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ABSTRACT: The twoguard problem asks whether two guards can walk to detect an unpredictable, moving target in a polygonal region P, no matter how fast the target moves, and if so, construct a walk schedule of the guards. For safety, two guards are required to always be mutually visible, and thus they move on the polygon boundary. In particular, a straight walk requires both guards to monotonically move on the boundary of P from beginning to end, one clockwise and the other counterclockwise.
The objective of this paper is to find an optimum straight walk such that the maximum distance between the two guards is minimized. We present an O(n2) time algorithm for optimizing this metric, where n is the number of vertices of the polygon P. Our result is obtained by investigating a number of new properties of the min–max walks and converting the problem of finding an optimum walk in the min–max metric into that of finding a shortest path between two nodes in a graph. This answers an open question posed by Icking and Klein. Theoretical Computer Science 05/2014; 532:73–79. DOI:10.1007/9783642297007_5 · 0.66 Impact Factor

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ABSTRACT: A simple polygon PP is LRLRvisible if there are two points ss, tt on the boundary of PP such that every point on the clockwise boundary of PP from ss to tt is visible from some point of the other boundary of PP from tt to ss and vice versa. We show that PP is not LRLRvisible if and only if it has kk nonredundant components such that each of them exactly intersects with k′k′ other components, where 0≤k′≤k−30≤k′≤k−3. Our characterization is obtained by investigating the structure of the considered nonredundant components and representing it by a set of directed chords of a circle. Furthermore, we develop a simple O(n)O(n) time algorithm for determining whether a given polygon with nn vertices is LRLRvisible as well as for reporting a pair or all pairs (s,t)(s,t) which admit LRLRvisibility. This greatly simplifies the existing algorithm for recognizing LRLRvisibility polygons. Also, our result can be used to simplify the existing solutions of other LRLRvisibility problems. Discrete Applied Mathematics 03/2014; 165:303–311. DOI:10.1016/j.dam.2012.10.030 · 0.80 Impact Factor

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ABSTRACT: We consider the problem of searching for a mobile intruder in a circular corridor by two mobile searchers, who hold one flashlight. A circular corridor is a polygon with one polygonal hole such that its outer and inner boundaries are mutually weakly visible. Both 1searchers always direct their flashlights at the inner boundary. The objective is to decide whether there exists a search schedule for two 1searchers to detect the intruder, no matter how fast he moves, and if so, generate a search schedule. We give a characterization of the circular corridors, which are searchable by two 1searchers. Based on our characterization, an O(nlogn) time algorithm is then presented to determine the searchability of a circular corridor, where n denotes the total number of vertices of the outer and inner boundaries. Moreover, a search schedule can be reported in time linear in its size, if it exists. Discrete Applied Mathematics 09/2011; 159(16):17931805. DOI:10.1016/j.dam.2010.10.007 · 0.80 Impact Factor

Source Available from: Haitao Wang
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ABSTRACT: Given $n$ points in a circular region $C$ in the plane, we study the problems
of moving the $n$ points to its boundary to form a regular $n$gon such that
the maximum (minmax) or the sum (minsum) of the Euclidean distances traveled
by the points is minimized. The problems have applications, e.g., in mobile
sensor barrier coverage of wireless sensor networks. The minmax problem
further has two versions: the decision version and optimization version. For
the minmax problem, we present an $O(n\log^2 n)$ time algorithm for the
decision version and an $O(n\log^3 n)$ time algorithm for the optimization
version. The previously best algorithms for the two problem versions take
$O(n^{3.5})$ time and $O(n^{3.5}\log n)$ time, respectively. For the minsum
problem, we show that a special case with all points initially lying on the
boundary of the circular region can be solved in $O(n^2)$ time, improving a
previous $O(n^4)$ time solution. For the general minsum problem, we present a
3approximation $O(n^2)$ time algorithm, improving the previous
$(1+\pi)$approximation $O(n^2)$ time algorithm. A byproduct of our techniques
is an algorithm for dynamically maintaining the maximum matching of a circular
convex bipartite graph; our algorithm can handle each vertex insertion or
deletion on the graph in $O(\log^2 n)$ time. This result is interesting in its
own right. Algorithmica 07/2011; 72(2). DOI:10.1007/9783642352614_36 · 0.79 Impact Factor

