A parallel circle-cover minimization algorithm (original) (raw)
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Parallel algorithms on circular-arc graphs
Computational Geometry, 1995
Given a set of n circular arcs, the problem of finding a minimum cut has been considered in the sequential model. Here we present a parallel algorithm in the EREW-PRAM model that runs in O(log n) time with O(n) processors if the arcs are not given already sorted and using O(n/log n) processors otherwise. On the hypercube model, we consider the minimum cut as well as the following problems on a set of n circular-arcs: the minimum dominating set, the minimum circle cover, the maximum independent set, and the minimum clique cover. We give a parallel algorithm of time complexity O(log n log log n) and processor complexity O(n) for the minimum dominating set problem based on the hypercube model. For the minimum cut sequence, minimum circle cover, minimum clique cover, and maximum independent set problems, we give parallel algorithms of time complexity O(log n log log n + log n log m) and processor complexity O(n) if the input is not given sorted, otherwise, the time complexity is O(log n log m); m is the size of the solution set. * Corresponding author. 0925-7721/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0925-772 1(94)00024-7
Efficient Speed-Up of the Smallest Enclosing Circle Algorithm
Informatica, 2022
The smallest enclosing circle is a well-known problem. In this paper, we propose modifications to speed-up the existing Weltzl’s algorithm. We perform the preprocessing to reduce as many input points as possible. The reduction step has lower computational complexity than the Weltzl’s algorithm and thus speed-ups its computation. Next, we propose some changes to Weltzl’s algorithm. In the end are summarized results, that show the speed-up for 106{10^{6}}106 input points up to 100 times compared to the original Weltzl’s algorithm. Even more, the proposed algorithm is capable to process significantly larger data sets than the standard Weltzl’s algorithm.
2012
The problem of packing circles into a domain of prescribed topology is considered. The circles need not have equal radii. The Collins-Stephenson algorithm computes such a circle packing. This algorithm is parallelized in two different ways and its performance is reported for a triangular, planar domain test case. The implementation uses the highly parallel graphics processing unit (GPU) on commodity hardware. The speedups so achieved are discussed based on a number of experiments.
Computational Geometry, 2003
A circle packing is a configuration P of circles realizing a specified pattern of tangencies. Radii of packings in the euclidean and hyperbolic planes may be computed using an iterative process suggested by William Thurston. We describe an efficient implementation, discuss its performance, and illustrate recent applications. A central role is played by new and subtle monotonicity results for "flowers" of circles.
An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphs
Journal of Applied Mathematics and Computing, 2005
The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents an O(n 2 ) time sequential algorithm and an O(n 2 /p + log n) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, where p and n represent respectively the number of processors and the number of vertices of the circular-arc graph.
Computing Radial Drawings on the Minimum Number of Circles
Journal of Graph Algorithms and Applications, 2005
A radial drawing is a representation of a graph in which the vertices lie on concentric circles of finite radius. In this paper we study the problem of computing radial drawings of planar graphs by using the minimum number of concentric circles. We assume that the edges are drawn as straight-line segments and that co-circular vertices can be adjacent. It is proven that the problem can be solved in polynomial time. The solution is based on a characterization of those graphs that admit a crossing-free straight-line radial drawing on k circles. For the graphs in this family, a linear time algorithm that computes a radial drawing on k circles is also presented.
Computing Minimum Cut Sets for Circular-Arc Graphs| a New Approach
1999
Let A be a set of n arcs on the unit circle. We present a new simple 2(n logn)-time algorithm for computing a minimum cut set for A (and a maximum independent subset of A). Our solution is based on a dynamic maintenance scheme for a set S of n intervals on the line, that enables us to update the current minimum cut set for S, following an insertion or a deletion of an interval, in time O(c logn), where c is the size of the current minimum cut set.
A polynomial time circle packing algorithm
Discrete Mathematics, 1993
Mohar, B., A polynomial time circle packing algorithm, Discrete Mathematics 117 (1993) 2577263. The Andreev-Koebe-Thurston circle packing theorem is generalized and improved in two ways. Simultaneous circle packing representations of the map and its dual map are obtained such that any two edges dual to each other cross at the right angle. The necessary and sufficient condition for a map to have such a primal-dual circle packing representation is that its universal cover graph is 3-connected. A polynomial time algorithm is obtained that given such a map M and a rational number E > 0 finds an a-approximation for the primal-dual circle packing representation of M. In particular, there is a polynomial time algorithm that produces simultaneous geodesic line convex drawings of a given map and its dual in a surface with constant curvature, so that only edges dual to each other cross.
New circular drawing algorithms
Proc. ITAT, 2004
Abstract. In the circular (other alternate concepts are outerplanar, convex and one-page) drawing one places vertices of a n−vertex m−edge connected graph G along a circle, and the edges are drawn as straight lines. The smallest possible number of crossings in such a drawing of ...
Solution Methodologies for the Smallest Enclosing Circle Problem
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Tribute. We would like to dedicate this paper to Elijah Polak. Professor Polak has made substantial contributions to a truly broad spectrum of topics in nonlinear optimization, including optimization for engineering design centering, multi-criteria optimization, optimal control, feasible directions methods, quasi-Newton and Newton methods, non-differential optimization, semi-infinite optimization, conjugate directions methods, gradient projection and reduced gradient methods, and barrier methods, among many other topics. His many and varied contributions to our field are important today and will influence the research in our field well into the future.