Fully Dynamic Orthogonal Range Reporting on RAM

Mortensen, Christian Worm

doi:10.1137/s0097539703436722

Cited by 37 publications

(14 citation statements)

References 24 publications

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“…Consequently, we obtain the current-record query time for dynamic orthogonal range reporting in any constant dimension d ≥ 3 with polylogarithmic update time: O((log n/ log log n) d−3 log n + k) query time with O(log d+6+ε n) amortized expected update time. Compare this to Mortensen's previous result [24], which has query time worse by almost a log factor, though with a better update bound: O((log n/ log log n) d−1 + k) query time with O(log d−1−1/8+ε n) update time. (See also [25] and references therein for more on dynamic orthogonal range reporting.)…”

Section: Introductionsupporting

confidence: 66%

“…For b = log ε n, the query time is thus O(log n + k). The 3-d j-sided orthogonal range reporting problem reduces to 3-d (j − 1)-sided range reporting by standard binary divide-and-conquer (e.g., see [24]), where the update time (but not the query time) increases by a logarithmic factor. Thus, 3-d general (6-sided) orthogonal range reporting reduces to 3-d dominance range reporting, where the update time increases by a log 3 n factor.…”

Section: Theorem 32mentioning

confidence: 99%

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Three Problems About Dynamic Convex Hulls

Chan

2012

Int. J. Comput. Geom. Appl.

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We present three results related to dynamic convex hulls:• A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log 1+ε n) for an arbitrarily small constant ε > 0. This improves the previous bound of O(log 3/2 n).• A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n + k) time with O(polylog n) expected amortized update time. A similar result holds for 3-dimensional orthogonal range reporting. For 3-dimensional halfspace range reporting, the query time increases to O(log 2 n/ log log n + k).• A semi-online dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O(log n) and amortized update time O(n ε ). As a corollary, we can solve the following problem in O(n 1+ε ) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.

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Section: Introductionsupporting

confidence: 66%

Section: Theorem 32mentioning

confidence: 99%

Three Problems About Dynamic Convex Hulls

Chan

2012

Int. J. Comput. Geom. Appl.

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“…Most of these work (see [13,43,44,45,53] for a sample) are about van-EmdeBoas-type results, with only a few exceptions (e.g., [49,68]). For instance, Karlsson [44] obtained an O(n lg lg U )-time algorithm for the L ∞ -Voronoi diagram in 2-d. Chew and Fortune [25] later showed how to construct the Voronoi diagram under any fixed convex polygonal metric in 2-d in O(n lg lg n) time after sorting the points along a fixed number of directions.…”

Section: (Almost) Orthogonal Problemsmentioning

confidence: 99%

Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time

Chan¹,

caron²,

traşcu³

2009

SIAM J. Comput.

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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w , we present a linear-space data structure that can answer point location queries in O(min{lg n/ lg lg n, lg U/ lg lg U }) time on the unit-cost RAM with word size w. This is the first result to beat the standard Θ(lg n) bound for infinite precision models.As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higher-dimensional extensions and applications are also discussed.Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this long-standing limitation, answering, for example, a question of Willard (SODA'92).

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“…Dietz [11] first describes this problem in the context of fully persistent arrays, and gives a solution yielding O(log log m) expected amortized time operations, where m := |L| + q i=1 |S i | is the total number of element occurrences in subsets. Mortensen [15] describes a solution that supports updates to the subsets in expected O(log log m) time, and all other operations in O(log log m) worst case time. In section 3 we describe a UR version.…”

Section: Theorem 23 (Selected Treap Propertiesmentioning

confidence: 99%

“…They use O(n log n) space and use O(1)-wise independent hash functions. Although better redundant data structures for these problems are known [16,17,3] (an O(log log n)-factor improvement), our data structures are the first to be uniquely represented. Furthermore they are quite simple, arguably simpler than previous redundant structures that match our bounds.…”

Section: Introductionmentioning

confidence: 99%

Uniquely Represented Data Structures for Computational Geometry

Blelloch

Golovin

Vassilevska

2008

Algorithm Theory – SWAT 2008

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We present new techniques for the construction of uniquely represented data structures in a RAM, and use them to construct efficient uniquely represented data structures for orthogonal range queries, line intersection tests, point location, and 2-D dynamic convex hull. Uniquely represented data structures represent each logical state with a unique machine state. Such data structures are strongly history-independent. This eliminates the possibility of privacy violations caused by the leakage of information about the historical use of the data structure. Uniquely represented data structures may also simplify the debugging of complex parallel computations, by ensuring that two runs of a program that reach the same logical state reach the same physical state, even if various parallel processes executed in different orders during the two runs.

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Fully Dynamic Orthogonal Range Reporting on RAM

Cited by 37 publications

References 24 publications

Three Problems About Dynamic Convex Hulls

Three Problems About Dynamic Convex Hulls

Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time

Uniquely Represented Data Structures for Computational Geometry

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