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ABSTRACT: For a given convex polyhedron P of n vertices inside a sphere Q, we study the problem of cutting P out of Q by a sequence of plane cuts. The cost of a plane cut is the area of the intersection of the plane with Q, and the objective is to find a cutting sequence that minimizes the total cost. We present three approximation solutions to this problem: an O(nlogn) time O(log 2 n)factor approximation, an O(n 1·5 logn) time O(logn)factor approximation, and an O(1)factor approximation with exponential running time. Our results significantly improve upon the previous O(n 3 ) time O(log 2 n)factor approximation solution. Frontiers in Algorithmics and Algorithmic Aspects in Information and Management  Joint International Conference, FAWAAIM 2011, Jinhua, China, May 2831, 2011. Proceedings; 01/2011

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ABSTRACT: A simple polygon P is LRvisible if there are two points s, t on the boundary of P such that every point on the clockwise boundary of P from s to t is visible from some point of the other boundary of P from t to s and visa versa. In this paper, we give a simple, explict characterization of LRvisibility polygons. It is obtained by mapping the structure of nonredundant components used in determining LRvisibility into a set of directed chords of a circle. Using our characterization, we further develop a simple O(n) time algorithm for determining whether a given polygon is LRvisible. This greatly simplifies the existing algorithms for determining whether a simple polygon is LRvisible and for reporting all pairs s and t which admit LRvisibility as well. Computational Geometry, Graphs and Applications  9th International Conference, CGGA 2010, Dalian, China, November 36, 2010, Revised Selected Papers; 01/2010

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ABSTRACT: Monitoring and surveillance are important aspects in modern wireless sensor networks. In applications of wireless sensor networks,
it often asks for the sensors to quickly move from the interior of a specified region to the region’s perimeter, so as to
form a barrier coverage of the region. The region is usually given as a simple polygon or even a circle. In comparison with
the traditional concept of full area coverage, barrier coverage requires fewer sensors for detecting intruders, and can thus
be considered as a good approximation of full area coverage.
In this paper, we present an O(n
2.5 logn) time algorithm for moving n sensors to the perimeter of the given circle such that the new positions of sensors form a regular ngon and the maximum of the distances travelled by mobile sensors is minimized. This greatly improves upon the previous time
bound O(n
3.5 logn). Also, we describe an O(n
4) time algorithm for moving n sensors, whose initial positions are on the perimeter of the circle, to form a regular ngon such that the sum of the travelled distances is minimized. This solves an open problem posed in [2]. Moreover, our algorithms
are simpler and have more explicit geometric flavor. Frontiers in Algorithmics, 4th International Workshop, FAW 2010, Wuhan, China, August 1113, 2010. Proceedings; 01/2010

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ABSTRACT: A polygon P admits a sweep if two mobile guards can detect an unpredictable, moving target inside P, no matter how fast the target moves. For safety, two guards are required to always be mutually visible, and thus, they should
move on the polygon boundary. Our objective in this paper is to find an optimum sweep such that the sum of the distances travelled
by the two guards in the sweep is minimized. We present an O(n
2) time and O(n) space algorithm, where n is the number of vertices of the given polygon. This new result is obtained by converting the problem of sweeping simple
polygons with two guards into that of finding a shortest path between two nodes in a graph of size O(n). Frontiers in Algorithmics, 4th International Workshop, FAW 2010, Wuhan, China, August 1113, 2010. Proceedings; 01/2010

Source Available from: sciencedirect.com
Xuehou Tan ·
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ABSTRACT: Given a simple polygon P with two vertices u and v, the threeguard problem asks whether three guards can move from u to v such that the first and third guards are separately on two boundary chains of P from u to v and the second guard is always kept to be visible from two other guards inside P. It is a generalization of the wellknown twoguard problem, in which two guards move on the boundary chains from u to v and are always kept to be mutually visible. In this paper, we introduce the concept of link2ray shots, which can be considered as ray shots under the notion of link2visibility. Then, we show a onetoone correspondence between the structure of the restrictions placed on the motion of two guards and the one placed on the motion of three guards, and generalize the solution for the twoguard problem to that for the threeguard problem. We can decide whether there exists a solution for the threeguard problem in O(nlogn) time, and if so generate a walk in O(nlogn+m) time, where n denotes the number of vertices of P and the size of the optimal walk. This improves upon the previous time bounds O(n2) and O(n2logn), respectively. Moreover, our results can be used to solve other more sophisticated geometric problems. Discrete Applied Mathematics 10/2008; 156(17156):33123324. DOI:10.1016/j.dam.2008.05.007 · 0.80 Impact Factor

