In this paper, we study the problem of moving n sensors on a line to form a
barrier coverage of a specified segment of the line such that the maximum
moving distance of the sensors is minimized. Previously, it was an open
question whether this problem on sensors with arbitrary sensing ranges is
solvable in polynomial time. We settle this open question positively by giving
an O(n2logn) time algorithm. For the special case when all sensors have
the same-size sensing range, the previously best solution takes O(n2) time.
We present an O(nlogn) time algorithm for this case; further, if all
sensors are initially located on the coverage segment, our algorithm takes
O(n) time. Also, we extend our techniques to the cycle version of the problem
where the barrier coverage is for a simple cycle and the sensors are allowed to
move only along the cycle. For sensors with the same-size sensing range, we
solve the cycle version in O(n) time, improving the previously best O(n2)
time solution.Comment: This version corrected an error in the proof of Lemma 2 in the
previous version and the version published in DCG 2013. Lemma 2 is for
proving the correctness of an algorithm (see the footnote of Page 9 for why
the previous proof is incorrect). Everything else of the paper does not
change. All algorithms in the paper are exactly the same as before and their
time complexities do not change eithe