Efficient computation of the convex hull on sets of points stored in a k-tree compact data structure

In this paper we give a practical and efficient output-sensitive algorithm for constructing the display of a polyhedral terrain. It runs in O((d+n) log 2 n) time and uses O(nα(n)) space, where d is the size of the final display, and α(n) is a (very slowly growing) functional inverse of Ackermann’s function. Our implementation is especially simple and practical, because we try to take full advantage of the specific geometrical properties of the terrain. The asymptotic speed of our algorithm has been improved upon theoretically by other authors, but at the cost of higher space usage and/or high overhead and complicated code. Our main data structure maintains an implicit representation of the convex hull of a set of points that can be dynamically updated in O( log 2 n) time. It is especially simple and fast in our application since there are no rebalancing operations required in the tree.

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Meliorated Approach for Efficient Computation of Shortest Path on Edge Based Data Structure

2009 IEEE International Advance Computing Conference ◽

10.1109/iadcc.2009.4809023 ◽

2009 ◽

Author(s):

Anand Gupta ◽

Ankur Maheshwari ◽

Gaurav Jain

Keyword(s):

Data Structure ◽

Shortest Path ◽

Efficient Computation ◽

Edge Based

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A Real-Time Algorithm for Convex Hull Construction and Delaunay Triangulation of Scattered Points Data

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.181-182.661 ◽

2011 ◽

Vol 181-182 ◽

pp. 661-666

Author(s):

Xue Ming He ◽

Yi Lu ◽

Cheng Gang Li ◽

Min Min Ni ◽

Chen Liang Hua

Keyword(s):

Data Structure ◽

Computational Geometry ◽

Convex Hull ◽

Surface Reconstruction ◽

Delaunay Triangulation ◽

Time Algorithm ◽

Important Data ◽

Geometry Design ◽

Data Points ◽

Angle Method

Convex hull is a very important data structure of computational geometry design. This paper presents an algorithm to construct the convex hull of a set of scattered points by coordinates and relative angle method. The algorithm determines the convex vertexes and eliminates some non-convex vertexes, which greatly reduces the searching scope and the complexity. Delaunay triangulation is widely used in 3D surface reconstruction. Due to its duality, Delaunay triangulation is usually constructed through Voronoi diagram. Delaunay triangulation is directly constructed in this paper. The algorithm is simple, stable and easy to implement, especially for less data points.

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TRIANGULATING A REGION BETWEEN ARBITRARY POLYGONS

International Journal of Computing ◽

10.47839/ijc.16.3.899 ◽

2017 ◽

pp. 160-165

Author(s):

Vasyl Tereshchenko ◽

Yaroslav Tereshchenko

Keyword(s):

Data Structure ◽

Convex Hull ◽

Efficient Algorithm ◽

Time Complexity ◽

Optimal Algorithm ◽

Simple Polygon ◽

Overall Design ◽

Simple Implementation ◽

Monotone Polygons ◽

Two Stages

The paper presents an optimal algorithm for triangulating a region between arbitrary polygons on the plane with time complexity O(N log⁡N ). An efficient algorithm is received by reducing the problem to the triangulation of simple polygons with holes. A simple polygon with holes is triangulated using the method of monotone chains and keeping overall design of the algorithm simple. The problem is solved in two stages. In the first stage a convex hull for m polygons is constructed by Graham’s method. As a result, a simple polygon with holes is received. Thus, the problem of triangulating a region between arbitrary polygons is reduced to the triangulation of a simple polygon with holes. In the next stage the simple polygon with holes is triangulated using an approach based on procedure of splitting polygon onto monotone polygons using the method of chains [15]. An efficient triangulating algorithm is received. The proposed algorithm is characterized by a very simple implementation, and the elements (triangles) of the resulting triangulation can be presented in the form of simple and fast data structure: a tree of triangles [17].

