منابع مشابه
On Separating Points by Lines
Given a set P of n points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate n points, picked randomly (and uniformly) in the unit square, is Θ̃(n), where Θ̃ hides polylogarithmic factors. In addition, we provide a fast approximation algorithm for computing the separa...
متن کاملSeparating Points by Parallel Hyperplanes - Characterization Problem
This paper deals with partitions of a discrete set S of points in a d-dimensional space, by h parallel hyperplanes. Such partitions are in a direct correspondence with multilinear threshold functions which appear in the theory of neural networks and multivalued logic. The characterization (encoding) problem is studied. We show that a unique characterization (encoding) of such multilinear partit...
متن کاملSeparating points by axis-parallel lines
We study the problem of separating n points in the plane, no two of whi h have the same xor yoordinate, using a minimum number of verti al and horizontal lines avoiding the points, so that ea h ell of the subdivision ontains at most one point. Extending previous NP-hardness results due to Freimer et al. we prove that this problem and some variants of it are APX-hard. We give a 2-approximation a...
متن کاملSeparating Bi-Chromatic Points by Parallel Lines
Given a 2-coloring of the vertices of a regular n-gon P , how many parallel lines are needed to separate the vertices into monochromatic subsets? We prove that bn/2c is a tight upper bound, and also provide anO(n log n) time algorithm to determine the direction that gives the minimum number of lines. If the polygon is a non-regular convex polygon, then n − 3 lines may be necessary, while n − 2 ...
متن کاملSeparating pairs of points in the plane by monotone subsequences
Let S be a finite set of points in R. Let k be a positive integer. A pair of points {a, b} of S is called k-linked if there exists a weakly monotone sequence with k + 1 points of S in which a and b are two endpoints. Let f(n, k) be the maximum integer t such that every n-set S ⊂ R has t k-linked pairs. It is known that f(n, k) = 0 if and only if n ≤ k. Let t(n, k) be (1/2) · ki=1( (n + i − 1)/k...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1974
ISSN: 0012-365X
DOI: 10.1016/0012-365x(74)90028-4