an axis-parallel rectangle enclosing the m-gon. The paper next gives a simple algorithm which solves a two-variable integer programming problem with m constraints in O(m log m + log n) time where n is the dimension of a convex polygon corresponding to the feasible solution space. This improves the best known algorithm in complexity and simplicity. The paper finally presents algorithms for counting the number of grid points in a triangle or a simple polygon.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1994-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133564152","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 : 1993-09-30DOI: 10.1142/S0218195998000163
O. Devillers, M. Golin
We discuss two variations of the two-dimensional post-office problem that arise when the post-offices are replaced by n postmen moving with constant velocities. The first variation addresses the question: given a point qo and time to who is the nearest postman to qo at time to? We present a randomized incremental data structure that answers the query in expected O(log2n) time. The second variation views a query point as a dog searching for a postman to bite and finds the postman that a dog running with speed vo could reach first. We show that if the dog is quicker than all of the postmen then the data structure developed for the first problem permits us to solve the second one in O(log2n) time as well.
{"title":"Dog Bites Postman: Point Location in the Moving Voronoi Diagram and Related Problems","authors":"O. Devillers, M. Golin","doi":"10.1142/S0218195998000163","DOIUrl":"https://doi.org/10.1142/S0218195998000163","url":null,"abstract":"We discuss two variations of the two-dimensional post-office problem that arise when the post-offices are replaced by n postmen moving with constant velocities. The first variation addresses the question: given a point qo and time to who is the nearest postman to qo at time to? We present a randomized incremental data structure that answers the query in expected O(log2n) time. The second variation views a query point as a dog searching for a postman to bite and finds the postman that a dog running with speed vo could reach first. We show that if the dog is quicker than all of the postmen then the data structure developed for the first problem permits us to solve the second one in O(log2n) time as well.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"143 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123243046","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 : 1991-12-16DOI: 10.1142/S0218195999000212
X. Tan, T. Hirata, Y. Inagaki
The problem of finding the shortest watchman route in a simple polygon P through a point s on its boundary is considered. A route is a watchman route if every point inside P can be seen from at least one point along the route. We present an incremental algorithm that constructs the shortest watchman route in O(n3) time for a simple polygon with n edges. This improves the previous O(n4) bound.
{"title":"Corrigendum to \"An Incremental Algorithm for Constructing Shortest Watchman Routes\"","authors":"X. Tan, T. Hirata, Y. Inagaki","doi":"10.1142/S0218195999000212","DOIUrl":"https://doi.org/10.1142/S0218195999000212","url":null,"abstract":"The problem of finding the shortest watchman route in a simple polygon P through a point s on its boundary is considered. A route is a watchman route if every point inside P can be seen from at least one point along the route. We present an incremental algorithm that constructs the shortest watchman route in O(n3) time for a simple polygon with n edges. This improves the previous O(n4) bound.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131701504","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 : 1991-12-16DOI: 10.1142/S0218195993000208
Taro Asano
We consider problems on intervals which can be solved by dynamic programming. Specifically, we give an efficient implementation of dynamic programming on intervals. As an application, an optimal sequential partition of a graph G=(V, E) can be obtained in O(m log n) time, where n = ¦V¦ and m = ¦E¦. We also present an O(n log n) time algorithm for finding a minimum weight dominating set of an interval graph G=(V, E), and an O(m log n) time algorithm for finding a maximum weight clique of a circular-arc graph G=(V, E), provided their intersection models of n intervals (arcs) are given.
{"title":"Dynamic Programming on Intervals","authors":"Taro Asano","doi":"10.1142/S0218195993000208","DOIUrl":"https://doi.org/10.1142/S0218195993000208","url":null,"abstract":"We consider problems on intervals which can be solved by dynamic programming. Specifically, we give an efficient implementation of dynamic programming on intervals. As an application, an optimal sequential partition of a graph G=(V, E) can be obtained in O(m log n) time, where n = ¦V¦ and m = ¦E¦. We also present an O(n log n) time algorithm for finding a minimum weight dominating set of an interval graph G=(V, E), and an O(m log n) time algorithm for finding a maximum weight clique of a circular-arc graph G=(V, E), provided their intersection models of n intervals (arcs) are given.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125139157","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 : 1991-12-16DOI: 10.1142/S0218195993000257
L. Guibas, J. Hershberger, Joseph S. B. Mitchell, J. Snoeyink
We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We also discuss additional topological constraints such as simplicity.
{"title":"Approximating Polygons and Subdivisions with Minimum Link Paths","authors":"L. Guibas, J. Hershberger, Joseph S. B. Mitchell, J. Snoeyink","doi":"10.1142/S0218195993000257","DOIUrl":"https://doi.org/10.1142/S0218195993000257","url":null,"abstract":"We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We also discuss additional topological constraints such as simplicity.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123273945","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 : 1991-12-16DOI: 10.1142/S0218195993000166
Hans-Peter Lenhof, M. Smid
Let P be a set of n points in the Euclidean plane and let C be a convex figure. In 1985, Chazelle and Edelsbrunner presented an algorithm, which preprocesses P such that for any query point q, the points of P in the translate C+q can be retrieved efficiently. Assuming that constant time suffices for deciding the inclusion of a point in C, they provided a space and query time optimal solution. Their algorithm uses O(n) space. A query with output size k can be solved in O(log n+k) time. The preprocessing step of their algorithm, however, has time complexity O(n2). We show that the usage of a new construction method for layers reduces the preprocessing time to O(n log n). We thus provide the first space, query time and preprocessing time optimal solution for this class of point retrieval problems. Besides, we present two new dynamic data structures for these problems. The first dynamic data structure allows on-line insertions and deletions of points in O((log n)2) time. In this dynamic data structure, a query with output size k can be solved in O(log n+k(log n)2) time. The second dynamic data structure, which allows only semi-online updates, has O((log n)2) amortized update time and O(log n+k) query time.