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COMPACT INTERVAL TREES: A DATA STRUCTURE FOR CONVEX HULLS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195991000025 ◽

1991 ◽

Vol 01 (01) ◽

pp. 1-22 ◽

Cited By ~ 19

Author(s):

LEONIDAS GUIBAS ◽

JOHN HERSHBERGER ◽

JACK SNOEYINK

Keyword(s):

Data Structure ◽

Convex Hull ◽

Compact Interval ◽

Common Tangent ◽

Convex Polygons ◽

Fixed Set ◽

Interval Tree ◽

Interval Trees ◽

Arrangements Of Lines ◽

The Common

In this paper, we investigate the problem of finding the common tangents of two convex polygons that intersect in two (unknown) points. First, we give a Θ( log 2n) bound for algorithms that store the polygons in independent arrays. Second, we show how to beat the lower bound if the vertices of the convex polygons are drawn from a fixed set of n points. We introduce a data structure called a compact interval tree that supports common tangent computations, as well as the standard binary-search-based queries, in O( log n) time apiece. Third, we apply compact interval trees to solve the subpath hull query problem: given a simple path, preprocess it so that we can find the convex hull of a query subpath quickly. With O(n log n) preprocessing, we can assemble a compact interval tree that represents the convex hull of a query subpath in O( log n) time. In order to represent arrangements of Lines implicitly, Edelsbrunner et al. used a less efficient structure, called bridge trees, to solve the subpath hull query problem. Our compact interval trees improve their results by a factor of O( log n). Thus, the present paper replaces the paper on bridge trees referred to by Edelsbrunner et al.

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ALGORITHMS FOR DISTANCE PROBLEMS IN PLANAR COMPLEXES OF GLOBAL NONPOSITIVE CURVATURE

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195914500010 ◽

2014 ◽

Vol 24 (01) ◽

pp. 1-38 ◽

Cited By ~ 1

Author(s):

DANIELA MAFTULEAC

Keyword(s):

Data Structure ◽

Convex Hull ◽

Metric Spaces ◽

Linear Time ◽

Single Point ◽

Simple Polygon ◽

Query Point ◽

Planar Complex ◽

Finite Set ◽

Set Of Points

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex [Formula: see text] with n vertices, one can construct in O(n2 log n) time a data structure [Formula: see text] of size O(n2) so that, given a point [Formula: see text], the shortest path γ(x, y) between x and the query point y can be computed in linear time. Our second algorithm computes the convex hull of a finite set of points in a CAT(0) planar complex. This algorithm is based on Toussaint's algorithm for computing the convex hull of a finite set of points in a simple polygon and it constructs the convex hull of a set of k points in O(n2 log n + nk log k) time, using a data structure of size O(n2 + k).

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THREE PROBLEMS ABOUT DYNAMIC CONVEX HULLS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195912600096 ◽

2012 ◽

Vol 22 (04) ◽

pp. 341-364 ◽

Cited By ~ 2

Author(s):

TIMOTHY M. CHAN

Keyword(s):

Data Structure ◽

Convex Hull ◽

Query Time ◽

Convex Hulls ◽

Lower Envelope ◽

Dynamic Data ◽

Line Segments ◽

3 Dimensional ◽

Small Constant ◽

Dynamic Data Structure

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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Memory-Adjustable Navigation Piles with Applications to Sorting and Convex Hulls

ACM Transactions on Algorithms ◽

10.1145/3452938 ◽

2021 ◽

Vol 17 (2) ◽

pp. 1-19

Author(s):

Omar Darwish ◽

Amr Elmasry ◽

Jyrki Katajainen

Keyword(s):

Data Structure ◽

Lower Bound ◽

Convex Hull ◽

Euclidean Plane ◽

State Of The Art ◽

Random Access ◽

Priority Queue ◽

Convex Hulls ◽

Limited Capacity ◽

Range Of Values

We consider space-bounded computations on a random-access machine, where the input is given on a read-only random-access medium, the output is to be produced to a write-only sequential-access medium, and the available workspace allows random reads and writes but is of limited capacity. The length of the input is N elements, the length of the output is limited by the computation, and the capacity of the workspace is O ( S ) bits for some predetermined parameter S ≥ lg N . We present a state-of-the-art priority queue—called an adjustable navigation pile —for this restricted model. This priority queue supports M inimum in O (1) time, C onstruct in O ( N ) time, and E xtract - min in O ( N / S + lg S ) time for any S ≥ lg N . The priority queue can be further augmented in O ( N ) time to deal with a batch of at most S elements in a specified range of values at a time, and allow to I nsert (activate) or E xtract (deactivate) an element among these elements, such that I nsert and E xtract take O ( N / S + lg S ) time for any S ≥ lg N . We show how to use our data structure to sort N elements and to compute the convex hull of N points in the Euclidean plane in O ( N 2 / S + N lg S ) time for any S ≥ lg N . Following a known lower bound for the space-time product of any branching program for finding unique elements, both our sorting and convex-hull algorithms are optimal. The adjustable navigation pile has turned out to be useful when designing other space-efficient algorithms, and we expect that it will find its way to yet other applications.

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