1 Introduction
Mobile sensors are being deployed in many application areas to enable easier information retrieval in the communication environments, from sensing and diagnostics to critical infrastructure monitoring (e.g. see Frasca et al. (2015); Gao et al. (2018); Kapelko (2018b); Kim et al. (2017); Li et al. (2016); Mohamed et al. (2017); Tian and Wang (2016) and Zhou et al. (2019)). Current reduction in manufacturing costs makes random displacement of the sensors more attractive. Even existing sensor displacement schemes cannot guarantee precise placement of sensors, so their initial deployment may be somewhat random.
A typical sensor is able to sense and thus cover a bounded region specified by its sensing radius, say Kumar et al. (2005). To monitor and protect a larger region against intruders every point of the region has to be within the sensing range of a sensor. It is also known that proximity between sensors affects the transmission and reception of signals and causes the degradation of performance Gupta and Kumar (2000). Therefore in order to avoid interference a critical value, say is established. It is assumed that for a given parameter two sensors interfere with each other during communication if their distance is less than (see Kapelko (2018d); Kranakis and Shaikhet (2014)). However, random deployment of the sensors might leave some gaps in the coverage of the area and the sensors may be too close to each other. Therefore, to attain coverage of the area and to avoid interference it is necessary to reallocate the sensors from their random locations to new positions. Clearly, the displacement of a team of sensors should be performed in the most efficient way.
The energy consumption for the displacement of a set of sensors is measured by the sum of the respective displacements to the power of the individual sensors. We define below the concept of total displacement.
Definition 1 (total displacement).
Let be a constant. Suppose the displacement of the th sensor is a distance . The total displacement is defined as the sum .
Motivation for this cost metric arises from the fact that the parameter in the exponents represents various conditions on the region lubrication, friction which affect the sensor movement.
This paper is concerned with the expected minimum total displacement of moving random mobile sensors from their original positions to new positions so as to achieve full coverage of a region and to avoid interference, i.e., every point in the region is within the range of at least one sensor while at the same time the sensors are not too close.
Fix We consider mobile sensors are placed independently at random with the uniform distribution in the dimensional unit cube
For the case of each sensor is occupied with omnidirectional antenna of identical sensing radius Thus, a sensor placed at location in the unit interval can cover any point at distance at most either to the left or right of (See Figure 1(a)).
For the case of each sensor has identical square sensing radius
Definition 2 (Square Sensing Radius).
We assume that a sensor located in position where can cover any point in the area delimited by the square with the vertices and and call the square sensing radius of the sensor. ^{2}^{2}2The concept of square sensing radius was introduced in the paper Kapelko and Kranakis (2015) which received the Best Paper Award at the 14th International Conference ADHOCNOW 2015.
Figure 1(b) illustrates square sensing radius .
However, in most cases sensing area of a sensor is a circular disk of radius but our investigation can be easily applied to this model by taking circle circumscribing the square. The upper bound result proved in the sequel for square sensing radius are obviously valid for circular disk of radius equal to
The sensors are required to move from their current random locations to new positions so as to satisfy the following scheduling requirement.
Definition 3 (C&i).
The coverage & interference problem requires:

Every point in the dimensional cube is within the range of a sensor, i.e. dimensional unit cube is completely covered.

Each pair of sensors is placed at Euclidean distance greater or equal to ^{3}^{3}3It is worth mentioning that in this paper when the sensors are displaced in the unit square they can move directly to the final locations with a shortened distance not only vertical and horizontal fashion as it was in Kapelko and Kranakis (2015, 2016a).
In this paper we investigate the problem of energy efficient displacement of the sensors so that in their final placement the sensor system satisfy coverage & interference requirement and the total displacement is minimized in expectation.
For the case of unit interval the threshold phenomena around the sensing radius and the interference distance for the expected minimum total displacement of sensors is discovered and explained
For the sensors placed in the unit square we discover and explain the threshold phenomena around the the square sensing radius and the interference distance for the expected minimum total displacement of sensors.
Throughout the paper, we will use the Landau asymptotic notation:

