Partitioning arrangements of lines II: Applications

In this paper we present efficient deterministic algorithms for various problems involving lines or segments in the plane, using the partitioning algorithm described in a companion paper [A3]. These applications include: (i) anO(m ^2/3 n ^2/3 · log^2/3 n · log^/3 (m/n)+(m+n) logn) algorithm to compute all incidences betweenm points andn lines, where is a constant <3.33; (ii) anO(m ^2/3 n ^2/3 · log^5/3 n · log^/3 (m/n)+(m+n) logn) algorithm to computem faces in an arrangement ofn lines; (iii) anO(n ^4/3 log^(+2)/3 n) algorithm to count the number of intersections in a set ofn segments; (iv) anO(n ^4/3 log^{( + 2)/3} n) algorithm to count red-blue intersections between two sets of segments, and (v) anO(n ^3/2 log^/3 n) algorithm to compute spanning trees with low stabbing number for a set ofn points. We also present an algorithm that, given set ofn points in the plane, preprocesses it, in timeO(nm log^+1/2 n), into a data structure of sizeO(m) forn lognmn ², so that the number of points ofS lying inside a query triangle can be computed inO((n/m) log^3/2 n) time.Work on this paper has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-83-20085, and by grants from the Digital Equipment Corporation and the IBM Corporation. A preliminary version of this paper appears in theProceedings of the 5th ACM Symposium on Computational Geometry, 1989, pp. 11–22.     

Usort: An efficient hybrid of Distributive Partitioning Sorting

A new hybrid of Distributive Partitioning Sorting is described and tested against Quicksort on uniformly distributed items. Pointer sort versions of both algorithms are also tested.     

An efficient deterministic optimization approach for rectangular packing problems

The rectangular packing problem aims to seek the best way of placing a given set of rectangular pieces within a large rectangle of minimal area. Such a problem is often constructed as a quadratic mixed-integer program. To find the global optimum of a rectangular packing problem, this study transforms the original problem as a mixed-integer linear programming problem by logarithmic transformations and an efficient piecewise linearization approach that uses a number of binary variables and constraints logarithmic in the number of piecewise line segments. The reformulated problem can be solved to obtain an optimal solution within a tolerable error. Numerical examples demonstrate the computational efficiency of the proposed method in globally solving rectangular packing problems.     

Triangles in arrangements of lines

A set of n nonconcurrent lines in the projective plane (called an arrangment) divides the plane into polygonal cells. It has long been a problem to find a nontrivial upper bound on the number of triangular regions. We show that $\text{5}\text{12}\text{n(n ? 1)}$ is such a bound. We also show that if no three lines are concurrent, then the number of quadrilaterals, pentagons and hexagons is at least cn².     

Triangles in arrangements of lines

A set of n lines in the projective plane divides the plane into a certain number of polygonal cells. We show that if we insist that all of these cells be triangles, then there are at most $\text{1}\text{3}\text{n(n ? 1) + 4 ?}\text{2}\text{7}\text{n}$ of them. We also observe that if no point of the plane belongs to more than two of the lines and n is at least 4, some of the cells must be either quadrangles or pentagons. We further show that for $\text{n ≧ 4}$, there is a set of n lines which divides the plane into at least $\text{1}\text{3}\text{n(n ? 3) + 4}$ triangles.     

Triangles in arrangements of lines

Givenn lines in the real projective plane, Grünbaum conjectures that, for n16, the numberp ₃ of triangular regions determined by the lines is at most 1/3n(n–1). We show that ifn7 thenp ₃ 8/21n(n–1)+2/7, we also point out that if no vertex is a point of intersection of exactly three of the lines, thenp ₃1/3n(n–1).Professor Gu died while on a visit to Poland in April 1997     

An efficient algorithm for voting sequences

The chaos theorems show that given almost any alternatives x and y, there exists voting sequence from x to y. However, proofs of such results have been purely existential; that is, there is no algorithm by which such a voting path can be constructed. In this paper, we present such an algorithm for one standard example. Furthermore, it is shown that the algorithm has the property that the voting sequence involves the fewest possible number of steps.     

An efficient algorithm for solving inequalities

An efficient algorithm for solving a finite system of inequalities in a finite number of iterations is described and analyzed.This work was supported by the UK Science and Engineering Research Council     

An efficient algorithm for linear programming

A simple but efficient algorithm is presented for linear programming. The algorithm computes the projection matrix exactly once throughout the computation unlike that of Karmarkar’s algorithm where in the projection matrix is computed at each and every iteration. The algorithm is best suitable to be implemented on a parallel architecture. Complexity of the algorithm is being studied.     

An efficient algorithm for enumeration of triangulations

We consider the problem of enumerating triangulations of n points in the plane in general position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in O(loglogn) time per triangulation. It improves the previous bound by almost linear factor.     

Quadrilaterals and pentagons in arrangements of lines

Let p _k denote the number of k-sided faces in an arrangement of n5 lines in the real projective plane. B. Grünbaum has shown that p ₄1/2n(n–3) and has conjectured that equality can occur only for simple arrangements. We prove this conjecture here. We also show that 4p ₄+5p ₅3n holds for every simple arrangement of n4 lines. This latter result is a strengthening of a theorem of T. O. Strommer.     

