Corrections to Lee's visibility polygon algorithm

We present a modification and extension of the (linear time) visibility polygon algorithm of Lee. The algorithm computes the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon, or in its blocked exterior (the cases of viewpoints on the boundary or in the free exterior being simple extensions of the interior case). We show by example that the original algorithm by Lee, and a more complex algorithm by El Gindy and Avis, can fail for polygons that wind sufficiently. We present a second version of the algorithm, which does not extend to the blocked exterior case.This work was partially supported by grants from the Central Research Fund of the University of Alberta and the Natural Sciences and Engineering Research Council of Canada.     

A linear-time algorithm for computing the voronoi diagram of a convex polygon

We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram ofn sites in the plane can be computed in (n) time when these sites form the vertices of a convex polygon in, say, counterclockwise order. This settles an open problem in computational geometry. Our techniques can also be used to obtain linear-time algorithms for computing the furthest-site Voronoi diagram and the medial axis of a convex polygon and for deleting a site from a general planar Voronoi diagram.This research began while the first and fourth authors were visiting the Mathematical Sciences Research Institute in Berkeley, California. Work by the fourth author was supported in part by NSF Grant No. 8120790.     

A linear time algorithm to remove winding of a simple polygon

In this paper, we present a linear time algorithm to remove winding of a simple polygon P with respect to a given point q inside P. The algorithm removes winding by locating a subset of Jordan sequence that is in the proper order and uses only one stack.     

AnO(n logn) algorithm for computing the link center of a simple polygon

We present an algorithm that determines the link center of a simplen-vertex polygonP inO(n logn) time. The link center of a simple polygon is the set of pointsx insideP at which the maximal link-distance fromx to any other point inP is minimized. The link distance between two pointsx andy insideP is defined to be the smallest number of straight edges in a polygonal path insideP connectingx andy. Using our algorithm we also obtain anO(n logn)-time solution to the problem of determining the link radius ofP. The link radius ofP is the maximum link distance from a point in the link center to any vertex ofP. Both results are improvements over theO(n ²) bounds previously established for these problems. The research of J.-R. Sack was supported by the Natural Sciences and Engineering Research Council of Canada.     

An algorithm for computing equilibria in a linear monetary economy

An algorithm is presented for computing equilibria in a linear monetary economy, that is, an exchange economy in which all individuals have linear utility functions and in which goods are bought and sold only in exchange for money. The algorithm computes the equilibrium prices by solving a finite sequence of linear programming problems.     

Interior proximal point algorithm for linear programs

An interior proximal point algorithm for finding a solution of a linear program is presented. The distinguishing feature of this algorithm is the addition of a quadratic proximal term to the linear objective function. This perturbation has allowed us to obtain solutions with better feasibility. Implementation of this algorithm shows that the algorithms. We also establish global convergence and local linear convergence of the algorithm.This research was supported by National Science Foundation Grants DCR-85-21228 and CCR-87-23091 and by Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-89-0410. It was conducted while the author was a Graduate Student at the Computer Sciences Department, University of Wisconsin, Madison, Wisconsin.     

A parallel interior point algorithm for linear programming on a network of transputers

Interior Point algorithms have become a very successful tool for solving large-scale linear programming problems. The Dual Affine algorithm is one of the Interior Point algorithms implemented in the computer program OB1. It is a good candidate for implementation on a parallel computer because it is very computing-intensive. A parallel Dual Affine algorithm is presented which is suitable for a parallel computer with a distributed memory. The algorithm obtains its speedup from parallel sparse linear algebra computations such as Cholesky factorisation, matrix multiplication, and triangular system solving, which form the bulk of the computing work. Efficient algorithms based on the grid distribution of matrices are presented for each of these computations. The algorithm is implemented in occam 2 on a square mesh of transputers. The resulting parallel program is connected to the sequentialFortran 77 program OB1, which performs the preprocessing and the postprocessing. Experimental results on a mesh of 400 transputers are given for a test set of seven realistic planning and scheduling problems from Shell and seven problems from the NETLIB LP collection; the results show a speedup of 88 for the largest problem.     