Source Available from: sciencedirect.com
Xuehou Tan ·
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ABSTRACT: We study the problem of searching for a mobile intruder in a polygonal region P with a door d (called a room) by a mobile searcher. The objective is to decide whether there exists a search schedule for the searcher to detect the intruder without allowing him to exit P through d, no matter how fast he moves, and if so, generate a search schedule. A searcher is called the ksearcher if he holds k flashlights and can see only along the rays of the flashlights emanating from his position, or two guards if two endpoints of the 1searcher's flashlight move on the polygon boundary continuously.In this paper, we develop a simple, unified solution to the room search problem. The characterizations of the ksearchable and twoguard walkable rooms are all given in terms of components and deadlocks. A study on the structure of nonredundant components and deadlocks gives critical visibility events which occur in any search schedule, and a vertex of P at which our search schedule ends. Our characterizations are not only simple but also lead to efficient algorithms for all decision problems and schedule reporting problems. Particularly, we present optimal O(n) time algorithms for determining the 1searchability and the twoguard walkability of a room, and an O(nlogn+m) time and O(n) space algorithm for generating a search schedule, if it exists, where n is the number of vertices of P and m(⩽n2) is the number of search instructions reported. Computational Geometry 05/2008; 40(140):4560. DOI:10.1016/j.comgeo.2007.04.001 · 0.48 Impact Factor

Source Available from: sciencedirect.com
Xuehou Tan ·
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ABSTRACT: Given a simple polygon P of n vertices, the watchman route problem asks for a shortest (closed) route inside P such that each point in the interior of P can be seen from at least one point along the route. In this paper, we present a simple, lineartime algorithm for computing a watchman route of length at most two times that of the shortest watchman route. The best known algorithm for computing a shortest watchman route takes O(n4logn) time, which is too complicated to be suitable in practice.This paper also involves an optimal O(n) time algorithm for computing the set of socalled essential cuts, which are the line segments inside the polygon P such that any route visiting them is a watchman route. It solves an intriguing open problem by improving the previous O(nlogn) time result, and is thus of interest in its own right. Theoretical Computer Science 09/2007; 384(1384):92103. DOI:10.1016/j.tcs.2007.05.021 · 0.66 Impact Factor

Xuehou Tan ·
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ABSTRACT: We study the problem of detecting a moving target using a group of k+1 (k is a positive integer) mobile guards inside a simple polygon. Our guards always form a simple polygonal chain within the polygon such that consecutive guards along the chain are mutually visible. In this paper, we introduce the notion of the linkk diagram of a polygon, which records the pairs of points on the polygon boundary such that the link distance between any of these pairs is at most k and a transition relation among minimumlink (⩽k) paths as well. An O(n2) time algorithm is then presented to compute the minimum number r* of guards required to detect the target, no matter how fast the target moves. Moreover, a sweep schedule can be reported in O(r*n2) time. Our results improve upon the known time bounds by a linear factor. Information Processing Letters 04/2007; 102(2102):6671. DOI:10.1016/j.ipl.2006.11.010 · 0.55 Impact Factor

Xuehou Tan ·
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ABSTRACT: Let P be a simple polygon, and let P be a set of disjoint convex polygons inside P, each sharing one edge with P. The zookeeper's route problem asks for a shortest route inside P that visits (but does not enter) each polygon in P. We present an O(n) time algorithm for computing a zookeeper's route of length at most 2 times that of the shortest zookeeper's route. Our result improves upon the previous approximation factor 6. Information Processing Letters 12/2006; 100(5):183187. DOI:10.1016/j.ipl.2006.06.005 · 0.55 Impact Factor

Xuehou Tan ·
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ABSTRACT: Given a simple polygon P of n vertices, the watchman route problem asks for a shortest (closed) route inside P such that each point in the interior of P can be seen from at least one point along the route. We present a simple, lineartime algorithm for computing a watchman
route of length at most 2 times that of the shortest watchman route. The best known algorithm for computing a shortest watchman
route takes O(n
4 log n) time, which is too complicated to be suitable in practice.
This paper also involves an optimal O(n) time algorithm for computing the set of socalled essential cuts, which are the line segments inside the polygon P such that any route visiting them is a watchman route. It solves an intriguing open problem by improving the previous O(n log n) time result, and is thus of interest in its own right. 05/2006: pages 181191;

Source Available from: ataserver.csie.ntu.edu.tw
Xuehou Tan ·
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ABSTRACT: Given a simple polygon P with n vertices and a starting point s on its boundary, the watchman route problem asks for a shortest route in P through s such that each point in the interior of the polygon can be seen from at least one point along the route. In this paper, we present a simple, lineartime algorithm for computing a watchman route of length at most times that of the shortest watchman route. The best known algorithm for computing the shortest watchman route through s takes O(n4) time. In addition, it is too complicated to be suitable in practice. Moreover, our approximation scheme can be applied to the zookeeper’s problem, which is a variant of the watchman route problem. Discrete Applied Mathematics 02/2004; 136(23136):363376. DOI:10.1016/S0166218X(03)004517 · 0.80 Impact Factor