if there exists a constant and integer such that for all

if there exists a constant and integer such that for all

if and only if and
1.1 Preliminary Results
In this subsection we distinguish two cases.
1.1.1 Sensors in the unit interval
Observe that in the case when the sensing radius and the interference distance the only way to achieve coverage & interference requirement is for the sensors to occupy the equidistant anchor positions , for The following exact asymptotic result was proved in Kapelko and Kranakis (2016b).
Theorem 4 (cf. Kapelko and Kranakis (2016b)).
Let be an even positive natural number. Assume that, mobile sensors are thrown uniformly and independently at random in the unit interval. The expected sum over all sensors from to where the contribution of the th sensor is its displacement from the current location to the anchor point raised to the th power is
The next theorem extends Theorem 4 to all real valued exponent
Theorem 5.
Fix Assume that, mobile sensors are thrown uniformly and independently at random in the unit interval. The expected total movement of all sensors, when the th sensor sorted in increasing order moves from their current random location to the equidistant anchor location , for , respectively, is
(1) 
Before proving Theorem 5, we briefly explain the main ideas.
When and is not an even integer the proof of asymptotic result (1
) lies in probability theory. It is indeed based on the following formula of absolute moments in terms of characteristic function.
Theorem 6 (cf. Von Bahr (1965), Ushakov (2011)).
Let
be a random variable with the distribution function
and the characteristic function Assume that where and is not an even integer. Let where is nonnegative integer. Thenwhere is the greatest integer less than or equal to
It turns out that combining estimations when
is positive even natural with the representation in Theorem 6, one obtains the desired asymptotic formula (1) for all positive real numbers1.1.2 Sensors in the unit square
Assume that, mobile sensors with the same square sensing radius are thrown uniformly and independently at random in the unit square Observe that to fullfil coverage & interference requirement the sensors have to occupy the following positions where and must be the square of a natural number.
It is known that expected total movement in this case is Namely, the following theorem was obtained in Talagrand (2014) a book related to these problems which develops modern methods to bound stochastic processes.
Theorem 7 (cf. Talagrand (2014), Chapter 4.3).
Let for some Assume that random variables are independently uniformly distributed in the unit square Consider the nonrandom points evenly distributed as follows: where Then
where the infimum is over all permutations of and where is the Euclidean distance.
The Euclidean Bipartite Matching Problem to find a permutation in the set of cardinality which minimizes the transportation cost can be solved running time, where is an arbitrary small positive constant (see Agarwal et al. (1995)).
We are now ready to extend Theorem 7 to the displacement to the power provided that
Theorem 8.
Fix
Let for some Assume that random variables are independently uniformly distributed in the unit square
Consider the nonrandom points evenly distributed as follows:
where
Then
where the infimum is over all permutations of and where is the Euclidean distance.
Before proving Theorem 8 we recall Jensen’s inequality for expectations. If is a convex function, then
(2) 
provided the expectations exists (see (Ross, 2002, Proposition 3.1.2)).
The general strategy of our combinatorial proof of Theorem 8 is to combine the result of Theorem 7 with Jensen’s inequality for expectations, as well as discrete Hölder inequality.
Proof.
( Theorem 8) Let
where permutations is the set of all permutations of the numbers
Fix Let be the permutation which gives us
(3) 
Applying discrete Hölder inequality we get
Hence
(4) 
Observe that
(5) 
Combining together equations (5), (4) and (3) we obtain
