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Optimal Search in Planar Subdivisions

David Kirkpatrick
1983 SIAM journal on computing (Print)  
The subdivision search problem is the following: given a subdivision S with n line segments and a query point P, determine which region of S contains P.  ...  A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions.  ...  Locating a set of points in a planar subdivision.  ... 
doi:10.1137/0212002 fatcat:xjol6hwh3na6hf45pvat6skppa

Adaptive Point Location in Planar Convex Subdivisions

Siu-Wing Cheng, Man-Kit Lau
2017 International journal of computational geometry and applications  
We present a planar point location structure for a convex subdivision S.  ...  planar point location queries in S to process the same query sequence.  ...  Conclusion We propose a self-adjusting point location structure for planar convex subdivisions.  ... 
doi:10.1142/s0218195917600019 fatcat:lvzdjx5bvjg23mnlvretxfgwde

A Space-Efficient Framework for Dynamic Point Location [chapter]

Meng He, Patrick K. Nicholson, Norbert Zeh
2012 Lecture Notes in Computer Science  
Let G be a planar subdivision with n vertices.  ...  We describe a general framework for converting dynamic data structures for planar point location into succinct geometric indexes, provided that the subdivision G to be maintained has bounded face size.  ...  Given a planar subdivision of n vertices, for a sufficiently large n, there exists an algorithm that can encode it as a permutation of its point set in O(n) time and such that the subdivision can be decoded  ... 
doi:10.1007/978-3-642-35261-4_57 fatcat:7slwvicdnja3xlqmrnqob6mop4

Parallel Transitive Closure and Point Location in Planar Structures

Roberto Tamassia, Jeffrey S. Vitter
1991 SIAM journal on computing (Print)  
Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location  ...  queries, and construction of visibility representations and drawings of planar graphs.  ...  We w ould like to thank the referees for several useful comments and suggestions.  ... 
doi:10.1137/0220045 fatcat:maxtox5mknar7fmbwq67ssdita

A NOTE ON COMMUNICATION-EFFICIENT DETERMINISTIC PARALLEL ALGORITHMS FOR PLANAR POINT LOCATION AND 2D VORONOÏ DIAGRAM

MOHAMADOU DIALLO, AFONSO FERREIRA, ANDREW RAU-CHAPLIN
2001 Parallel Processing Letters  
In this note we describe deterministic parallel algorithms for planar point location and for building the Vorono Diagram of n co-planar points.  ...  They are communicatione cient since they require, respectively, O(1) and O(log p) communication steps and O( n log n p ) local computation per step.  ...  In this note we address the case where the planar subdivision is convex and where many points are to be located instead of only one.The problem of planar multi-point location on a convex subdivision is  ... 
doi:10.1142/s0129626401000622 fatcat:c7mo2kx5nrhlljgykkxt43xnye

Page 5659 of Mathematical Reviews Vol. , Issue 91J [page]

1991 Mathematical Reviews  
The most important operation is point location: for a given point p, it returns the polygon of the planar subdivision which contains p.  ...  The point location time is optimal. The data structure basically decomposes the planar subdivision into a tree of trapezoids. It uses space O(N + nlog N).  ... 

Point-Location in The Arrangement of Curves [article]

Sepideh Aghamolaei, Mohammad Ghodsi
2020 arXiv   pre-print
An arrangement of n curves in the plane is given. The query is a point q and the goal is to find the face of the arrangement that contains q.  ...  We design a data structure for solving the point location problem queries in O(log C(n)+log S(n)) time using O(T(n)+S(n)log(S(n))) preprocessing time, if a polygonal subdivision of total size S(n), with  ...  Finding a point in a planar subdivision and returning the face that contain that point is called point location.  ... 
arXiv:2007.11451v2 fatcat:udqcs4ot7rdovjtf2mxqxnfohm

Communication-efficient deterministic parallel algorithms for planar point location and 2d Voronoi Diagram [chapter]

