Finding the convex hull of a simple polygon in linear time

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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A linear time algorithm for obtaining the convex hull of a simple polygon

Pattern Recognition ◽

10.1016/0031-3203(83)90075-4 ◽

1983 ◽

Vol 16 (6) ◽

pp. 587-592 ◽

Cited By ~ 10

Author(s):

S.K. Ghosh ◽

R.K. Shyamasundar

Keyword(s):

Convex Hull ◽

Linear Time ◽

Simple Polygon ◽

Time Algorithm ◽

Linear Time Algorithm

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A New Linear Time Algorithm for Computing the Convex Hull of a Simple Polygon

Operations Research ’93 ◽

10.1007/978-3-642-46955-8_61 ◽

1994 ◽

pp. 263-265

Author(s):

W. Hochstättler ◽

S. Kromberg ◽

C. Moll

Keyword(s):

Convex Hull ◽

Linear Time ◽

Simple Polygon ◽

Time Algorithm ◽

Linear Time Algorithm

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How to Keep an Eye on Small Things

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195920500053 ◽

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.

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