## EFFICIENT POINT LOCATION IN A CONVEX SPATIAL CELL-COMPLEX*

### BibTeX

@MISC{P_efficientpoint,

author = {Franco P and Preparatat and Roberto Tamassia},

title = {EFFICIENT POINT LOCATION IN A CONVEX SPATIAL CELL-COMPLEX*},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper a new approach is proposed to point-location in a three-dimensional cellcomplex 7, which may be viewed as a nontrivial generalization of a corresponding two-dimensional technique due to Sarnak and Tarjan. Specifically, in a space-sweep of 7), the intersections of the sweep-plane with P occurring in a given slab, i.e., between two consecutive vertices, are topologically conformal planar subdivisions. If the sweep direction is viewed as time, the descriptions of the various slabs are distinct "versions " of a two-dimensional point-location data structure, dynamically updated each time a vertex is swept. Combining the persistence-addition technique of Driscoll, Sarnak, Sleator, and Tarjan [J. Comput. System. Sci., 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 a convex cell-complex with N facets is obtained. Key words, point location, convex cell complex, computational geometry, analysis of algorithms AMS(MOS) subject classifications. 68U05, 68Q25, 68P05, 68P10

### Citations

688 | Algorithms in Combinatorial Geometry - Edelsbrunner - 1987 |

230 |
A Dichromatic Framework for Balanced Trees
- Guibas, Sedgewick
- 1978
(Show Context)
Citation Context ...t at an amortized space cost of 0(1) per update step and a constant factor in the amortized time per operation. Considerable simplifications arise when the ephemeral data structure is a redblack tree =-=[7]-=-. In this case the node in-degree is at most 1, and rebalancing after an insertion/deletion requires O(1) rotations. 3. The ephemeral planar point location structure. In this section we describe algor... |

172 | Planar point location using persistent search trees - Sarnak, Tarjan - 1986 |

106 | A deterministic view of random sampling and its use in geometry
- Chazelle, Friedman
- 1990
(Show Context)
Citation Context ...een applied to the cell complex determined by an arrangement of n planes, and yields query time O(log n) but uses space O(n4/logn). The space bound has been recently improved by Chazelle and Friedman =-=[2]-=- to O(n3), by a modification of the random sampling technique of Clarkson [3]. Chazelle’s earlier Canal Tree technique [1] trades space for query time, and achieves space O(n3) and query time O(log 2 ... |

104 |
Cederbaum I. An algorithm for planarity testing of graphs
- Lempel, Even
- 1967
(Show Context)
Citation Context ...ondence between monotone subdivisions and planar st-graphs, the topological underpinning of the total order on the regions of 7 can be found in the theory of planar st-graphs and planar lattices [8], =-=[10]-=-, [15]. We shall use, where appropriate, a string notation to illustrate the order -, so- EFFICIENT SPATIAL POINT LOCATION 271 that AB denotes that the partial subdivision described by the string A p... |

61 |
Location of a Point in a Planar Subdivision and its Applications
- Lee, Preparata
- 1977
(Show Context)
Citation Context ...ement of a convex polyhedron). Any planar section of ’ is a planar convex subdivision (with the analogous exception of at most one nonconvex region), which is a special case of a monotone subdivision =-=[9]-=-. We shall now review the essentials of the dynamic point-location method of Preparata and Tamassia [13] for planar monotone subdivisions, in order to bring into sharper focus the requirements for the... |

49 |
Searching and storing similar lists
- Cole
- 1986
(Show Context)
Citation Context ...-dimensional counterpart, spatial point-location has not yet received extensive attention. In all reported research, the cell-complex satisfies some restrictive condition. Cole’s Similar Lists method =-=[4]-=- has been applied to the cell complex determined by an arrangement of n planes, and yields query time O(log n) but uses space O(n4/logn). The space bound has been recently improved by Chazelle and Fri... |

26 |
How to Search in History
- Chazelle
- 1985
(Show Context)
Citation Context ...n4/logn). The space bound has been recently improved by Chazelle and Friedman [2] to O(n3), by a modification of the random sampling technique of Clarkson [3]. Chazelle’s earlier Canal Tree technique =-=[1]-=- trades space for query time, and achieves space O(n3) and query time O(log 2 n). The same technique can be applied to a convex cell-complex with N facets, yielding query time O(log 2 N) and space O(N... |

23 | Fully dynamic point location in a monotone subdivision
- Preparata, Tamassia
- 1989
(Show Context)
Citation Context ...esearch and Defense Advanced Research Projects Agency contract N00014-91-J-4052 and ARPA order 8225, and by Cadre Technologies, Inc. 267268 F.P. PREPARATA AND ROBERTO TAMASSIA Preparata and Tamassia =-=[13]-=-. Our result is a new method with query time O(log 2 N) and space O(N log 2 N) for a convex spatial cell-complex with N facets. The general methodology of [5] is designed to add persistence to a dynam... |

23 | Parallel transitive closure and point location in planar structures
- Tamassia, Vitter
- 1991
(Show Context)
Citation Context ...he regions of P(z). We use the extension of the order to the set of vertices, edges, and regions of a monotone subdivision introduced in [15], and the local characterization of such ordering given in =-=[16]-=-. First, consider the expansion of vertex v into a triangle r. The update of the extended order is performed by simply replacing v with r and its vertices and edges. Hence, the sequence of the regions... |

20 |
Tarjan, Making data structures persistent
- Driscoll, Samak, et al.
- 1986
(Show Context)
Citation Context ...s be acyclic. Two recent results can be profitably combined to provide a new attractive approach to spatial point-location: the persistence-addition technique of Driscoll, Sarnak, Sleator, and Tarjan =-=[5]-=- and the dynamic planar point-location technique of Received by the editors December 26, 1989; accepted for publication (in revised form) March 26, 1991. An extended abstract of this paper was present... |

19 |
Dynamic maintenance of planar digraphs, with applications. Algorithmica
- Tamassia, Preparata
- 1990
(Show Context)
Citation Context ...e between monotone subdivisions and planar st-graphs, the topological underpinning of the total order on the regions of 7 can be found in the theory of planar st-graphs and planar lattices [8], [10], =-=[15]-=-. We shall use, where appropriate, a string notation to illustrate the order -, so- EFFICIENT SPATIAL POINT LOCATION 271 that AB denotes that the partial subdivision described by the string A precede... |

3 |
Planar point location revisited: A guided tour of a decade of research
- Preparata
- 1990
(Show Context)
Citation Context ...U05, 68Q25, 68P05, 68P10 1. Introduction. Point-location in three-dimensional space, called spatial pointlocation, is a natural generalization of the well-known planar point location (see, e.g., [6], =-=[11]-=-, [12]). The space is partitioned into polyhedral regions, called cells, and the resulting subdivision is frequently referred to as a cell-complex. The problem is so stated: Given a cell-complex 7 and... |

1 |
New applications of random sampling in computational geometry
- KSON
- 1987
(Show Context)
Citation Context ... yields query time O(log n) but uses space O(n4/logn). The space bound has been recently improved by Chazelle and Friedman [2] to O(n3), by a modification of the random sampling technique of Clarkson =-=[3]-=-. Chazelle’s earlier Canal Tree technique [1] trades space for query time, and achieves space O(n3) and query time O(log 2 n). The same technique can be applied to a convex cell-complex with N facets,... |