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.

POLYGON DECOMPOSITION AND THE ORTHOGONAL ART GALLERY PROBLEM

2007 ◽  
Vol 17 (02) ◽  
pp. 105-138 ◽  
Author(s):  
CHRIS WORMAN ◽  
J. MARK KEIL
Keyword(s):  
Polynomial Time ◽  
Art Gallery ◽  
Minimum Cardinality ◽  
Decomposition Problem ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition

A decomposition of a polygon P is a set of polygons whose geometric union is exactly P. We study a polygon decomposition problem that is equivalent to the Orthogonal Art Gallery problem. Two points are r-visible if the orthogonal bounding rectangle for p and q lies within P. A polygon P is an r-star if there exists a point k ∈ P such that for each point q ∈ P, q is r-visible from k. In this problem we seek a minimum cardinality decomposition of a polygon into r-stars. We show how to compute the minimum r-star cover of an orthogonal polygon in polynomial time.

OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS

1993 ◽  
Vol 03 (01) ◽  
pp. 85-105 ◽  
Author(s):  
SVANTE CARLSSON ◽  
BENGT J. NILSSON ◽  
SIMEON NTAFOS
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Np Hard ◽  
Art Gallery ◽  
Art Galleries ◽  
Art Gallery Problem ◽  
Natural Connection ◽  
Route Problem ◽  
Minimum Number

A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path or route which provide a natural connection between the art gallery problem, the m-watchmen routes problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen routes problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n3+n2m2)-time algorithm to compute the best set of m watchmen in a histogram.

Temporal Structure and the Sacred in Woolf’s To the Lighthouse

2017 ◽  
Author(s):  
Katie Gemmill
Keyword(s):  
Linear Time ◽  
Temporal Structure ◽  
Narrative Structure ◽  
Dimensional Structure ◽  
The Novel ◽  
Everyday Experience ◽  
To The Lighthouse ◽  
Complex Theory ◽  
Novel Reading

Critics frequently agree that in the “Time Passes” section of To the Lighthouse, Woolf transcends linear  time. In his article “History, Time and the Novel: reading Woolf’s To the Lighthouse”, however, Dominick LaCapra proposes a more complex theory of temporality, arguing that time has a two­  dimensional structure made up of a horizontal diachronic dimension, and a vertical synchronic dimension. The diachronic dimension comprises discrete “epochmaking events”, while the synchronic dimension  seemingly immobilizes a particular moment in defiance of linear time. Woolf’s narrative focuses on the synchronic dimension of time, thus subverting the traditional narrative structure that focuses on plot­  driving events that occur on the diachronic temporal plane. I believe that the thematic prominence of time and the sacred in “Time Passes” is not arbitrary; in fact, I argue that it is Woolf’s innovative conception of temporal structure that allows her to engage so profoundly with themes of the sacred. The synchronic dimension of time provides an escape from the limitations that linear time imposes on our experience of the sacred; in other words, the synchronic dimension is what allows Woolf seemingly to immobilize an  experience, to meditate on it in depth, and to convey more effectively the sacred nature of that experience. Throughout this section of To the Lighthouse Woolf suggests that by reframing how we exist in time, we can more readily feel the sacred that permeates everyday experience, and thus connect more intensely with  existence

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.

Sign in / Sign up

Export Citation Format

Share Document