## A Unified Approach to Dynamic Point Location, Ray Shooting, and Shortest Paths in Planar Maps (1992)

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Citations: | 20 - 6 self |

### BibTeX

@MISC{Chiang92aunified,

author = {Yi- Jen Chiang and Franco P. Preparata and Roberto Tamassia},

title = { A Unified Approach to Dynamic Point Location, Ray Shooting, and Shortest Paths in Planar Maps },

year = {1992}

}

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### Abstract

We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with 7 ~ vertices, and apply it to the development of a unified dynamic data structure that supports point-location, ray-shooting, and shortest-path queries in M. The space requirement is O(nlog n). Point-location queries take time O(log 7~). Ray-shooting and shortest-path queries take time O(log3 TZ) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log3 n) time (amortized for vertex updates).

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Citation Context ...tion can be viewed as a sequence of four elementary splits and joins. Three special types of data structures will be used in this paper: biased binary trees [3], B B[ot]-trees [20], and dynamic trees =-=[30]-=-. A biased binary tree [3] is a binary search tree whose leaves store weighted items. Let w be the sum of all weights. We have that the depth of a leaf with weight wi is at most 2, and each of the fol... |

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Citation Context ...ith the pioneering work in planar point location of the 1970s 10], 18], over the years several techniques have been developed, culminating in asymptotically time- and spaceoptimal methods [12], [17], =-=[29]-=- that are also of sufficiently practical flavor. Such methods, however, refer to the static case where no alteration of the map is allowed during its use. Due to the obvious importance of the dynamic ... |

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Citation Context ... maximal monotone subpath of a solid path P and specify the implementation of ltree(Q) and rtree(Q). We use pruned trees augmented with chain trees as secondary structures. Our scheme uses ideas from =-=[22]-=- and [16]. Trees ltree(Q) and rtree(Q) are implemented by means of pruned trees with respect to 3). Let/ be a node of ltree(Q) (nodes of rtree(Q) are handled identically) and v the parent of/z. Node/ ... |

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Citation Context .... Starting with the pioneering work in planar point location of the 1970s 10], 18], over the years several techniques have been developed, culminating in asymptotically time- and spaceoptimal methods =-=[12]-=-, [17], [29] that are also of sufficiently practical flavor. Such methods, however, refer to the static case where no alteration of the map is allowed during its use. Due to the obvious importance of ... |

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Citation Context ...articular, a simple rotation can be viewed as a sequence of four elementary splits and joins. Three special types of data structures will be used in this paper: biased binary trees [3], B B[ot]-trees =-=[20]-=-, and dynamic trees [30]. A biased binary tree [3] is a binary search tree whose leaves store weighted items. Let w be the sum of all weights. We have that the depth of a leaf with weight wi is at mos... |

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Citation Context ...try, see Chiang and Tamassia [9]. Algorithmic research on shortest-path and ray-shooting queries has also experienced steady progress, resulting in time-optimal techniques for the static setting [1], =-=[5]-=-, [7], [16], *Received by the editors January 8, 1992; accepted for publication (in revised form) August 3, 1994. An extended abstract of this paper was presented at the 4th ACM-SIAM Symposium on Disc... |

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Citation Context ...ving its root. In particular, a simple rotation can be viewed as a sequence of four elementary splits and joins. Three special types of data structures will be used in this paper: biased binary trees =-=[3]-=-, B B[ot]-trees [20], and dynamic trees [30]. A biased binary tree [3] is a binary search tree whose leaves store weighted items. Let w be the sum of all weights. We have that the depth of a leaf with... |

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Citation Context ...rtest-path queries, respectively, in a simple polygon with n vertices. No polylog-time method was previously known in a dynamic setting, although a polylog-time ray-shooting technique by Reif and Sen =-=[28]-=-, designed for monotone polygons, may be extensible to the general case. Sublinear-time techniques are known only for ray-shooting queries [1], [7], with O (/- polylog(n)) query/update time; they supp... |

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Citation Context ...he map is allowed during its use. Due to the obvious importance of the dynamic setting, in recent years considerable attention has been devoted to the development of dynamic point-location algorithms =-=[2]-=-, [6], [8], [14], [15], [21 ], [25], [26], [31 ]. All the known dynamic point-location results are for connected maps, since maintaining region names in a disconnected map would require solving half-p... |