{"title":"An optimal construction method for generalized convex layers","authors":"Hans-Peter Lenhof, M. Smid","doi":"10.1142/S0218195993000166","DOIUrl":"https://doi.org/10.1142/S0218195993000166","url":null,"abstract":"Let P be a set of n points in the Euclidean plane and let C be a convex figure. In 1985, Chazelle and Edelsbrunner presented an algorithm, which preprocesses P such that for any query point q, the points of P in the translate C+q can be retrieved efficiently. Assuming that constant time suffices for deciding the inclusion of a point in C, they provided a space and query time optimal solution. Their algorithm uses O(n) space. A query with output size k can be solved in O(log n+k) time. The preprocessing step of their algorithm, however, has time complexity O(n2). We show that the usage of a new construction method for layers reduces the preprocessing time to O(n log n). We thus provide the first space, query time and preprocessing time optimal solution for this class of point retrieval problems. Besides, we present two new dynamic data structures for these problems. The first dynamic data structure allows on-line insertions and deletions of points in O((log n)2) time. In this dynamic data structure, a query with output size k can be solved in O(log n+k(log n)2) time. The second dynamic data structure, which allows only semi-online updates, has O((log n)2) amortized update time and O(log n+k) query time.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127080399","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 : 1991-12-16DOI: 10.1142/S021819599300021X
Ravi Janardan
Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, S. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α and Β be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(α log2n) time, and reports in O(α log2n) time an approximation, ŵ, to the width such that (hat W/W leqslant sqrt {1 + tan ^2 tfrac{pi }{{4alpha }}}). The algorithm for the diameter problem uses O(Βn) space, supports updates in O(Βlogn) time, and reports in O(Β) time an approximation, D, to the diameter such that (hat D/D geqslant sin left( {tfrac{beta }{{beta + 1}}tfrac{pi }{2}} right)). Thus, for instance, even for α as small as 5, ŵ/W≤1.01, and for Β as small as 11, D/D≥.99. All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.
{"title":"On maintaining the width and diameter of a planar point-set online","authors":"Ravi Janardan","doi":"10.1142/S021819599300021X","DOIUrl":"https://doi.org/10.1142/S021819599300021X","url":null,"abstract":"Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, S. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α and Β be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(α log2n) time, and reports in O(α log2n) time an approximation, ŵ, to the width such that (hat W/W leqslant sqrt {1 + tan ^2 tfrac{pi }{{4alpha }}}). The algorithm for the diameter problem uses O(Βn) space, supports updates in O(Βlogn) time, and reports in O(Β) time an approximation, D, to the diameter such that (hat D/D geqslant sin left( {tfrac{beta }{{beta + 1}}tfrac{pi }{2}} right)). Thus, for instance, even for α as small as 5, ŵ/W≤1.01, and for Β as small as 11, D/D≥.99. All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128935379","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 : 1991-06-01DOI: 10.1142/S0218195993000117
F. Preparata, J. Vitter
In this paper we give a simple and efficient output-sensitive algorithm for constructing the display of a polyhedral terrain. It runs in $O((d + n)log^2 n)$ time, where $d$ is the size of the final display. The main data structure maintains an implicit representation of the convex hull of a set of points that can be dynamically updated in $O(log^2 n)$ time. It is especially simple and fast in our application since there are no rebalancing operations required in the tree.
{"title":"A Simplified Technique for Hidden-Line Elimination in Terrains","authors":"F. Preparata, J. Vitter","doi":"10.1142/S0218195993000117","DOIUrl":"https://doi.org/10.1142/S0218195993000117","url":null,"abstract":"In this paper we give a simple and efficient output-sensitive algorithm for constructing the display of a polyhedral terrain. It runs in $O((d + n)log^2 n)$ time, where $d$ is the size of the final display. The main data structure maintains an implicit representation of the convex hull of a set of points that can be dynamically updated in $O(log^2 n)$ time. It is especially simple and fast in our application since there are no rebalancing operations required in the tree.","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"171 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120875419","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 : 1991-03-01DOI: 10.1007/3-540-54233-7_173
M. Bern, D. Eppstein, Frances F. Yao
{"title":"The expected extremes in a Delaunay triangulation","authors":"M. Bern, D. Eppstein, Frances F. Yao","doi":"10.1007/3-540-54233-7_173","DOIUrl":"https://doi.org/10.1007/3-540-54233-7_173","url":null,"abstract":"","PeriodicalId":285210,"journal":{"name":"International Journal of Computational Geometry and Applications","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116020280","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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