Implicitly representing arrangements of lines or segments

Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n ²) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with (n) space¹ and (n ^3/2) preprocessing time, we can answer face queries in (n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path , form a data structure from which we can find the convex hull of any subpath of quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in (n ^1/3)+O(K) time, given(n ^4/3) space and (n ^5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time (m ^2/3 n ^2/3+n), which is nearly optimal.The first author is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and National Science Foundation Grant CCR-8714565. Work on this paper by the fifth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. The sixth author was supported in part by a National Science Foundation Graduate Fellowship. This work was begun while the non-DEC authors were visiting at the DEC Systems Research Center.     

An efficient algorithm to allocate shelf space

Shelf space is one of the most important resources of a retail firm. This paper formulates a model and proposes an approach which is similar to the algorithm used for solving a knapsack problem. Subject to given constraints, the proposed heuristic allocates shelf space item by item according to a descending order of sales profit for each item per display area or length. Through the use of simulations, the performances of objective value and the computational efficiency of this method are evaluated. Three options are also proposed for improving the heuristics. Compared to an optimal method, the improved heuristic is shown to be a very efficient algorithm which allocates shelf space at near-optimal levels.     

An efficient algorithm for the three-guard problem

Given a simple polygon P with two vertices u and v, the three-guard problem asks whether three guards can move from u to v such that the first and third guards are separately on two boundary chains of P from u to v and the second guard is always kept to be visible from two other guards inside P. It is a generalization of the well-known two-guard problem, in which two guards move on the boundary chains from u to v and are always kept to be mutually visible. In this paper, we introduce the concept of link-2-ray shots, which can be considered as ray shots under the notion of link-2-visibility. Then, we show a one-to-one correspondence between the structure of the restrictions placed on the motion of two guards and the one placed on the motion of three guards, and generalize the solution for the two-guard problem to that for the three-guard problem. We can decide whether there exists a solution for the three-guard problem in O(nlogn) time, and if so generate a walk in O(nlogn+m) time, where n denotes the number of vertices of P and the size of the optimal walk. This improves upon the previous time bounds O(n²) and O(n²logn), respectively. Moreover, our results can be used to solve other more sophisticated geometric problems.     

An efficient heuristic algorithm for minimum matching

We give a new heuristic algorithm for minimum matching problems and apply it to the Euclidean problem with random vertices in 2 dimensions. The algorithm is based on simulated annealing and performs in practice faster than previous heuristic algorithms yielding suboptimal solutions of the same good quality. From configurations with up toN=20.000 vertices in the unit square we estimate that the length of a minimum matching scales asymptotically asLN with (=0.3123±0.0016.

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Zusammenfassung Wir stellen einen neuen heuristischen Algorithmus für minimale Matching-Probleme vor und wenden diesen auf das euklidische Problem mit zufÄlliger Punkteverteilung in 2 Dimensionen an. Auf Simulated Annealing basierend lÄuft der Algorithmus schneller als frühere heuristische Algorithmen und erreicht dabei suboptimale Lösungen gleich guter QualitÄt. Aus Konfigurationen mit bis zuN=20.000 Punkten im Einheitsquadrat schÄtzen wir, da\ für die LÄnge des minimalen Matchings asymptotischLN mit=0.3123±0.0016 gilt.

     

An efficient algorithm for maximal margin clustering

Maximal margin based frameworks have emerged as a powerful tool for supervised learning. The extension of these ideas to the unsupervised case, however, is problematic since the underlying optimization entails a discrete component. In this paper, we first study the computational complexity of maximal hard margin clustering and show that the hard margin clustering problem can be precisely solved in O(n ^d+2) time where n is the number of the data points and d is the dimensionality of the input data. However, since it is well known that many datasets commonly ‘express’ themselves primarily in far fewer dimensions, our interest is in evaluating if a careful use of dimensionality reduction can lead to practical and effective algorithms. We build upon these observations and propose a new algorithm that gradually increases the number of features used in the separation model in each iteration, and analyze the convergence properties of this scheme. We report on promising numerical experiments based on a ‘truncated’ version of this approach. Our experiments indicate that for a variety of datasets, good solutions equivalent to those from other existing techniques can be obtained in significantly less time.     

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development

We present a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem. The functions involved are assumed to be nonconvex and twice continuously differentiable. The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree. A novel branching scheme is developed such that classical branch-and-bound is applied to both spaces without violating the hierarchy in the decisions and the requirement for (global) optimality in the inner problem. To achieve this, the well-known features of branch-and-bound algorithms are customized appropriately. For instance, two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. The proposed bounding problems do not grow in size during the algorithm and are obtained from the corresponding problems at the parent node.     

An efficient algorithm for product computations on computer

This paper presents an algorithm for computing the product of any two polynomials in the U-algebra of the split three dimensional simple Lie algebra L over an algebraically closed field of prime characteristic K, and expressing the product as a linear combination of standard monomials. All coefficients are taken from the prime field.