scholarly journals ALGORITHMS FOR DISTANCE PROBLEMS IN PLANAR COMPLEXES OF GLOBAL NONPOSITIVE CURVATURE

2014 ◽  
Vol 24 (01) ◽  
pp. 1-38 ◽  
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).

MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

1995 ◽  
Vol 05 (03) ◽  
pp. 243-256 ◽  
Author(s):  
DAVID RAPPAPORT
Keyword(s):  
Convex Hull ◽  
General Problem ◽  
Line Segment ◽  
Simple Polygon ◽  
Fixed Number ◽  
Line Segments ◽  
Geometric Objects ◽  
Finite Set ◽  
Minimum Perimeter ◽  
Set Of Points

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

scholarly journals Finding the convex hull of a simple polygon in linear time

1986 ◽  
Vol 19 (6) ◽  
pp. 453-458 ◽  
Author(s):  
S.Y. Shin ◽  
T.C. Woo
Keyword(s):  
Convex Hull ◽  
Linear Time ◽  
Simple Polygon

A SIMPLIFIED TECHNIQUE FOR HIDDEN-LINE ELIMINATION IN TERRAINS

1993 ◽  
Vol 03 (02) ◽  
pp. 167-181 ◽  
Author(s):  
FRANCO P. PREPARATA ◽  
JEFFREY SCOTT VITTER
Keyword(s):  
Data Structure ◽  
Convex Hull ◽  
Full Advantage ◽  
Implicit Representation ◽  
Geometrical Properties ◽  
Asymptotic Speed ◽  
Hidden Line ◽  
The Cost ◽  
Set Of Points

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.

scholarly journals A Universal Separable Diversity

2017 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
David Bryant ◽  
André Nies ◽  
Paul Tupper
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Urysohn Space ◽  
Fundamental Properties ◽  
Finite Set ◽  
Whole Space ◽  
Set Of Points ◽  
Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

Comments on convex hull of a finite set of points in two dimensions

1979 ◽  
Vol 9 (3) ◽  
pp. 141-142 ◽  
Author(s):  
Ferenc Dévai ◽  
Tibor Szendrényi
Keyword(s):  
Convex Hull ◽  
Two Dimensions ◽  
Finite Set ◽  
Set Of Points

scholarly journals TRIANGULATING A REGION BETWEEN ARBITRARY POLYGONS

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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