Two-Guard Walkability of Simple Polygons

1998 ◽  
Vol 08 (01) ◽  
pp. 85-116 ◽  
Author(s):  
L. H. Tseng ◽  
P. Heffernan ◽  
D. T. Lee
Keyword(s):  
Linear Space ◽  
Simple Polygon ◽  
The Other ◽  
Simple Polygons

A pair of points s and g on the boundary of a simple polygon P admits a walk if two guards can simultaneously walk along the two boundary chains of P from s to g such that they are always visible to each other. The walk is a counter-walk if one guard moves from s to g while the other moves from g to s in the same direction along the boundary and they are always visible to each other. The (counter-)walk is straight if no backtracking is necessary during the (counter-)walk. In this paper, we show that, given a polygon with n vertices, to test if there exists (s,g) that admits a (straight) (counter-)walk can be solved in time O(n log n) and in linear space. Also we compute all (s,g)'s that admit a (straight) walk in O(n log n) time and all vertex pairs that admit a (straight) counter-walk in O(n log n + m), where m is O(n2).

scholarly journals Fixed points of quasi-nonexpansive mappings

1972 ◽  
Vol 13 (2) ◽  
pp. 167-170 ◽  
Author(s):  
W. G. Dotson
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Linear Space ◽  
Complex Plane ◽  
Normed Linear Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
The Other ◽  
Commuting Mappings ◽  
Analytic Mappings

A self-mapping T of a subset C of a normed linear space is said to be non-expansive provided ║Tx — Ty║ ≦ ║x – y║ holds for all x, y ∈ C. There has been a number of recent results on common fixed points of commutative families of nonexpansive mappings in Banach spaces, for example see DeMarr [6], Browder [3], and Belluce and Kirk [1], [2]. There have also been several recent results concerning common fixed points of two commuting mappings, one of which satisfies some condition like nonexpansiveness while the other is only continuous, for example see DeMarr [5], Jungck [8], Singh [11], [12], and Cano [4]. These results, with the exception of Cano's, have been confined to mappings from the reals to the reals. Some recent results on common fixed points of commuting analytic mappings in the complex plane have also been obtained, for example see Singh [13] and Shields [10].

THE TWO GUARDS PROBLEM

1992 ◽  
Vol 02 (03) ◽  
pp. 257-285 ◽  
Author(s):  
CHRISTIAN ICKING ◽  
ROLF KLEIN
Keyword(s):  
Linear Space ◽  
Simple Polygon ◽  
Minimum Length

Given a simple polygon in the plane with two distinguished vertices, s and g, is it possible for two guards to simultaneously walk along the two boundary chains from s to g in such a way that they are always mutually visible? We decide this question in time O (n log n) and in linear space, where n is the number of edges of the polygon. Moreover, we compute a walk of minimum length within time O(n log n+k), where k is the size of the output, and we prove that this is optimal.

Determining Weak Visibility of a Polygon from an Edge in Parallel

1998 ◽  
Vol 08 (03) ◽  
pp. 277-304
Author(s):  
Danny Z. Chen
Keyword(s):  
Computational Model ◽  
Parallel Algorithm ◽  
Simple Polygon ◽  
Sequential Algorithms ◽  
Simple Polygons ◽  
Crew Pram

The problem of determining the weak visibility of an n-vertex simple polygon P from an edge e of P is that of deciding whether every point in P is weakly visible from e. In this paper we present an optimal parallel algorithm for solving this problem. Our algorithm runs in O( log n) time using O(n/ log n) processors in the CREW PRAM computational model, and is very different from the sequential algorithms for this problem. We also show how to solve optimally, in parallel, several other problems that are related to the weak visibility of simple polygons.

scholarly journals NLP Formulation for Polygon Optimization Problems

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 24 ◽  
Author(s):  
Saeed Asaeedi ◽  
Farzad Didehvar ◽  
Ali Mohades
Keyword(s):  
Genetic Algorithm ◽  
Optimization Problems ◽  
Theoretical Perspective ◽  
Simple Polygon ◽  
Programming Models ◽  
Minimum Area ◽  
Simple Polygons ◽  
Maximum Angle ◽  
Optimal Approach ◽  
Set Of Points

In this paper, we generalize the problems of finding simple polygons with minimum area, maximum perimeter, and maximum number of vertices, so that they contain a given set of points and their angles are bounded by α + π where α ( 0 ≤ α ≤ π ) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate these three generalized problems as nonlinear programming models, and then present a genetic algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.

PARALLEL COMPUTATION OF INTERNAL AND EXTERNAL FARTHEST NEIGHBORS IN SIMPLE POLYGONS

1992 ◽  
Vol 02 (02) ◽  
pp. 175-190 ◽  
Author(s):  
SUMANTA GUHA
Keyword(s):  
Parallel Algorithms ◽  
Parallel Computation ◽  
Shortest Path ◽  
Simple Polygon ◽  
Divide And Conquer ◽  
Simple Polygons ◽  
Crew Pram ◽  
Serial Algorithm

We present efficient parallel algorithms for two problems in simple polygons: the all-farthest neighbors problem and the external all-farthest neighbors problem. The all-farthest neighbors problem is that of computing, for each vertex p of a simple polygon P, a point ψ(p) in P farthest from p when the distance between p and ψ(p) is measured by the shortest path between them constrained to lie inside P. The external all-farthest neighbors problem is that of computing, for each vertex p of a simple polygon P, a point ϕ(p) on (the boundary of) P farthest from p when the distance between p and ϕ(p) is measured by the shortest path between them constrained to lie outside (the interior of) P. Both our algorithms run in O( log 2 n) time on a CREW PRAM with O(n) processors. Our divide-and-conquer method for the external all-farthest neighbors problem, in fact, leads to a new O(n log n) time serial algorithm that matches the currently best serial algorithm for this problem, but is simpler.

