THREE PROBLEMS ABOUT DYNAMIC CONVEX HULLS

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+εn) for an arbitrarily small constant ε > 0. This improves the previous bound of O( log 3/2n). • 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 2n/ 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(nε). As a corollary, we can solve the following problem in O(n1+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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A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries

Journal of the ACM ◽

10.1145/1706591.1706596 ◽

2010 ◽

Vol 57 (3) ◽

pp. 1-15 ◽

Cited By ~ 15

Author(s):

Timothy M. Chan

Keyword(s):

Data Structure ◽

Nearest Neighbor ◽

Convex Hulls ◽

Dynamic Data ◽

Dynamic Data Structure ◽

Nearest Neighbor Queries

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AN OPTIMAL CONSTRUCTION METHOD FOR GENERALIZED CONVEX LAYERS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195993000166 ◽

1993 ◽

Vol 03 (03) ◽

pp. 245-267

Author(s):

HANS-PETER LENHOF ◽

MICHIEL SMID

Keyword(s):

Data Structure ◽

Optimal Solution ◽

Construction Method ◽

Query Time ◽

Query Point ◽

Dynamic Data ◽

Time Optimal ◽

New Construction ◽

On Line ◽

Dynamic Data Structure

Let P be a set of n points in the Euclidean plane and let C be a convex figure. In 1985, Chazelle and Edelsbrunner presented an algorithm, which preprocesses P such that for any query point q, the points of P in the translate C+q can be retrieved efficiently. Assuming that constant time suffices for deciding the inclusion of a point in C, they provided a space and query time optimal solution. Their algorithm uses O(n) space. A query with output size k can be solved in O( log n+k) time. The preprocessing step of their algorithm, however, has time complexity O(n2). We show that the usage of a new construction method for layers reduces the preprocessing time to O(n log n). We thus provide the first space, query time and preprocessing time optimal solution for this class of point retrieval problems. Besides, we present two new dynamic data structures for these problems. The first dynamic data structure allows on-line insertions and deletions of points in O(( log n)2) time. In this dynamic data structure, a query with output size k can be solved in O( log n+k( log n)2) time. The second dynamic data structure, which allows only semi-online updates, has O(( log n)2) amortized update time and O( log n+k) query time.

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Persistent triangulations

Journal of Functional Programming ◽

10.1017/s0956796801004087 ◽

2001 ◽

Vol 11 (5) ◽

pp. 441-466 ◽

Cited By ~ 3

Author(s):

GUY BLELLOCH ◽

HAL BURCH ◽

KARL CRARY ◽

ROBERT HARPER ◽

GARY MILLER ◽

...

Keyword(s):

Data Structure ◽

Convex Hull ◽

Randomized Algorithm ◽

Computation Time ◽

Simplicial Complexes ◽

Constant Factor ◽

Running Time ◽

3 Dimensional ◽

Small Constant ◽

Triangulated Surfaces

Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These representations of triangulations make it difficult to support persistence, including ‘multiple futures’, the ability to use a data structure in several unrelated ways in a given computation; ‘time travel’, the ability to move freely among versions of a data structure; or parallel computation, the ability to operate concurrently on a data structure without interference. We present a purely functional interface and representation of triangulated surfaces, and more generally of simplicial complexes in higher dimensions. In addition to being persistent in the strongest sense, the interface more closely matches the mathematical definition of triangulations (simplicial complexes) than do interfaces based on mutable representations. The representation, however, comes at the cost of requiring O(lg n) time for traversing or adding triangles (simplices), where n is the number of triangles in the surface. We show both analytically and experimentally that for certain important cases, this extra cost does not seriously affect end-to-end running time. Analytically, we present a new randomized algorithm for 3-dimensional Convex Hull based on our representations for which the running time matches the Ω(n lg n) lower-bound for the problem. This is achieved by using only O(n) traversals of the surface. Experimentally, we present results for both an implementation of the 3-dimensional Convex Hull and for a terrain modeling algorithm, which demonstrate that, although there is some cost to persistence, it seems to be a small constant factor.

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