In this paper, we first remodel the line coverage as a 1D discrete problem with co-linear targets. Then, an order-based greedy algorithm, called OGA, is proposed to solve the problem optimally. It will be shown that the existing order in the 1D modeling, and especially the resulted Markov property of the selected sensors can help design greedy algorithms such as OGA. These algorithms demonstrate optimal/efficient performance and have lower complexity compared to the state-of-the-art. Furthermore, it is demonstrated that the conventional continuous line coverage problem can be converted to an equivalent discrete problem and solved optimally by OGA. Next, we formulate the well-known weak barrier coverage problem as an instance of the continuous line coverage problem (i.e. a 1D problem) as opposed to the conventional 2D graph-based models. We demonstrate that the equivalent discrete version of this problem can be solved optimally and faster than the state-of-the-art methods using an extended version of OGA, called K-OGA. Moreover, an efficient local algorithm, called LOGM, is proposed to mend barrier gaps due to sensor failure. In the case of m gaps, LOGM is proved to select at most 2m-1 sensors more than the optimal while being local and implementable in distributed fashion. We demonstrate the optimal/efficient performance of the proposed algorithms via extensive simulations.
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