Algorithms for Covering Barrier Points by Mobile Sensors with Line Constraint

Princy Jain, Haitao Wang
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Abstract

We study the problem of covering barrier points by mobile sensors. Each sensor is represented by a point in the plane with the same covering range [Formula: see text] so that any point within distance [Formula: see text] from the sensor can be covered by the sensor. Given a set [Formula: see text] of [Formula: see text] points (called “barrier points”) and a set [Formula: see text] of [Formula: see text] points (representing the “sensors”) in the plane, the problem is to move the sensors so that each barrier point is covered by at least one sensor and the maximum movement of all sensors is minimized. The problem is NP-hard. In this paper, we consider two line-constrained variations of the problem and present efficient algorithms that improve the previous work. In the first problem, all sensors are given on a line [Formula: see text] and are required to move on [Formula: see text] only while the barrier points can be anywhere in the plane. We propose an [Formula: see text] time algorithm for the problem. We also consider the weighted case where each sensor has a weight; we give an [Formula: see text] time algorithm for this case. In the second problem, all barrier points are on [Formula: see text] while all sensors are in the plane but are required to move onto [Formula: see text] to cover all barrier points. We also solve the weighted case in [Formula: see text] time.
移动传感器覆盖障碍点的线性约束算法
我们研究的是移动传感器覆盖障碍点的问题。每个传感器由平面上的一个点表示,该点的覆盖范围[公式:见正文]相同,因此传感器可以覆盖距离[公式:见正文]内的任何一点。给定平面上一组[公式:见正文][公式:见正文]点(称为 "障碍点")和一组[公式:见正文][公式:见正文]点(代表 "传感器"),问题是移动传感器,使每个障碍点至少被一个传感器覆盖,并使所有传感器的最大移动量最小。该问题具有 NP 难度。在本文中,我们考虑了该问题的两个线性约束变体,并提出了改进前人工作的高效算法。在第一个问题中,所有传感器都给定在一条直线上[计算公式:见正文],并且只要求在[计算公式:见正文]上移动,而障碍点可以是平面上的任何位置。我们为这个问题提出了一种[公式:见正文]时间算法。我们还考虑了加权情况,即每个传感器都有一个权重;我们给出了这种情况下的[公式:见正文]时间算法。在第二个问题中,所有障碍点都在[公式:见正文]上,而所有传感器都在平面上,但需要移动到[公式:见正文]上以覆盖所有障碍点。我们也可以在[公式:见正文]时间内解决加权情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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