Disjoint Sets Of Points Research Articles

Spanning Trees in Multipartite Geometric Graphs

Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let $$n=|R\cup B|$$ . A bichromatic spanning tree is a spanning tree in the complete bipartite geometric graph with bipartition (R, B). The minimum (respectively maximum) bichromatic spanning tree problem is the problem of computing a bichromatic spanning tree of minimum (respectively maximum) total edge length. (1) We present a simple algorithm that solves the minimum bichromatic spanning tree problem in $$O(n\log ^3 n)$$ time. This algorithm can easily be extended to solve the maximum bichromatic spanning tree problem within the same time bound. It also can easily be generalized to multicolored point sets. (2) We present $$\Theta (n\log n)$$ -time algorithms that solve the minimum and the maximum bichromatic spanning tree problems. (3) We extend the bichromatic spanning tree algorithms and solve the multicolored version of these problems in $$O(n\log n\log k)$$ time, where k is the number of different colors (or the size of the multipartition in a complete multipartite geometric graph).

Plane Bichromatic Trees of Low Degree

Let R and B be two disjoint sets of points in the plane such that $$|B|\le |R|$$ , and no three points of $$R\cup B$$ are collinear. We show that the geometric complete bipartite graph K(R, B) contains a non-crossing spanning tree whose maximum degree is at most $$\max \,\{3, \lceil (|R|-1)/|B|\rceil + 1\}$$ ; this is the best possible upper bound on the maximum degree. This proves two conjectures made by Kaneko, 1998, and solves an open problem posed by Abellanas et al. at the Graph Drawing Symposium, 1996.

The sub‐Riemannian cut locus of H‐type groups

In the present paper we prove that the sub‐Riemannian cut locus at the origin of a wide class of nilpotent groups of step two, called H‐type groups, corresponds to the center of the group. We obtain this result by completely describing the sub‐Riemannian geodesics in the group, and using these to obtain three disjoint sets of points in the group determined by the number of geodesics joining them to the origin.

A Quantitative Variant of the Multi-colored Motzkin–Rabin Theorem

We prove a quantitative version of the multi-colored Motzkin---Rabin theorem in the spirit of Barak et al. (Proceedings of the National Academy of Sciences, 2012): Let $$V_1,\ldots ,V_n \subset {\mathbb {R}}^d$$V1,?,Vn?Rd be $$n$$n disjoint sets of points (of $$n$$n `colors'). Suppose that for every $$V_i$$Vi and every point $$v \in V_i$$v?Vi there are at least $$\delta |V_i|$$?|Vi| other points $$u \in V_i$$u?Vi so that the line connecting $$v$$v and $$u$$u contains a third point of another color. Then the union of the points in all $$n$$n sets is contained in a subspace of dimension bounded by a function of $$n$$n and $$\delta $$? alone.

Range search on tuples of points

Range search is a fundamental query-retrieval problem, where the goal is to preprocess a given set of points so that the points lying inside a query object (e.g., a rectangle, or a ball, or a halfspace) can be reported efficiently. This paper considers a new version of the range search problem: Several disjoint sets of points are given along with an ordering of the sets. The goal is to preprocess the sets so that for any query point q and a distance δ, all the ordered sequences (i.e., tuples) of points can be reported (one per set and consistent with the given ordering of the sets) such that, for each tuple, the total distance traveled starting from q and visiting the points of the tuple in the specified order is no more than δ. The problem has applications in trip planning, where the objective is to visit venues of multiple types starting from a given location such that the total distance traveled satisfies a distance constraint. Efficient solutions are given for the fixed distance and variable distance versions of this problem, where δ is known beforehand or is specified as part of the query, respectively.

Support Vector Machine Polyhedral Separability in Semisupervised Learning

We introduce separation margin maximization, a characteristic of the Support Vector Machine technique, into the approach to binary classification based on polyhedral separability and we adopt a semisupervised classification framework. In particular, our model aims at separating two finite and disjoint sets of points by means of a polyhedral surface in the semisupervised case, that is, by exploiting information coming from both labeled and unlabeled samples. Our formulation requires the minimization of a nonconvex nondifferentiable error function. Numerical results are presented on several data sets drawn from the literature.