Mohamadou Diallol, Afonso Ferreira, Andrew Rau-Chaplin
1998 Lecture Notes in Computer Science  
In this paper we describe deterministic parallel algorithms for planar point location and for building the Voronoi Diagram of n co-planar points.  ...  Part of this work was done while the second author was with the LIP at the ENS Lyon and while the authors visited each other in Lyon and in Halifax.  ...  Planar Point Location The problem of planar multi-point location on a convex subdivision is stated as follows: Locate O(n) points in a planar convex subdivision de ned by O(n) edges.  ... 
doi:10.1007/bfb0028576 fatcat:nglf7j24crecxhz6o2ts6y5a5i

Efficient Point Location in a Convex Spatial Cell-Complex

Franco P. Preparata, Roberto Tamassia
1992 SIAM journal on computing (Print)  
., 38 (1989), pp. 86-124] with the recently discovered dynamic structure for planar point-location in monotone subdivisions, a method with query time O(log N) and space O(N log N) for point-location in  ...  planar point-location technique of  ...  FIG. 2 . 2 Dynamic point location technique: (a) monotone subdivision T; (b) subdivision T* and its chains of proper edges; (c) separator tree of T*. Insert and Delete.  ... 
doi:10.1137/0221020 fatcat:ncjgpriervfwniioblqtv76ubq

Star-shaped polyhedron point location with orthogonal walk algorithm

Roman Soukal, Ivana Kolingerová
2010 Procedia Computer Science  
polyhedron and a modification of the planar Orthogonal walk algorithm.  ...  The point location problem is one of the most frequent tasks in computational geometry.  ...  Váša and Mr. Hlaváček for their feedback and inspiring discussions. This work is supported by the Grant Agency of the Czech Republic -the project 201/09/0097.  ... 
doi:10.1016/j.procs.2010.04.025 fatcat:s3rjwt7uy5ef7p6fuzghbawqu4

Dynamic maintenance of planar digraphs, with applications

Roberto Tamassia, Franco P. Preparata
1990 Algorithmica  
provides the theoretical underpinning and unifying viewpoint for three significant applications: point location in a planar monotone subdivision, transitive-closure query in planar sr-graphs, and contact-chain  ...  Applications to planar point location, transitive closure, and contact chains are described in Section 4.  ... 
doi:10.1007/bf01840401 fatcat:e5ixzwxdszbvph6z4lsfbipdg4

Succinct Geometric Indexes Supporting Point Location Queries [chapter]

Prosenjit Bose, Eric Y. Chen, Meng He, Anil Maheshwari, Pat Morin
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time 1 .  ...  We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries.  ...  Point Location in Planar Subdivisions We now generalize the techniques of Section 3 to general planar subdivisions.  ... 
doi:10.1137/1.9781611973068.70 fatcat:ocva2u2vwvclxerdvnedmwxqi4

Succinct geometric indexes supporting point location queries

Prosenjit Bose, Eric Y. Chen, Meng He, Anil Maheshwari, Pat Morin
2012 ACM Transactions on Algorithms  
Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time 1 .  ...  We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries.  ...  Point Location in Planar Subdivisions We now generalize the techniques of Section 3 to general planar subdivisions.  ... 
doi:10.1145/2151171.2151173 fatcat:aop6jhrcqrfd3lklm62oewp35u

2D Fractional Cascading on Axis-aligned Planar Subdivisions [article]

Peyman Afshani, Pingan Cheng
2020 arXiv   pre-print
In 2D fractional cascading, the input includes a planar subdivision for every vertex of G and the query is a point q and a subgraph π and the goal is to locate the cell containing q in all the subdivisions  ...  In this paper, we show that it is possible to circumvent the lower bound of Chazelle and Liu for axis-aligned planar subdivisions.  ...  Given a planar subdivision S of complexity n, in the planar point location problem, we are asked to preprocess S such that given any query point q in the plane, we can find the face f in S containing q  ... 
arXiv:2009.05541v2 fatcat:4fzeakxnuzebdmyf3npxg3iwfa

Succinct Geometric Indexes Supporting Point Location Queries [article]

Prosenjit Bose, Eric Y. Chen, Meng He, Anil Maheshwari, Pat Morin
2008 arXiv   pre-print
Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time.  ...  We then generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries.  ...  Second, it is not known how to support point location in planar triangulations or planar subdivisions using implicit data structures.  ... 
arXiv:0805.4147v1 fatcat:o27rcrfifjej5hs7l56mjkq7je
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