Dynamic Point Location in General Subdivisions

Baumgarten, Hanna; Jung, Hermann; Mehlhorn, Kurt

doi:10.1006/jagm.1994.1040

Cited by 39 publications

(48 citation statements)

References 15 publications

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“…The space saving technique, is influenced from the work of Baumgarten et al [4], that present a linear space implementation for dynamic point location in general subdivisions. Intuitively, we save space by inserting the segments of S into a large fan-out interval tree, where only a fraction of the segments is also inserted into a segment tree.…”

Section: How To Use Only Linear Spacementioning

confidence: 99%

“…A similar generalization has been made by Mortensen [17]. Finally, we reduce the space to linear, using a technique of Baumgarten et al [4], in which we store all the segments in an interval tree, where only a carefully chosen subset of the segments is stored in a segment tree.…”

Section: Introductionmentioning

confidence: 99%

“…Baumgarten et al [4] describe a data structure for the vertical ray shooting version of the general problem, and therefore also solves our problem. This solution requires linear space, supports queries in O(log n log log n) worst-case time, insertions in O(log n log log n) amortized time, and deletions in O(log 2 n) amortized time.…”

Section: Introductionmentioning

confidence: 99%

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Optimal dynamic vertical ray shooting in rectilinear planar subdivisions

Giora

Kaplan

2009

ACM Trans. Algorithms

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In this paper we consider the dynamic vertical ray shooting problem, that is the task of maintaining a dynamic set S of n non intersecting horizontal line segments in the plane subject to a query that reports the first segment in S intersecting a vertical ray from a query point. We develop a linearsize structure that supports queries, insertions and deletions in O(log n) worst-case time. Our structure works in the comparison model and uses a RAM . IntroductionIn this paper we consider data structures for the dynamic vertical ray shooting problem. In this problem we maintain a dynamic set S of n non intersecting horizontal line segments in the plane such that we can efficiently report the segment in S immediately above a given point. The vertical ray shooting problem is in fact a version of the dynamic rectilinear planar point location problem. In particular, given a subdivision of the plane by horizontal and vertical line segments, our data structure allows to find the rectangle containing a query point. The dynamic rectilinear planar point location problem is a special case of the general dynamic planar point location problem in which segments are not restricted to be horizontal or vertical. Obtaining a linear space data structure with logarithmic query and update time for dynamic planar point location is a central open question in algorithms and computational geometry. Although the restriction of segments to be horizontal is strong, an optimal algorithm for this special case was not known prior to our work.We present a data structure in the RAM model of computation, that requires linear space and supports updates and queries in O(log n) worst-case time. Our data structure does not make any assumptions on the segments. That is, we manipulate the segments only by comparisons. In this sense our result is optimal, since by an easy reduction from sorting, at least one of the operations takes Ω(log n) time.

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Section: How To Use Only Linear Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal dynamic vertical ray shooting in rectilinear planar subdivisions

Giora

Kaplan

2009

ACM Trans. Algorithms

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“…In the dynamic version of this problem, updates manipulate the geometry of the subdivision. Preparata and Tamassia [26] give an algorithm that runs in time O(log 2 n) per operation; this was improved to query time O(log n) by Baumgarten, Jung, and Mehlhorn [5]. The lower bound for this problem in [19] applies only to algorithms returning the name of the region containing the queried point.…”

Section: Dynamic Graph Algorithmsmentioning

confidence: 99%

“…We also present bounds for planar point location in monotone subdivisions [5,26], reachability in upward planar digraphs [28], and incremental parsing of balanced parentheses [11]. We show that these problems require time Ω(log n/ log log n) per operation .…”

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confidence: 99%

New Lower Bound Techniques for Dynamic Partial Sums and Related Problems

Husfeldt¹,

Rauhe²

2003

SIAM J. Comput.

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Abstract. We study the complexity of the dynamic partial sum problem in the cell-probe model. We give the model access to nondeterministic queries and prove that the problem remains hard. We give the model access to the right answer ±1 as an oracle and prove that the problem remains hard. This suggests which kind of information is hard to maintain.From these results, we derive a number of lower bounds for dynamic algorithms and data structures: We prove lower bounds for dynamic algorithms for existential range queries, reachability in directed graphs, planarity testing, planar point location, incremental parsing, and fundamental data structure problems like maintaining the majority of the prefixes of a string of bits. We prove a lower bound for reachability in grid graphs in terms of the graph's width. We characterize the complexity of maintaining the value of any symmetric function on the prefixes of a bit string.

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On-line Planar Graph Embedding

Tamassia

1996

Journal of Algorithms

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We present a dynamic data structure for the incremental construction of a planar embedding of a planar graph. The data structure supports the following Ž . operations: i testing if a new edge can be added to the embedding without Ž . introducing crossing; and ii adding vertices and edges. The time complexity of Ž . Ž . each operation is O log n amortized for edge insertion , and the memory space Ž . and preprocessing time are O n , where n is the current number of vertices of the graph. ᮊ

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Dynamic Point Location in General Subdivisions

Abstract: We prove that the dilation of an m X n toroidal mesh in an m~vertex path equals 2min{m,n}, if m:f:. n and 2n -1, if m = n.

Cited by 39 publications

References 15 publications

Optimal dynamic vertical ray shooting in rectilinear planar subdivisions

Optimal dynamic vertical ray shooting in rectilinear planar subdivisions

New Lower Bound Techniques for Dynamic Partial Sums and Related Problems

On-line Planar Graph Embedding

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