The h-vector of the union of two sets of points in the projective plane

Given two h-vectors, handh0, we study which are the possible h-vectors for the union of two disjoint sets of points in P2, respectively associated to h and h0and how they can be constructed. We wil ...

On Polygons Enclosing Point Sets II

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56 |Conv(B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 , ... , p n and S a set of m points contained in the interior of P, m ≤ n − 1. Then there is a convex decomposition {P 1 , ... , P n } of P such that all points from S lie on the boundaries of P 1 , ... , P n , and each P i contains a whole edge of P on its boundary.

Steiner triple systems of order 19 and 21 with subsystems of order 7

Steiner triple systems (STSs) with subsystems of order 7 are classified. For order 19, this classification is complete, but for order 21 it is restricted to Wilson-type systems, which contain three subsystems of order 7 on disjoint point sets. The classified STSs of order 21 are tested for resolvability; none of them is doubly resolvable.

SOME LOWER BOUNDS ON GEOMETRIC SEPARABILITY PROBLEMS

We obtain lower bounds in the algebraic computation tree model for deciding the separability of two disjoint point sets. In particular, we show Ω(n log n) time lower bounds for separability by means of strips, wedges, wedges with apices on a given line, fixed-slopes double wedges, and triangles, which match the complexity of the existing algorithms, and therefore prove their optimality.

On the Intersection Number of Matchings and Minimum Weight Perfect Matchings of Multicolored Point Sets

Let P and Q be disjoint point sets with 2r and 2s elements respectively, and M1 and M2 be their minimum weight perfect matchings (with respect to edge lengths). We prove that the edges of M1 and M2 intersect at most |M1|+|M2|−1 times. This bound is tight. We also prove that P and Q have perfect matchings (not necessarily of minimum weight) such that their edges intersect at most min{r,s} times. This bound is also sharp.

SEMI-BALANCED PARTITIONS OF TWO SETS OF POINTS AND EMBEDDINGS OF ROOTED FORESTS

Let m be a positive integer and let R1, R2 and B be three disjoint sets of points in the plane such that no three points of R1 ∪ R2 ∪ B lie on the same line and |B| = (m-1)|R1|+m|R2|. Put g = |R1∪R2|. Then there exists a subdivision X1∪X2∪⋯∪Xg of the plane into g disjoint convex polygons such that (i) |Xi ∩ (R1 ∪ R2)| = 1 for all 1 ≤ i ≤ g; and (ii) |Xi∩B| = m-1 if |Xi∩R1| = 1, and |Xi∩B| = m if |Xi∩R2| = 1. This partition is called a semi-balanced partition, and our proof gives an O(n4) time algorithm for finding the above semi-balanced partition, where n = |R1| + |R2| + |B|. We next apply the above result to the following theorem: Let T1,…,Tg be g disjoint rooted trees such that |Ti| ∈ {m,m+1} and vi is the root of Ti for all 1 ≤ i ≤ g. Let P be a set of |T1|+⋯+|Tg| points in the plane in general position that contains g specified points p1,…,pg. Then the rooted forest T1 ∪ ⋯ ∪ Tg can be straight-line embedded onto P so that each vi corresponds to pi for every 1 ≤ i ≤ g.

RED-BLUE SEPARABILITY PROBLEMS IN 3D

In this paper we study the problems of separability of two disjoint point sets in 3D by multiple criteria extending some notions on separability of two disjoint point sets in the plane.

On plane spanning trees and cycles of multicolored point sets with few intersections

Let P 1 , … , P k be a collection of disjoint point sets in R 2 in general position. We prove that for each 1 ⩽ i ⩽ k we can find a plane spanning tree T i of P i such that the edges of T 1 , … , T k intersect at most ( k − 1 ) ( n − k ) + 1 2 k ( k − 1 ) , where n is the number of points in P 1 ∪ ⋯ ∪ P k . If the intersection of the convex hulls of P 1 , … , P k is nonempty, we can find k spanning cycles such that their edges intersect at most ( k − 1 ) n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 8 n times, where | P ∪ Q | = n (the weight of an edge is its length).

