Algorithms on Minimizing the Maximum Sensor Movement for Barrier Coverage of a Linear Domain

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 $O(n^2 \log n)$ time algorithm. For the special case when all sensors have the same-size sensing range, the previously best solution takes $O(n^2)$ time. We present an $O(n \log n)$ 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 $O(n^2)$ 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

Separating Overlapped Intervals on a Line

Given n intervals on a line ℓ, we consider the problem of moving these intervals on ℓ such that no two intervals overlap and the maximum moving distance of the intervals is minimized. The difficulty for solving the problem lies in determining the order of the intervals in an optimal solution. By interesting observations, we show that it is sufficient to consider at most n candidate lists of ordered intervals. Further, although explicitly maintaining these lists takes Ω(n2) time and space, by more observations and a pruning technique, we present an algorithm that can compute an optimal solution in O(n log n) time and O(n) space. We also prove an Ω(n log n) time lower bound for solving the problem, which implies the optimality of our algorithm

Lean, Green, and Lifetime Maximizing Sensor Deployment on a Barrier

Mobile sensors are located on a barrier represented by a line segment, and each sensor has a single energy source that can be used for both moving and sensing. Sensors may move once to their desired destinations and then coverage/communication is commenced. The sensors are collectively required to cover the barrier or in the communication scenario set up a chain of communication from endpoint to endpoint. A sensor consumes energy in movement in proportion to distance traveled, and it expends energy per time unit for sensing in direct proportion to its radius raised to a constant exponent. The first focus is of energy efficient coverage. A solution is sought which minimizes the sum of energy expended by all sensors while guaranteeing coverage for a predetermined amount of time. The objective of minimizing the maximum energy expended by any one sensor is also considered. The dual model is then studied. Sensors are equipped with batteries and a solution is sought which maximizes the coverage lifetime of the network, i.e. the minimum lifetime of any sensor. In both of these models, the variant where sensors are equipped with predetermined radii is also examined. Lastly, the problem of maximizing the lifetime of a wireless connection between a transmitter and a receiver using mobile relays is considered. These problems are mainly examined from the point of view of approximation algorithms due to the hardness of many of them