Xuehou Tan ·
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ABSTRACT: Given a simple polygon P with two vertices u and v, the twoguard problem asks if two guards can move on the boundary chains of P from u to v, one clockwise and one counterclockwise, such that they are mutually visible. By a close study of the structure of the restrictions placed on the motion of two guards, we present a simpler solution to the twoguard problem. The main goal of this paper is to extend the solution for the twoguard problem to that for the threeguard problem, in which the first and third guards move on the boundary chains of P from u to v and the second guard is always kept to be visible from them inside P. By introducing the concept of link2ray shots, we show a onetoone correspondence between the structure of the restrictions placed on the motion of two guards and the one placed on the motion of three guards. We can decide if there exists a solution for the threeguard problem in O(n log n) time, and if so generate a walk in O(n log n + m) time, where n denotes the number of vertices of P and m ( n2) the size of the optimal walk. Algorithms and Computation, 15th International Symposium, ISAAC 2004, Hong Kong, China, December 2022, 2004, Proceedings; 01/2004

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ABSTRACT: Let P be a simple polygon, and let P be a set of disjoint convex polygons inside P, each sharing one edge with P. The safari route problem asks for a shortest route inside P that visits each polygon in P. In this paper, we first present a dynamic programming algorithm with running time O(n3) for computing the shortest safari route in the case that a starting point on the route is given, where n is the total number of vertices of P and polygons in P. (Ntafos in [Comput. Geom. 1 (1992) 149–170] claimed a more efficient solution, but as shown in Appendix A of this paper, the time analysis of Ntafos' algorithm is erroneous and no time bound is guaranteed for his algorithm.) The restriction of giving a starting point is then removed by a bruteforce algorithm, which requires O(n4) time. The solution of the safari route problem finds applications in watchman routes under limited visibility. Information Processing Letters 08/2003; 87(4):179186. DOI:10.1016/S00200190(03)002849 · 0.55 Impact Factor

Xuehou Tan ·
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ABSTRACT: We consider the problem of searching for a moving target with unbounded speed in a dark polygonal region by a searcher. The
searcher continuously moves on the polygon boundary and can see only along the rays of the flashlights emanating from his
position at a time. We present necessary and sufficient conditions for a polygon of n vertices to be searchable from the boundary. Our two main results are the following:
1
We present an O(n log n) time and O(n) space algorithm for testing the searchability of simple polygons. Moreover, a search schedule can be reported in time linear
in its size I, if it exists. For the searcher having full 360° vision, I n, and for the searcher having only one flashlight, I n
2. Our result improves upon the previous O(n
2) time and space solution, given by LaValle et al [5]. Also, the linear bound for the searcher having full 360° vision solves
an open problem posed by Suzuki et al [7].
2
We show the equivalence of the abilities of the searcher having only one flashlight and the one having full 360° vision. Although
the same result has been obtained by Suzuki et al [7], their proof is long and complicated, due to lack of the characterization
of boundary search. Combinatorial Geometry and Graph Theory, IndonesiaJapan Joint Conference,IJCCGGT 2003, Bandung, Indonesia, September 1316, 2003, Revised Selected Papers; 01/2003

Xuehou Tan ·
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ABSTRACT: The watchman route problem deals with finding a shortest route in a simple polygon of n vertices such that each point in the interior of the polygon can be seen from at least one point along the route. In this paper, we show that the shortest watchman route in a simple polygon is unique, except for very special cases where there is an infinite number of shortest routes of equal length, and present an O(n5) time solution to the watchman route problem. Our result improves upon the previous O(n6) time bound. Information Processing Letters 01/2001; 77(177):2733. DOI:10.1016/S00200190(00)001460 · 0.55 Impact Factor

Xuehou Tan ·
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ABSTRACT: Let P be a simple polygon, and let P be a set of disjoint convex polygons inside P, each sharing one edge with P. The zookeeper's route problem asks for a shortest route inside P that visits (but does not enter) each polygon in P. We present an O(n2) time algorithm for computing a shortest zookeeper's route, on which no starting point is specified. Information Processing Letters 01/2001; 77(1):2326. DOI:10.1016/S00200190(00)001447 · 0.55 Impact Factor