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### Finding the shortest boundary guard of a simple polygon

Bor-Kuan Lu, Fang-Rong Hsu, Chuan Yi Tang
2001 Theoretical Computer Science
We propose an algorithm to ÿnd the shortest boundary guard of a simple polygon P in O(n log n) time, where n is the number of vertices of P. c 2001 Published by Elsevier Science B.V.  ...  In this paper, we consider a new type of guard, boundary guard, which is a guard capable of moving along a boundary of a polygon and every interior point of the polygon can be seen by the mobile guard.  ...  There always exists a boundary guard of any simple polygon; in worst cases, a mobile guard traverse the whole boundary of a simple polygon. The paper is organized into six sections.  ...

### Page 5221 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews
We propose an algorithm to find the shortest boundary guard of a simple polygon 7 in O(nlogn) time, where n is the number of vertices of F.” 2002g:68156 68U10 94A08 94A24 Akman, Emir Tufan (TR-ISTUN-EL  ...  Martin Henk (A-TUWN-AN; Vienna) 2002g:68155 68U05 52B55 68T40 Lu, Bor-Kuan (RC-NTHU-CS; Hsinchu); Hsu, Fang-Rong ; Tang, Chuan Yi (RC-NTHU-CS; Hsinchu) Finding the shortest boundary guard of a simple polygon  ...

### A Linear-time Algorithm for Orthogonal Watchman Route Problem with Minimum Bends [article]

Hamid Hoorfar, Alireza Bagheri
2017 arXiv   pre-print
In the other words, in the watchman route problem we must compute a shortest watchman route inside a simple polygon of n vertices such that all the points interior to the polygon and on its boundary are  ...  Given an orthogonal polygon P with n vertices, the goal of the watchman route problem is finding a path S of the minimum length in P such that every point of the polygon P is visible from at least one  ...  The layout of the area is represented usually by a simple polygon and the objective is to find a shortest closed curve such that all the points within the polygon and on its boundary are visible to at  ...

### Open Guard Edges and Edge Guards in Simple Polygons [chapter]

Csaba D. Tóth, Godfried T. Toussaint, Andrew Winslow
2012 Lecture Notes in Computer Science
An open edge of a simple polygon is the set of points in the relative interior of an edge.  ...  We characterize open guard edges, and derive an algorithm that finds all open guard edges of a simple n-gon in O(n) time in the RAM model of computation.  ...  So all open edge guards are on the boundary of the outer polygon. By the result in [8] there are at most 3 closed guard edges on the outer boundary of a polygon with holes.  ...

### Review of visibility algorithms in the plane by Subir Kumar Ghosh (Cambridge University Press, 2007)

Alice M. Dean
2009 ACM SIGACT News
Posed by Victor Klee in 1973, it asks how many stationary guards are required to see all points in an art gallery represented by a simple, n-sided polygon.  ...  Questions such as this one, of visibility within a polygon, are the subject of Visibility Algorithms in the Plane, by S. Ghosh.  ...  This query algorithm is presented, as well as query algorithms for the ray-shooting problem (given q ∈ P , find the point on the boundary of P closest to q), Euclidean shortest path, and minimum link path  ...

### A Note on Edge Guards in Art Galleries [article]

R. Nandakumar
2012 arXiv   pre-print
We present a heuristic algorithm to arrange edge guards to guard only the inward side of the walls of any N-vertex simple polygonal gallery using at most roof (N/4) edge guards - a weakened version of  ...  Toussaint's conjecture on the number of edge guards that can guard an entire simple polygon.  ...  Algorithm Our approach to arranging edge guards to guard the boundary of the simple polygon is: Add edge guards incrementally such that after the r'th guard has been added, at least 4r edges of the polygon  ...

### Shortest paths in simple polygons with polygon-meet constraints

2004 Information Processing Letters
We study a constrained version of the shortest path problem in simple polygons, in which the path must visit a given target polygon.  ...  We provide a worst-case optimal algorithm for this problem and also present a method to construct a subdivision of the simple polygon to efficiently answer queries to retrieve the shortest polygon-meeting  ...  Acknowledgement We would like to thank the anonymous referee for his/her valuable remarks.  ...

### New Similarity Measures between Polylines with Applications to Morphing and Polygon Sweeping

Efrat, Guibas, Sariel Har-Peled, Mitchell, Murali
2002 Discrete & Computational Geometry
Our computation of these metrics relies on two closely-related combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon.  ...  The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon.  ...  Acknowledgments The authors wish to thank Pankaj Agarwal, Helmut Alt, Danny Halperin and Micha Sharir for helpful discussions concerning the problems studied in this paper.  ...

### k-Transmitter Watchman Routes [article]

Bengt J. Nilsson, Christiane Schmidt
2022 arXiv   pre-print
We show that even in simple polygons the shortest k-transmitter watchman route problem for a discrete set of points S⊂ P is NP-complete and cannot be approximated to within a logarithmic factor (unless  ...  We consider the watchman route problem for a k-transmitter watchman: standing at point p in a polygon P, the watchman can see q∈ P if pq intersects P's boundary at most k times – q is k-visible to p.  ...  Special Observations for k-Transmitters The first observation is that, while for 0-transmitter watchmen guarding a simple polygon's boundary is enough to guard all of the polygon, this does not hold for  ...

### Page 579 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews
Summary: “Given an n-vertex simple polygon P, the problem of computing the shortest weakly visible subedge of P is that of find- ing a shortest line segment s on the boundary of P such that P is weakly  ...  visible subedge of a simple polygon.  ...

### Watchman routes under limited visibility

Simeon Ntafos
1992 Computational geometry
If we only want to see the boundary of the polygon we have the d-watchman problem which can be stated as follows: Given a simple polygon P and a point s on its boundary, find a shortest route in the interior  ...  Let s be a point on the boundary of P. The safari route problem is to find a shortest route in the interior of the polygon from s back to s and such that the route visits each polygon in P'.  ...

### Randomized planning for short inspection paths

T. Danner, L.E. Kavraki
2000 Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)
This paper addresses the following inspection problem: given a known workspace and a robot with vision capabilities compute a short path path for the robot such that each point on boundary of the workspace  ...  Autonomous inspection, such as by a flying camera, or a virtual reality architectural walkthrough, could be guided by a solution to the above inspection problem.  ...  Murali, Jean-Claude Latombe, and the members of our research group, the Physical Computing Group at Rice, for their comments.  ...

### Pursuit-evasion in a two-dimensional domain

Andrew Beveridge, Yiqing Cai
2017 Ars Mathematica Contemporanea
In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game.  ...  Second, we consider the game played in a compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Euler characteristic.  ...  We thank the anonymous referee for helping us to improve the exposition.  ...

### Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary [article]

Andrew Beveridge, Yiqing Cai
2015 arXiv   pre-print
In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game.  ...  Second, we consider the game played in a compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Euler characteristic.  ...  Acknowledgments This work was completed while the first author was a long-term visitor at the Institute for Mathematics and its Applications.  ...

### Capturing an evader in polygonal environments with obstacles: The full visibility case

Deepak Bhadauria, Kyle Klein, Volkan Isler, Subhash Suri
2012 The international journal of robotics research
The bound is independent of the number of vertices or holes in the polygonal environment.  ...  Suppose an unpredictable evader is free to move around in a polygonal environment of arbitrary complexity that is under full camera surveillance.  ...  ., the new polygon is a subset of the previous one. 3. the paths guarded by the pursuers forming the boundary of P i+1 are both the shortest paths in P i+1 .  ...
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