13,748 research outputs found
On the Displacement for Covering a dimensional Cube with Randomly Placed Sensors
Consider sensors placed randomly and independently with the uniform
distribution in a dimensional unit cube (). The sensors have
identical sensing range equal to , for some . We are interested in
moving the sensors from their initial positions to new positions so as to
ensure that the dimensional unit cube is completely covered, i.e., every
point in the dimensional cube is within the range of a sensor. If the
-th sensor is displaced a distance , what is a displacement of minimum
cost? As cost measure for the displacement of the team of sensors we consider
the -total movement defined as the sum , for some
constant . We assume that and are chosen so as to allow full
coverage of the dimensional unit cube and .
The main contribution of the paper is to show the existence of a tradeoff
between the dimensional cube, sensing radius and -total movement. The
main results can be summarized as follows for the case of the dimensional
cube.
If the dimensional cube sensing radius is and
, for some , then we present an algorithm that uses
total expected movement (see Algorithm 2 and
Theorem 5).
If the dimensional cube sensing radius is greater than
and is a natural
number then the total expected movement is
(see Algorithm 3 and Theorem 7).
In addition, we simulate Algorithm 2 and discuss the results of our
simulations
Algorithms on Minimizing the Maximum Sensor Movement for Barrier Coverage of a Linear Domain
In this paper, we study the problem of moving 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 time algorithm. For the special case when all sensors have
the same-size sensing range, the previously best solution takes time.
We present an time algorithm for this case; further, if all
sensors are initially located on the coverage segment, our algorithm takes
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 time, improving the previously best
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
Barrier Coverage with Non-uniform Lengths to Minimize Aggregate Movements
Given a line segment I=[0,L], the so-called barrier, and a set of n sensors with varying ranges positioned on the line containing I, the barrier coverage problem is to move the sensors so that they cover I, while minimising the total movement. In the case when all the sensors have the same radius the problem can be solved in O(n log n) time (Andrews and Wang, Algorithmica 2017). If the sensors have different radii the problem is known to be NP-hard to approximate within a constant factor (Czyzowicz et al., ADHOC-NOW 2009).
We strengthen this result and prove that no polynomial time rho^{1-epsilon}-approximation algorithm exists unless P=NP, where rho is the ratio between the largest radius and the smallest radius. Even when we restrict the number of sensors that are allowed to move by a parameter k, the problem turns out to be W[1]-hard. On the positive side we show that a ((2+epsilon)rho+2/epsilon)-approximation can be computed in O(n^3/epsilon^2) time and we prove fixed-parameter tractability when parameterized by the total movement assuming all numbers in the input are integers
Geometric Algorithms for Intervals and Related Problems
In this dissertation, we study several problems related to intervals and develop efficient algorithms for them. Interval problems have many applications in reality because many objects, values, and ranges are intervals in nature, such as time intervals, distances, line segments, probabilities, etc. Problems on intervals are gaining attention also because intervals are among the most basic geometric objects, and for the same reason, computational geometry techniques find useful for attacking these problems. Specifically, the problems we study in this dissertation includes the following: balanced splitting on weighted intervals, minimizing the movements of spreading points, dispersing points on intervals, multiple barrier coverage, and separating overlapped intervals on a line. We develop efficient algorithms for these problems and our results are either first known solutions or improve the previous work.
In the problem of balanced splitting on weighted intervals, we are given a set of n intervals with non-negative weights on a line and an integer k ≥ 1. The goal is to find k points to partition the line into k + 1 segments, such that the maximum sum of the interval weights in these segments is minimized. We give an algorithm that solves the problem in O(n log n) time. Our second problem is on minimizing the movements of spreading points. In this problem, we are given a set of points on a line and we want to spread the points on the line so that the minimum pairwise distance of all points is no smaller than a given value δ. The objective is to minimize the maximum moving distance of all points. We solve the problem in O(n) time. We also solve the cycle version of the problem in linear time. For the third problem, we are given a set of n non-overlapping intervals on a line and we want to place a point on each interval so that the minimum pairwise distance of all points are maximized. We present an O(n) time algorithm for the problem. We also solve its cycle version in O(n) time. The fourth problem is on multiple barrier coverage, where we are given n sensors in the plane and m barriers (represented by intervals) on a line. The goal is to move the sensors onto the line to cover all the barriers such that the maximum moving distance of all sensors is minimized. Our algorithm for the problem runs in O(n2 log n log log n + nm log m) time. In a special case where the sensors are all initially on the line, our algorithm runs in O((n + m) log(n + m)) time. Finally, for the problem of separating overlapped intervals, we have a set of n intervals (possibly overlapped) on a line and we want to move them along the line so that no two intervals properly intersect. The objective is to minimize the maximum moving distance of all intervals. We propose an O(n log n) time algorithm for the problem.
The algorithms and techniques developed in this dissertation are quite basic and fundamental, so they might be useful for solving other related problems on intervals as well
Algorithms for Covering Barrier Points by Mobile Sensors with Line Constraint
In this thesis, we develop efficient algorithms for the problem of covering barrier points by mobile sensors. Each sensor is represented by a point in the plane with the same covering range r so that any point within distance r from the sensor can be covered by the sensor. Given a set B of m points (called “barrier points”) and a set S of n points (representing the “sensors”) in the plane, the problem is to move the sensors so that each barrier point is covered by at least one sensor and the maximum movement of all sensors is minimized. The problem is NP-hard. In this thesis, we consider two line-constrained variations of the problem and present efficient algorithms that improve the previous work. In the first problem, all sensors are given on a line l and are required to move on l only while the barrier points can be anywhere in the plane. We propose an O((n+m) log(n+m)) time algorithm for the problem. We also consider the weighted case where each sensor has a weight; we give an O((m+n) log2(m+n)) time algorithm for this case. In the second problem, all barrier points are on l while all sensors are in the plane but are required to move to l to cover all barrier points. We solve the weighted case in O(mlogm+nlog2n) time
Maximizing Barrier Coverage Lifetime with Mobile Sensors
Sensor networks are ubiquitously used for detection and tracking and as a
result covering is one of the main tasks of such networks. We study the problem
of maximizing the coverage lifetime of a barrier by mobile sensors with limited
battery powers, where the coverage lifetime is the time until there is a
breakdown in coverage due to the death of a sensor. Sensors are first deployed
and then coverage commences. Energy is consumed in proportion to the distance
traveled for mobility, while for coverage, energy is consumed in direct
proportion to the radius of the sensor raised to a constant exponent. We study
two variants which are distinguished by whether the sensing radii are given as
part of the input or can be optimized, the fixed radii problem and the variable
radii problem. We design parametric search algorithms for both problems for the
case where the final order of the sensors is predetermined and for the case
where sensors are initially located at barrier endpoints. In contrast, we show
that the variable radii problem is strongly NP-hard and provide hardness of
approximation results for fixed radii for the case where all the sensors are
initially co-located at an internal point of the barrier
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