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Publications (3)0 Total impact

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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 (min-max) or the sum (min-sum) 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 min-max problem further has two versions: the decision version and optimization version. For the min-max 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 min-sum 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 min-sum problem, we present a 3-approximation $O(n^2)$ time algorithm, improving the previous $(1+\pi)$-approximation $O(n^2)$ time algorithm. A by-product 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.
    07/2011;
  • Xuehou Tan, Gangshan Wu
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    ABSTRACT: This paper presents the following approximation algorithms for computing a minimum cost sequence of planes to cut a convex polyhedron P of n vertices out of a sphere Q: an O(n logn) time O(log2 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(log2 n)-factor approximation solution.
    Frontiers in Algorithmics and Algorithmic Aspects in Information and Management - Joint International Conference, FAW-AAIM 2011, Jinhua, China, May 28-31, 2011. Proceedings; 01/2011
  • Xuehou Tan, Gangshan Wu
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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 n-gon 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 n-gon 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 11-13, 2010. Proceedings; 01/2010