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Citation Context ...p is allowed during its use. Due to the obvious importance of the dynamic setting, in recent years considerable attention has been devoted to the development of dynamic point-location algorithms [2], =-=[6]-=-, [8], [14], [15], [21 ], [25], [26], [31 ]. All the known dynamic point-location results are for connected maps, since maintaining region names in a disconnected map would require solving half-planar... |

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Citation Context ...Due to the obvious importance of the dynamic setting, in recent years considerable attention has been devoted to the development of dynamic point-location algorithms [2], [6], [8], [14], [15], [21 ], =-=[25]-=-, [26], [31 ]. All the known dynamic point-location results are for connected maps, since maintaining region names in a disconnected map would require solving half-planar range searching in a dynamic ... |

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Citation Context ...ing diagonals be involved in any update. In this paper we combine the feature just stated with the underpinnings of the trapezoid method, whose search efficiency both in theory [4], [23] and practice =-=[11]-=- is well established. This leads to the adoption of horizontal normalizing diagonals, called lids. The method rests on three major components: 1. a normalization structure that transforms a connected ... |

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Citation Context ...ed during its use. Due to the obvious importance of the dynamic setting, in recent years considerable attention has been devoted to the development of dynamic point-location algorithms [2], [6], [8], =-=[14]-=-, [15], [21 ], [25], [26], [31 ]. All the known dynamic point-location results are for connected maps, since maintaining region names in a disconnected map would require solving half-planar range sear... |

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Citation Context ...to onerous updates, such as when the insertion of an edge causes the removal of a very large number of normalizing diagonals. A rather complicated and only partially documented technique due to Fries =-=[13]-=- is reported to assure that only a logarithmic number of normalizing diagonals be involved in any update. In this paper we combine the feature just stated with the underpinnings of the trapezoid metho... |

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Citation Context ...see Chiang and Tamassia [9]. Algorithmic research on shortest-path and ray-shooting queries has also experienced steady progress, resulting in time-optimal techniques for the static setting [1], [5], =-=[7]-=-, [16], *Received by the editors January 8, 1992; accepted for publication (in revised form) August 3, 1994. An extended abstract of this paper was presented at the 4th ACM-SIAM Symposium on Discrete ... |

1 |
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Citation Context ...allowed during its use. Due to the obvious importance of the dynamic setting, in recent years considerable attention has been devoted to the development of dynamic point-location algorithms [2], [6], =-=[8]-=-, [14], [15], [21 ], [25], [26], [31 ]. All the known dynamic point-location results are for connected maps, since maintaining region names in a disconnected map would require solving half-planar rang... |

1 |
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Citation Context ...ting with the pioneering work in planar point location of the 1970s 10], 18], over the years several techniques have been developed, culminating in asymptotically time- and spaceoptimal methods [12], =-=[17]-=-, [29] that are also of sufficiently practical flavor. Such methods, however, refer to the static case where no alteration of the map is allowed during its use. Due to the obvious importance of the dy... |

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Citation Context ...t N00014-91-J-4052, ARPA order 8225. tDepartment of Computer Science, Brown University, Providence, RI 02912-1910 ({yjc,franco,rt}@cs. brown.edu). 207208 Y.-J. CHIANG, E R PREPARATA, AND R. TAMASSIA =-=[19]-=-. In particular, the linear-space data structures of Chazelle and Guibas [5] and of Guibas and Hershberger 16] support in O (log n) time ray-shooting and shortest-path queries, respectively, in a simp... |

1 | Range searching in a set of line segments - RMARS - 1985 |

1 |
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Citation Context ...number of normalizing diagonals be involved in any update. In this paper we combine the feature just stated with the underpinnings of the trapezoid method, whose search efficiency both in theory [4], =-=[23]-=- and practice [11] is well established. This leads to the adoption of horizontal normalizing diagonals, called lids. The method rests on three major components: 1. a normalization structure that trans... |

1 |
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Citation Context ...mark. In a concrete situation where vertices are a priori restricted to a fixed set of ordinates, tree 3; is static; if we then implement the trees ltree and rtree by means of contracted binary trees =-=[27]-=- of depth < log IYI (whose maintenance requires no rotation), then the update times become O(log 2 n log IYI), in the worst case. The following arc two additional types of queries that can be supporte... |

1 | An incremental reconstruction methodfor dynamic planar point location - TAMASSIA - 1991 |