Linear-Time 3-Approximation Algorithm for the r-Star Covering Problem

2008 ◽  
pp. 157-168 ◽  
Author(s):  
Andrzej Lingas ◽  
Agnieszka Wasylewicz ◽  
Paweł Żyliński
Keyword(s):  
Approximation Algorithm ◽  
Linear Time ◽  
Covering Problem

LINEAR-TIME 3-APPROXIMATION ALGORITHM FOR THE r-STAR COVERING PROBLEM

2012 ◽  
Vol 22 (02) ◽  
pp. 103-141 ◽  
Author(s):  
ANDRZEJ LINGAS ◽  
AGNIESZKA WASYLEWICZ ◽  
PAWEŁ ŻYLIŃSKI
Keyword(s):  
Approximation Algorithm ◽  
Linear Time ◽  
Covering Problem ◽  
The Novel ◽  
Art Gallery ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition ◽  
Orthogonal Polygons ◽  
Complexity Status

The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has Õ(n17)-time complexity, where Õ(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons.

Graph matching

2020 ◽  
pp. 291-358
Author(s):  
Rob H. Bisseling
Keyword(s):  
Approximation Algorithm ◽  
Parallel Algorithm ◽  
Graph Matching ◽  
Linear Time ◽  
Total Weight ◽  
Mathematical Representation ◽  
Optimal Weight ◽  
Graph Problems ◽  
Positive Weights ◽  
Local Dominance

This chapter explores parallel algorithms for graph matching. Here, a graph is the mathematical representation of a network, with vertices representing the nodes of the network and edges representing their connections. The edges have positive weights, and the aim is to find a matching with maximum total weight. The chapter first presents a sequential, parallelizable approximation algorithm based on local dominance that guarantees attaining at least half the optimal weight in near-linear time. This algorithm, coupled with a vertex partitioning, is the basis for developing a parallel algorithm. The BSP approach is shown to be especially advantageous for graph problems, both in developing a parallel algorithm and in proving it correct. The basic parallel algorithm is enhanced by giving preference to local matches when breaking ties and by adding a load-balancing mechanism. The scalability of the parallel algorithm is put to the test using graphs of up to 150 million edges.

scholarly journals A generalized Robinson-Foulds distance for labeled trees

BMC Genomics ◽  
2020 ◽  
Vol 21 (S10) ◽  
Author(s):  
Samuel Briand ◽  
Christophe Dessimoz ◽  
Nadia El-Mabrouk ◽  
Manuel Lafond ◽  
Gabriela Lobinska
Keyword(s):  
Approximation Algorithm ◽  
Phylogenetic Trees ◽  
Linear Time ◽  
Crucial Aspect ◽  
Labeled Trees ◽  
Link Type

Abstract Background The Robinson-Foulds (RF) distance is a well-established measure between phylogenetic trees. Despite a lack of biological justification, it has the advantages of being a proper metric and being computable in linear time. For phylogenetic applications involving genes, however, a crucial aspect of the trees ignored by the RF metric is the type of the branching event (e.g. speciation, duplication, transfer, etc). Results We extend RF to trees with labeled internal nodes by including a node flip operation, alongside edge contractions and extensions. We explore properties of this extended RF distance in the case of a binary labeling. In particular, we show that contrary to the unlabeled case, an optimal edit path may require contracting “good” edges, i.e. edges shared between the two trees. Conclusions We provide a 2-approximation algorithm which is shown to perform well empirically. Looking ahead, computing distances between labeled trees opens up a variety of new algorithmic directions.Implementation and simulations available at https://github.com/DessimozLab/pylabeledrf.

scholarly journals A linear-time approximation algorithm for weighted matchings in graphs

2005 ◽  
Vol 1 (1) ◽  
pp. 107-122 ◽  
Author(s):  
Doratha E. Drake Vinkemeier ◽  
Stefan Hougardy
Keyword(s):  
Approximation Algorithm ◽  
Linear Time ◽  
Time Approximation

scholarly journals Fast Fréchet Distance Between Curves with Long Edges

2019 ◽  
Vol 29 (02) ◽  
pp. 161-187
Author(s):  
Joachim Gudmundsson ◽  
Majid Mirzanezhad ◽  
Ali Mohades ◽  
Carola Wenk
Keyword(s):  
Data Structure ◽  
Approximation Algorithm ◽  
Special Class ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Fréchet Distance ◽  
Frechet Distance ◽  
Polygonal Curves

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve.

scholarly journals FINDING PLANAR REGIONS IN A TERRAIN – IN PRACTICE AND WITH A GUARANTEE

2005 ◽  
Vol 15 (04) ◽  
pp. 379-401 ◽  
Author(s):  
STEFAN FUNKE ◽  
THEOCHARIS MALAMATOS ◽  
RAHUL RAY
Keyword(s):  
Approximation Algorithm ◽  
Unit Disk ◽  
Related Problem ◽  
Linear Time ◽  
Real Data ◽  
Simple Approximation ◽  
Data Sets ◽  
Fixed Angle ◽  
Planar Regions

We consider the problem of computing large connected regions in a triangulated terrain of size n for which the normals of the triangles deviate by at most some small fixed angle. In previous work an exact near-quadratic algorithm was presented, but only a heuristic implementation with no guarantee was practicable. We present a new approximation algorithm for the problem which runs in O(n/∊2) time and—apart from giving a guarantee on the quality of the produced solution—has been implemented and shows good performance on real data sets representing fracture surfaces consisting of around half a million triangles. Further we present a simple approximation algorithm for a related problem: given a set of n points in the plane, determine the placement of the unit disk which contains most points. This algorithm runs in linear time as well.

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