ALTERNATING HAMILTON CYCLES WITH MINIMUM NUMBER OF CROSSINGS IN THE PLANE

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n2 log n) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

Balanced partitions of two sets of points in the plane

We prove the following theorem. Let m≥2 and q≥1 be integers and let S and T be two disjoint sets of points in the plane such that no three points of S∪ T are on the same line, | S|=2 q and | T|= mq. Then S∪ T can be partitioned into q disjoint subsets P 1, P 2,…, P q satisfying the following two conditions: (i) conv( P i )∩conv( P j )= φ for all 1≤ i< j≤ q, where conv( P i ) denotes the convex hull of P i ; and (ii) | P i ∩ S|=2 and | P i ∩ T|= m for all 1≤ i≤ q.

AN EFFICIENT PARALLEL ALGORITHM FOR THE ASSIGNMENT PROBLEM ON THE PLANE∗

Abstract We consider a parallel algorithm for minimum weight perfect matching on complete bipartite graphs induced by two disjoint sets of points on the plane. The algorithm is designed for the EREW PRAM or EREW RPRAM model of parallel computation with p (a function of n) processors. For points on the Euclidean plane, the algorithm executes in O(n 3/ p 2 + n2log n + n 2.5log n/p) time for p < equation pending We achieve a perfect speedup algorithm when p equation pending

Multicategory discrimination via linear programming

A single linear program is proposed for discriminating between the elements of κ disjoint point sets in the n-dimensional real space Rn . When the conical hulls of the κ sets are (κ−1)-point disjoint in Rn +1, a κ-piece piecewise-linear surface generated by the linear program completely separates the κ sets. This improves on a previous linear programming approach which required that each set be linearly separable from the remaining κ−1 sets. When the conical hulls of the κ sets are not (κ;−1)-point disjoint, the proposed linear program generates an error-minimizing piecewise-linear separator for the κ Sets. For this case it is shown that the null solution is never a unique solver of the linear program and occurs only under the rather rare condition when the mean of each point set equals the mean of the means of the other κ−l sets. This makes the proposed linear computational programming formulation useful for approximately discriminating between κ sets that are not piecewise-linear separable. Computational results are reported for three previously available databases.

Bilinear separation of two sets inn-space

The NP-complete problem of determining whether two disjoint point sets in then-dimensional real spaceRn can be separated by two planes is cast as a bilinear program, that is minimizing the scalar product of two linear functions on a polyhedral set. The bilinear program, which has a vertex solution, is processed by an iterative linear programming algorithm that terminates in a finite number of steps a point satisfying a necessary optimality condition or at a global minimum. Encouraging computational experience on a number of test problems is reported.

An algorithm for calculating minimum Euclidean distance between two geographic features

An efficient algorithm is presented for determining the shortest Euclidean distance between two features of arbitrary shape that are represented in quadtree form. These features may be disjoint point sets, lines, or polygons. It is assumed that the features do not overlap. Features also may be intertwined and polygons may be complex (i.e. have holes). Utilizing a spatial divide-and-conquer approach inherent in the quadtree data model, the basic rationale is to narrow-in on portions of each feature quickly that are on a facing edge relative to the other feature, and to minimize the number of point-to-point Euclidean distance calculations that must be performed. Besides offering an efficient, grid-based alternative solution, another unique and useful aspect of the current algorithm is that is can be used for rapidly calculating distance approximations at coarser levels of resolution. The overall process can be viewed as a top-down parallel search. Using one list of leafcode addresses for each of the two features as input, the algorithm is implemented by successively dividing these lists into four sublists for each descendant quadrant. The algorithm consists of two primary phases. The first determines facing adjacent quadrant pairs where part or all of the two features are separated between the two quadrants, respectively. The second phase then determines the closest pixel-level subquadrant pairs within each facing quadrant pair at the lowest level. The key element of the second phase is a quick estimate distance heuristic for further elimination of locations that are not as near as neighboring locations.