Abstract. We consider n mobile sensors located on a line containing a barrier represented by a finite line segment. Sensors form a wireless sensor network and are able to move within the line. An intruder traversing the barrier can be detected only when it is within the sensing range of at least one sensor. The sensor network establishes barrier coverage of the segment if no intruder can penetrate the barrier from any direction in the plane without being detected. Starting from arbitrary initial positions of sensors on the line we are interested in finding final positions of sensors that establish barrier coverage and minimize the maximum distance traversed by any sensor. We distinguish several variants of the problem, based on (a) whether or not the sensors have identical ranges, (b) whether or not complete coverage is possible and (c) in the case when complete coverage is impossible, whether or not the maximal coverage is required to be contiguous. For the case of n sensors with identical range, when complete coverage is impossible, we give linear time optimal algorithms that achieve maximal coverage, both for the contiguous and non-contiguous case. When complete coverage is possible, we give an O(n 2 ) algorithm for an optimal solution, a linear time approximation scheme with approximation factor 2, and a (1 + ) PTAS. When the sensors have unequal ranges we show that a variation of the problem is NP-complete and identify some instances which can be solved with our algorithms for sensors with unequal ranges.

Abstract. A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned.

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Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed 1. The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [9] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most 5.740 and also proved that any algorithm has evacuation time at least 3 + π 4 + √ 2 ≈ 5.199. We improve both the upper and lower bounds on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most 5.628 and show that any algorithm has evacuation time at least 3 + π 6 + √ 3 ≈ 5.255. To achieve the upper bound, we designed an algorithm which non-intuitively proposes a forced meeting between the two robots, even if the exit has not been found by either of them.We study the problem of two robots that start at the same time at the centre of a unit disk and attempt to reach an exit placed at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose within the disk. Indeed, they can take short-cuts by moving in the interior of the disk if desired. We assume their maximum speed is 1. The robots can communicate with each other only if they are at the same point at the same time: we call this communication model face-to-face communication. Our goal is to schedule the trajectories of the robots so as to minimize the evacuation time, which is the time it takes both robots to reach the exit (for the worst case location of the exit).

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