Minimum polygonal separation
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TL;DR: It is shown that for k = Θ(n), Ω(n log n) is a lower bound to the running time of any algorithm for this problem, and exhibit two algorithms of distinctly different flavors.
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Abstract: In this paper we study the problem of polygonal separation in the plane, ie, finding a convex polygon with minimum number k of sides separating two given finite point sets (k-separator), if it exists We show that for k = Θ(n), Ω(n log n) is a lower bound to the running time of any algorithm for this problem, and exhibit two algorithms of distinctly different flavors The first relies on an O(n log n)-time preprocessing task, which constructs the convex hull of the internal set and a nested star-shaped polygon determined by the external set; the k-separator is contained in the annulus between the boundaries of these two polygons and is constructed in additional linear time The second algorithm adapts the prune-and-search approach, and constructs, in each iteration, one side of the separator; its running time is O(kn), but the separator may have one more side than the minimum
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References
Linear-Time Algorithms for Linear Programming in $R^3 $ and Related Problems
TL;DR: A linear-time algorithm is given for the classical problem of finding the smallest circle enclosing n given points in the plane, which disproves a conjecture by Shamos and Hoey that this problem requires Ω(n log n) time.
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Finding the intersection of two convex polyhedra
TL;DR: An algorithm to test whether their intersection is empty, and if so to find a separating plane, and to construct their intersection polyhedron is developed, which runs in timeO (n log n), where n is the sum of the numbers of vertices of the two polyhedra.
351
A Linear Algorithm for Determining the Separation of Convex Polyhedra
TL;DR: This work presents a linear algorithm for constructing a pair of points that realize the separation of two convex polyhedra in three dimensions based on a simple hierarchical description of polyhedRA that is of interest in its own right.
249
•Proceedings Article
Classifying Learnable Geometric Concepts with the Vapnik-Chervonenkis Dimension (Extended Abstract)
Anselm Blumer,Andrzej Ehrenfeucht,David Haussler,Manfred K. Warmuth +3 more
- 01 Jan 1986
TL;DR: It is shown that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned.
165
Finding minimal convex nested polygons
TL;DR: This work considers the problem of finding a polygon nested between two given convex polygons that has a minimal number of vertices, and presents an O(n log k) algorithm for solving the problem, where n is the total number of Vertices of the given polygons.
81