Dynamic rectangular intersection with priorities

Kaplan, Haim; Molad, Eyal; Tarjan, Robert E.

doi:10.1145/780632.780635

Cited by 14 publications

(23 citation statements)

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“…The results in this paper combine and extend results from two conference abstracts [1,14]. Our main result is a linear-size data structure for the stabbing-semigroup problem, in the pointer-machine model [19] of computation.…”

Section: Introductionsupporting

confidence: 75%

“…Kaplan, Molad, and Tarjan [14] showed that if we redefine A v to be the set of those ancestors of v for which L vu = ∅, then the size of the structure reduces to O(n log log n) without affecting the query or update time. Furthermore by maintaining only the top part of T explicitly and storing the intervals of nodes of depth larger than n/ log n separately, the space can be made linear and the insertion time O(log n), but the deletion time remains O(log n log log n).…”

Section: Lemmamentioning

confidence: 99%

“…Previously an Ω(log n/ log log n) lower bound was known [3] for the problem. Our structure can be extended to higher dimensions using segment trees in a standard way [14], by paying a penalty of an O(log n) factor in both time and space for each additional dimension, but the results may not be optimal in two or higher dimensions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Optimal Dynamic Data Structure for Stabbing-Semigroup Queries

Agarwal¹,

Arge²,

Kaplan³

et al. 2012

SIAM J. Comput.

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Let S be a set of n intervals in R, and let (S, +) be any commutative semigroup. We assign a weight ω(s) ∈ S to each interval in S. For a point x ∈ R, let S(x) ⊆ S be the set of intervals that contain x. Given a point q ∈ R, the stabbing-semigroup query asks for computing s∈S(q) ω(s). We propose a linear-size dynamic data structure, under the pointermachine model, that answers queries in worst-case O(log n) time, and supports both insertions and deletions of intervals in amortized O(log n) time. It is the first data structure that attains the optimal O(log n) bound for all three operations. Furthermore, our structure can easily be adapted to external memory, where we obtain a linear-size structure that answers queries and supports updates in O(log B n) I/Os, where B is the disk block size.For the restricted case of nested family of intervals (every pair of intervals are either disjoint or one contains the other), we present a simpler solution based on dynamic trees.

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Section: Introductionsupporting

confidence: 75%

Section: Lemmamentioning

confidence: 99%

See 1 more Smart Citation

An Optimal Dynamic Data Structure for Stabbing-Semigroup Queries

Agarwal¹,

Arge²,

Kaplan³

et al. 2012

SIAM J. Comput.

Self Cite

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show abstract

“…The best result we know with logarithmic query time is by Kaplan, Molad, and Tarjan [15], and requires O(log 2 n) time for updates.…”

Section: Introductionmentioning

confidence: 99%

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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“…McCreight's paper on priority search trees [19] refers to enumeration in increasing order along the yaxis but it does not indeed discuss how to report the items in sorted order along the y-axis. An indirect form of ranking can be found in the (dynamic) rectangular intersection with priorities [15] and in the document listing problem [21].…”

Section: Introductionmentioning

confidence: 99%

Rank-Sensitive Data Structures

Bialynicka-Birula

Grossi

2005

String Processing and Information Retrieval

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Abstract. Output-sensitive data structures result from preprocessing n items and are capable of reporting the items satisfying an on-line query in O(t(n) + ℓ) time, where t(n) is the cost of traversing the structure and ℓ ≤ n is the number of reported items satisfying the query. In this paper we focus on rank-sensitive data structures, which are additionally given a ranking of the n items, so that just the top k best-ranking items should be reported at query time, sorted in rank order, at a cost of O(t(n) + k) time. Note that k is part of the query as a parameter under the control of the user (as opposed to ℓ which is query-dependent). We explore the problem of adding rank-sensitivity to data structures such as suffix trees or range trees, where the ℓ items satisfying the query form O(polylog(n)) intervals of consecutive entries from which we choose the top k best-ranking ones. Letting s(n) be the number of items (including their copies) stored in the original data structures, we increase the space by an additional term of O(s(n) lg ǫ n) memory words of space, each of O(lg n) bits, for any positive constant ǫ < 1. We allow for changing the ranking on the fly during the lifetime of the data structures, with ranking values in 0 . . . O(n). In this case, query time becomes O(t(n)+k) plus O(lg n/ lg lg n) per interval; each change in the ranking and each insertion/deletion of an item takes O(lg n) time; the additional term in space occupancy increases to O(s(n) lg n/ lg lg n).

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Dynamic rectangular intersection with priorities

Cited by 14 publications

References 0 publications

An Optimal Dynamic Data Structure for Stabbing-Semigroup Queries

An Optimal Dynamic Data Structure for Stabbing-Semigroup Queries

Optimal dynamic vertical ray shooting in rectilinear planar subdivisions

Rank-Sensitive Data Structures

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