We introduce and study a new problem concerning the exploration of a geometric domain by mobile robots. Consider a line segment [0, I] and a set of n mobile robots r1, r2, . . . , rn placed at one of its endpoints. Each robot has a searching speed si and a walking speed wi, where si < wi. We assume that each robot is aware of the number of robots of the collection and their corresponding speeds. At each time moment a robot ri either walks along a portion of the segment not exceeding its walking speed wi or searches a portion of the segment with the speed not exceeding si. A search of segment [0, I] is completed at the time when each of its points have been searched by at least one of the n robots. We want to develop mobility schedules (algorithms) for the robots which complete the search of the segment as fast as possible. More exactly we want to maximize the speed of the mobility schedule (equal to the ratio of the segment length versus the time of the completion of the schedule). We analyze first the offline scenario when the robots know the length of the segment that is to be searched. We give an algorithm producing a mobility schedule for arbitrary walking and searching speeds and prove its optimality. Then we propose an online algorithm, when the robots do not know in advance the actual length of the segment to be searched. The speed S of such algorithm is defined aswhere S(IL) denotes the speed of searching of segment IL = [0, L]. We prove that the proposed online algorithm is 2-competitive. The competitive ratio is shown to be better in the case when the robots' walking speeds are all the same.

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We consider the following Antenna Orientation Problem: Given a connected Unit Disk Graph (UDG) formed by n identical omnidirectional sensors, what is the optimal range (or radius) which is necessary and sufficient for a given antenna beamwidth (or angle) φ so that after replacing the omnidirectional sensors by directional antennas of beamwidth φ it is possible to find an appropriate orientation of each antenna so that the resulting graph is strongly connected? In this paper we study beamwidth/range tradeoffs for the Antenna Orientation Problem.Namely, for the full range of angles in the interval [0, 2π ] we compare the antenna range provided by an orientation algorithm to the optimal possible for the given beamwidth. We propose new antenna orientation algorithms that ensure improved bounds for given angle ranges and analyze their complexity. We also examine the Antenna Orientation Problem with Constant Stretch Factor, where we wish to optimize both the transmission range and the hop-stretch factor of the induced communication network. We present approximations to this problem for antennas with angles π /2 ≤ φ ≤ 2π .

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