Passing to the expectations and using Jensen inequality ( see (2)) for and we get the following estimation
(6) 
Putting together Theorem 7 and inequality (6) we obtain
Therefore
This finally completes the proof of Theorem 8. ∎
1.2 Related Work
There are extensive studies dealing with both coverage (e.g., see Abbasi et al. (2009); Ammari and Das (2012); Bhattacharya et al. (2009); Ghosh and Das (2008); Khedr et al. (2018); Saipulla et al. (2009); Wang et al. (2006)) and interference problems (e.g., see Burkhart and Zollinger (2004); Devroye and Morin (2018); Halldórsson and Tokuyama (2008); Jain and Qiu (2005); Kimura and Saito (2018); Kranakis et al. (2010)). Closely related to barrier and area coverage the matching problem is also of interest in the research community (e.g., see Ajtai et al. (1984); Gao et al. (2009); Kapelko (2018c); Talagrand (2014))
An important setting in considerations for coverage of domain is when the sensors are displaced at random with the uniform distribution. Some authors proposed using several rounds of random displacement to achieve complete coverage of domain Eftekhari et al. (2013b); Yan and Qiao (2010). Another approach is to use the relocating sensors Czyzowicz et al. (2009); Eftekhari et al. (2013a).
More importantly, our work is closely related to the papers Kapelko and Kranakis (2016a, b), where the authors considered the expected total displacement for coverage problem where the sensors are randomly placed in the unit interval Kapelko and Kranakis (2016b) and in the higher dimension Kapelko and Kranakis (2016a). Both papers study performance bounds for some algorithms, using Chernoff’s inequality. The methods used in these papers have the limitations  the most important and difficult cases when the sensing radius is close to and the square sensing radius is close to were not included in Kapelko and Kranakis (2016a, b). Moreover, in the paper Kapelko and Kranakis (2016a) the sensors can move only along to the axes. Hence, the analysis of coverage problem in Kapelko and Kranakis (2016a) is incomplete.
This paper we study the most important cases for the threshold phenomena, when the sensing radius is close to i.e. and the square sensing radius is close to i.e. for both coverage and interference, provided that is arbitrary small constant independent on
Compared to the coverage problem, the complex scheduling requirement not only ensures coverage, but also avoids interference and is more reasonable when providing good communication within the network.
It is worth mentioning that, in this paper in two dimensions the sensors can move directly to the final locations with a shortened distance not only vertical and horizontal fashion as it was in Kapelko and Kranakis (2016a) for the unit square.
Hence, our picture of the threshold phenomena is complete.
Finally, it is worth mentioning that, our work is related to the papers Kapelko (2018a, d), where the author investigated the maximum of the expected sensor’s displacement (the time required) for coverage with interference on the line Kapelko (2018a) and for the power consumption Kapelko (2018d). In Kapelko (2018a, d) it is assumed that the sensors are initially deployed on the according to the arrival times of the Poisson process with arrival rate and coverage (connectivity) is in the sense that there are no uncovered points from the origin to the last rightmost sensor.
1.3 Contribution and Outline of the Paper
In this paper we give the complete picture of the threshold phenomena for coverage simultaneously with interference in one dimension, as well as in two dimension.
Let us recall that in two dimensions the sensors can move directly to the final locations with a shortened distance not only vertical and horizontal fashion.
Let be a constant. Fix Let be arbitrary small constants independent on
Assume that mobile sensors with identical sensing range are placed independently at random in the with the uniform distribution in the unit dimensional cube
Table summarizes our main contribution in one dimension.