OPTIMAL PARALLEL ALGORITHMS FOR TRIANGULATED SIMPLE POLYGONS

1995 ◽  
Vol 05 (01n02) ◽  
pp. 145-170 ◽  
Author(s):  
JOHN HERSHBERGER
Keyword(s):  
Convex Hull ◽  
Parallel Algorithms ◽  
Shortest Path ◽  
Simple Polygon ◽  
Implicit Representation ◽  
Shortest Path Tree ◽  
Simple Polygons ◽  
Pram Model ◽  
Crew Pram ◽  
Path Queries

We provide optimal parallel solutions to several shortest path and visibility problems set in triangulated simple polygons. Let P be a triangulated simple polygon with n vertices, preprocessed to support shortest path queries. We can find the shortest path tree from any point inside P in O(log n) time using O(n/log n) processors. In the game bounds, we can preprocess P for shooting queries (a query can be answered in O(log n) time by a uniprocessor). Given a set S of m points inside P, we can find an implicit representation of the relative convex hull of S in O(log(nm)) time with O(m) processors. If the relative convex hull has k edges, we can explicitly produce these edges in O(log(nm)) time with O(k/log(nm)) processors. All of these algorithms are deterministic and use the CREW PRAM model.

ON THE TIME BOUND FOR CONVEX DECOMPOSITION OF SIMPLE POLYGONS

2002 ◽  
Vol 12 (03) ◽  
pp. 181-192 ◽  
Author(s):  
MARK KEIL ◽  
JACK SNOEYINK
Keyword(s):  
Simple Polygon ◽  
Time And Space ◽  
Convex Decomposition ◽  
Simple Polygons ◽  
Minimum Number

We show that a decomposition of a simple polygon having n vertices, r of which are reflex, into a minimum number of convex regions without the addition of Steiner vertices can be computed in O(n + r2 min {r2, n}) time and space. A Java demo is available at .

scholarly journals Query-Points Visibility Constraint Minimum Link Paths in Simple Polygons

2021 ◽  
Vol 182 (3) ◽  
pp. 301-319
Author(s):  
Mohammad Reza Zarrabi ◽  
Nasrollah Moghaddam Charkari
Keyword(s):  
Shortest Path ◽  
Simple Polygon ◽  
Query Point ◽  
Simple Polygons ◽  
Link Distance ◽  
Shortest Path Map ◽  
Number Of Faces

We study the query version of constrained minimum link paths between two points inside a simple polygon P with n vertices such that there is at least one point on the path, visible from a query point. The method is based on partitioning P into a number of faces of equal link distance from a point, called a link-based shortest path map (SPM). Initially, we solve this problem for two given points s, t and a query point q. Then, the proposed solution is extended to a general case for three arbitrary query points s, t and q. In the former, we propose an algorithm with O(n) preprocessing time. Extending this approach for the latter case, we develop an algorithm with O(n3) preprocessing time. The link distance of a q-visible path between s, t as well as the path are provided in time O(log n) and O(m + log n), respectively, for the above two cases, where m is the number of links.

Infinite Doubly Stochastic Matrices

1962 ◽  
Vol 5 (1) ◽  
pp. 1-4 ◽  
Author(s):  
J.R. Isbell
Keyword(s):  
Linear Space ◽  
Convex Set ◽  
The Other ◽  
Locally Convex Spaces ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Incorrect Assertion ◽  
Doubly Stochastic Matrices ◽  
The Axiom Of Choice ◽  
Locally Convex

This note proves two propositions on infinite doubly stochastic matrices, both of which already appear in the literature: one with an unnecessarily sophisticated proof (Kendall [2]) and the other with the incorrect assertion that the proof is trivial (Isbell [l]). Both are purely algebraic; so we are, if you like, in the linear space of all real doubly infinite matrices A = (aij).Proposition 1. Every extreme point of the convex set of ail doubly stochastic matrices is a permutation matrix.Kendall's proof of this depends on an ingenious choice of a topology and the Krein-Milman theorem for general locally convex spaces [2]. The following proof depends on practically nothing: for example, not on the axiom of choice.

Min-/Max-Volume Roofs Induced by Bisector Graphs of Polygonal Footprints of Buildings

2018 ◽  
Vol 28 (04) ◽  
pp. 309-340
Author(s):  
Günther Eder ◽  
Martin Held ◽  
Peter Palfrader
Keyword(s):  
Piecewise Linear ◽  
Simple Polygon ◽  
Maximum Volume ◽  
Drain Water ◽  
Plane Sweep ◽  
Simple Polygons

Piecewise-linear terrains (“roofs”) over simple polygons were first studied by Aichholzer et al. (J. UCS 1995) in their work on straight skeletons of polygons. We show how to construct a roof over the polygonal footprint of a building that has minimum or maximum volume among all roofs that drain water. Our algorithm for computing such a roof extends the standard plane-sweep approach known from the theory of straight skeletons by additional events. For both types of roofs our algorithm runs in [Formula: see text] time for a simple polygon with [Formula: see text] vertices.

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