GUARD PLACEMENT FOR MAXIMIZING L-VISIBILITY EXTERIOR TO A CONVEX POLYGON

2009 ◽  
Vol 19 (04) ◽  
pp. 357-370
Author(s):  
DEBABRATA BARDHAN ◽  
SANSANKA ROY ◽  
SANDIP DAS
Keyword(s):  
Shortest Path ◽  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Placement Problem ◽  
Maximum Area

Two points a and b are said to be L-visible among a set of polygonal obstacles if the length of the shortest path from a to b avoiding these obstacles is no more than L. For a given convex polygon P with n vertices, Gewali et al.1 addressed the guard placement problem on the boundary of P that covers the maximum area outside to the polygon under L-visibility with P as obstacle. Their proposed algorithm runs in O(n) time if [Formula: see text], where π(P) denotes the perimeter of P. They conjectured that if [Formula: see text], then the problem can be solved in subquadratic time. In this paper, we settle the conjecture in the affirmative sense, by proposing an easy to implement linear time algorithm for any arbitrary value of L.

AN APPROXIMATE MORPHING BETWEEN POLYLINES

2005 ◽  
Vol 15 (02) ◽  
pp. 193-208 ◽  
Author(s):  
SERGEY BEREG
Keyword(s):  
Shortest Path ◽  
Optimization Problem ◽  
Linear Time ◽  
Medial Axis ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Geodesic Path ◽  
Minimum Width ◽  
Approximation Factor ◽  
Previous Algorithm

We consider the problem of continuously transforming or morphing one simple polyline into another polyline so that every point p of the initial polyline moves to a point q of the final polyline using the geodesic shortest path from p to q. The width of a morphing is defined as the longest geodesic path between corresponding points of the polylines. The optimization problem is to compute a morphing that minimizes the width. We present a linear-time algorithm for finding a morphing with width guaranteed to be at most two times the minimum width of a morphing. This improves the previous algorithm10 by a factor of log n. We develop a linear-time algorithm for computing a medial axis separator. We also show that the approximation factor is less than two for κ-straight polylines.

scholarly journals On the Maximal Shortest Paths Cover Number

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1592
Author(s):  
Iztok Peterin ◽  
Gabriel Semanišin
Keyword(s):  
Shortest Path ◽  
Linear Time ◽  
Shortest Paths ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Np Hard ◽  
Minimum Cardinality ◽  
Graph Parameters

A shortest path P of a graph G is maximal if P is not contained as a subpath in any other shortest path. A set S⊆V(G) is a maximal shortest paths cover if every maximal shortest path of G contains a vertex of S. The minimum cardinality of a maximal shortest paths cover is called the maximal shortest paths cover number and is denoted by ξ(G). We show that it is NP-hard to determine ξ(G). We establish a connection between ξ(G) and several other graph parameters. We present a linear time algorithm that computes exact value for ξ(T) of a tree T.

A PARALLEL ALGORITHM FOR ENCLOSED AND ENCLOSING TRIANGLES

1992 ◽  
Vol 02 (02) ◽  
pp. 191-214 ◽  
Author(s):  
SHARAT CHANDRAN ◽  
DAVID M. MOUNT
Keyword(s):  
Parallel Algorithms ◽  
Parallel Algorithm ◽  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
New Techniques ◽  
Crcw Pram

We consider the problems of computing the largest area triangle enclosed within a given n-sided convex polygon and the smallest area triangle which encloses a given convex polygon. We show that these problems are closely related by presenting a single sequential linear time algorithm which essentially solves both problems simultaneously. We also present a cost-optimal parallel algorithm that solves both of these problems in O( log log n) time using n/ log log n processors on a CRCW PRAM. In order to achieve these bounds we develop new techniques for the design of parallel algorithms for computational problems involving the rotating calipers method.

QUADRANGULAR REFINEMENTS OF CONVEX POLYGONS WITH AN APPLICATION TO FINITE-ELEMENT MESHES

2000 ◽  
Vol 10 (01) ◽  
pp. 1-40 ◽  
Author(s):  
MATTHIAS MÜLLER-HANNEMANN ◽  
KARSTEN WEIHE
Keyword(s):  
Finite Element ◽  
Convex Polygon ◽  
Linear Time ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Convex Polygons ◽  
Minimum Number ◽  
Three Dimensional Space

We present a linear–time algorithm that decomposes a convex polygon conformally into a minimum number of strictly convex quadrilaterals. Moreover, we characterize the polygons that can be decomposed without additional vertices inside the polygon, and we present a linear–time algorithm for such decompositions, too. As an application, we consider the problem of constructing a minimum conformal refinement of a mesh in the three–dimensional space, which approximates the surface of a workpiece. We prove that this problem is strongly [Formula: see text] –hard, and we present a linear-time algorithm with a constant approximation ratio of four.

AN OPTIMAL DATA STRUCTURE FOR SHORTEST RECTILINEAR PATH QUERIES IN A SIMPLE RECTILINEAR POLYGON

1996 ◽  
Vol 06 (02) ◽  
pp. 205-225 ◽  
Author(s):  
SVEN SCHUIERER
Keyword(s):  
Data Structure ◽  
Shortest Path ◽  
Linear Time ◽  
Time Algorithm ◽  
Query Time ◽  
Linear Time Algorithm ◽  
Link Length ◽  
N Storage ◽  
Path Queries ◽  
Rectilinear Polygon

We present a data structure that allows to preprocess a rectilinear polygon with n vertices such that, for any two query points, the shortest path in the rectilinear link or L1-metric can be reported in time O( log n+k) where k is the link length of the shortest path. If only the distance is of interest, the query time reduces to O( log n). Furthermore, if the query points are two vertices, the distance can be reported in time O(1) and a shortest path can be constructed in time O(1+k). The data structure can be computed in time O(n) and needs O(n) storage. As an application we present a linear time algorithm to compute the diameter of a simple rectilinear polygon w.r.t. the L1-metric.

COMPUTING A SHORTEST WEAKLY EXTERNALLY VISIBLE LINE SEGMENT FOR A SIMPLE POLYGON

1999 ◽  
Vol 09 (01) ◽  
pp. 81-96 ◽  
Author(s):  
BINAY K. BHATTACHARYA ◽  
ASISH MUKHOPADHYAY ◽  
GODFRIED T. TOUSSAINT
Keyword(s):  
Line Segment ◽  
Convex Polygon ◽  
Linear Time ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Suitable Generalization

A simple polygon P is said to be weakly extrenally visible from a line segment L if it lies outside P and for every point p on the boundary of P there is a point q on L such that no point in the interior of [Formula: see text] lies inside P. In this paper, a linear time algorithm is proposed for computing a shortest line segment from which P is weakly externally visible. This is done by a suitable generalization of a linear time algorithm which solves the same problem for a convex polygon.

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

1989 ◽  
Vol 4 (6) ◽  
pp. 591-604 ◽  
Author(s):  
Alok Aggarwal ◽  
Leonidas J. Guibas ◽  
James Saxe ◽  
Peter W. Shor
Keyword(s):  
Voronoi Diagram ◽  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm
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