A fast las vegas algorithm for triangulating a simple polygon

We present a randomized algorithm that triangulates a simple polygon onn vertices inO(n log*n) expected time. The averaging in the analysis of running time is over the possible choices made by the algorithm; the bound holds for any input polygon.Research partially supported by the National Science Foundation under Grant No. DCR-8605962.     

An algorithm for computing the spectral structure of a singular linear matrix pencil

An algorithm is proposed for computing the structure of the Kronecker canonical form for a singular linear matrix pencil. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 159, pp. 23–32, 1987.     

Computing the maximum clique in the visibility graph of a simple polygon

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n²e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P. We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.     

A customized proximal point algorithm for convex minimization with linear constraints

This paper demonstrates a customized application of the classical proximal point algorithm (PPA) to the convex minimization problem with linear constraints. We show that if the proximal parameter in metric form is chosen appropriately, the application of PPA could be effective to exploit the simplicity of the objective function. The resulting subproblems could be easier than those of the augmented Lagrangian method (ALM), a benchmark method for the model under our consideration. The efficiency of the customized application of PPA is demonstrated by some image processing problems.     

A polynomial path-following interior point algorithm for general linear complementarity problems

Linear Complementarity Problems (LCPs) belong to the class of \mathbbNP{\mathbb{NP}} -complete problems. Therefore we cannot expect a polynomial time solution method for LCPs without requiring some special property of the coefficient matrix. Our aim is to construct interior point algorithms which, according to the duality theorem in EP (Existentially Polynomial-time) form, in polynomial time either give a solution of the original problem or detects the lack of property P_*([(k)\tilde]){\mathcal{P}_*(\tilde\kappa)} , with arbitrary large, but apriori fixed [(k)\tilde]{\tilde\kappa}). In the latter case, the algorithms give a polynomial size certificate depending on parameter [(k)\tilde]{\tilde{\kappa}} , the initial interior point and the input size of the LCP). We give the general idea of an EP-modification of interior point algorithms and adapt this modification to long-step path-following interior point algorithms.     

An interior point algorithm for semi-infinite linear programming

We consider the generalization of a variant of Karmarkar's algorithm to semi-infinite programming. The extension of interior point methods to infinite-dimensional linear programming is discussed and an algorithm is derived. An implementation of the algorithm for a class of semi-infinite linear programs is described and the results of a number of test problems are given. We pay particular attention to the problem of Chebyshev approximation. Some further results are given for an implementation of the algorithm applied to a discretization of the semi-infinite linear program, and a convergence proof is given in this case.     

A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment

We consider the problem of planning the motion of an arbitraryk-sided polygonal robotB, free to translate and rotate in a polygonal environmentV bounded byn edges. We present an algorithm that constructs a single component of the free configuration space ofB in timeO((kn) ^2+ɛ), for any ɛ>0. This algorithm, combined with some standard techniques in motion planning, yields a solution to the underlying motion-planning problem, within the same running time. Work on this paper by Dan Halperin has been supported by a Rothschild Postdoctoral Fellowship, by a grant from the Stanford Integrated Manufacturing Association (SIMA), by NSF/ARPA Grant IRI-9306544, and by NSF Grant CCR-9215219. Work on this paper by Micha Sharir has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary and extended version of the paper has appeared as: D. Halperin and M. Sharir, Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment,Proc. 34th IEEE Symp. on Foundations of Computer Science, 1993, pp. 382–391. Part of the work on the paper was carried out while Dan Halperin was at Tel Aviv University.     

Two linear-time algorithms for computing the minimum length polygon of a digital contour

The Minimum Length Polygon (MLP) is an interesting first order approximation of a digital contour. For instance, the convexity of the MLP is characteristic of the digital convexity of the shape, its perimeter is a good estimate of the perimeter of the digitized shape. We present here two novel equivalent definitions of MLP, one arithmetic, one combinatorial, and both definitions lead to two different linear time algorithms to compute them. This paper extends the work presented in Provençal and Lachaud (2009) [26], by detailing the algorithms and providing full proofs. It includes also a comparative experimental evaluation of both algorithms showing that the combinatorial algorithm is about 5 times faster than the other. We also checked the multigrid convergence of the length estimator based on the MLP.     

Triangulating a simple polygon in linear time

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm. The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8700917.