Theorem  

5  


15 

As the sensing radius increases from to and the interference distance decreases from to

It is the sharp decline (the threshold) from to in the expected minimum total displacement for all powers
Table summarizes our main contribution in two dimension.



Theorem  








if  16 

As the square sensing radius increases from to and the interference distance decreases from to

It is the sharp decline (the threshold) from to in the expected minimum total displacement for all powers
We also present 3 randomized algorithms. It is worthwhile to mention that, however the algorithms are simple but the analysis is challenging. In Section 2
we discover and prove new statistical properties of Beta distribution with special positive integers parameters (see Lemma
10 and Lemma 11).The overall organization of the paper is as follows. In Section 2 we present some preliminary results that will be used in the sequel. Section 3 deals with sensors in the unit interval. In Section 4 we investigate sensors in the unit square. Section 5 deals with simulation results. The final section contains conclusions and directions for future work.
2 Preliminaries
In this section we introduce some basic concepts and notation that will be used in the sequel. We also present three lemmas which will be helpful in proving our main results. In this paper, in the crucial one dimensional scenario, the n mobile sensors are thrown independently at random with the uniform distribution in the unit interval Let be the position of the th sensor after sorting the initial random locations of sensors with respect to the origin of the interval i.e. the th order statistics of the uniform distribution on the unit interval. It is known that the random variable obeys the Beta distribution with parameters (see Arnold et al. (2008)).
Assume that are positive integers. The Beta distribution (see of Mathematical Functions ) with parameters is the continuous distribution on
with probability density function
given by(7) 
The cumulative distribution function of the Beta distribution with parameters
is(8) 
denotes the incomplete Beta function.
Moreover, the incomplete Beta function with is related to the binomial distribution by
(9) 
(see (of Mathematical Functions, , Identity 8.17.5) for and ), as well as the binomial identity
(10) 
The following inequality which relates binomial and Poisson distribution was discovered by Yu. V. Prohorov (see
(LeCam, 1965, Theorem 2), Prohorov (1953)).(11) 
where is integer which satisfies
We will also use the classical Stirling’s approximation for factorial (see (Feller, 1968, page 54))
(12) 
We use the following notation
(13) 
for positive parts of
We are now ready to give some useful properties of Beta distribution in the following sequences of lemmas.
Lemma 9.
Let Assume that random variable obeys the Beta distribution with parameters provided that are positive integers and Then
Proof.
Lemma 10.
Let be a constant. Fix independent on Let Assume that random variable obeys the Beta distribution with parameters provided that are positive integers and Then
(15) 
(16) 
Lemma 11.
Let be a constant. Fix independent on Let Assume that random variable obeys the Beta distribution with parameters provided that are positive integers and Then
(17) 
Lemma 12.
Fix Assume that the sensor movement is the finite sum of movements for i.e. Then
where is some constant which depend only on fixed and
Proof.
Firstly we recall two elementary inequalities.
Fix Let Then
(18) 
Notice that, Inequality (18) is the consequence of the fact that is convex over for
3 Sensors in 1D
In this section, we study the expected total displacement to achieve coverage & interference requirement when mobile sensors are thrown independently at random with the uniform distribution in the unit interval
3.1 Analysis of Algorithm 1
Fix Let be arbitrary small independent on and let
This subsection is concerned with reallocating of the random sensors within the unit interval to achieve only the following property:

The distance between consecutive sensors is greater than or equal to and less than or equal to

The first leftmost sensor is in the distance less than or equal to from the origin.
We present basic and energy efficient algorithm (see Algorithm 1). Theorem 13 states that the expected total displacement of algorithm is in when and Algorithm 1
is very simple but the asymptotic analysis is not totally trivial.
^{5}^{5}5We note that asymptotic analysis of Algorithm 1 is crucial in deriving the threshold phenomena.In the proof of Theorem 13 we combine together combinatorial techniques with probabilistic analysis of Beta distribution (see Equation (16) in Lemma 10 and Equation (17) in Lemma 11). The estimations for Beta distribution with special positive integers parameters in Lemma 10 and Lemma 11 are new to the best author knowledge.
Theorem 13.
Let be a constant. Fix independent on Assume that mobile sensors are thrown uniformly and independently at random in the unit interval. Then Algorithm 1 for and reallocate the random sensors within the unit interval so that:

The distance between consecutive sensors is greater than or equal to and less than or equal to

The first leftmost sensor is in the distance less than or equal to from the origin.

The expected total displacement in ^{6}^{6}6This theorem is valid regardless of the sensing radius.
Before starting the proof of Theorem 13, we briefly discuss one technical issue in the steps (24) of Algorithm 1. Let be the locations of sensors after Algorithm 1. It is possible there exists with the following property for all and for all Then to avoid interference to achieve the property that the distance between consecutive sensors is greater than or equal to , we have to deactivate some set of sensors. Namely,

if then for all the sensors will not sense any longer,

if then for all the sensors will not sense any longer.
We are now ready to give the proof of Theorem 13.
Proof.
(Theorem 13)
Let and There are two cases to consider.
Case 1: The steps () of Algorithm (1)
We observe that Algorithm 1 is the sequence of the two phases: and During phase Algorithm 1 moves the sensors at the new positions. Then in phase Algorithm 1 leaves the sensors at the same positions.
Consider the phase as specified above. Let for some

The sensors move right to left. Observe that the sensors have to move cumulatively, namely for the sensor move right to left to the position The displacement to the power is

The sensors move left to right. Notice that the sensors have to move cumulatively, namely for the sensors move left to right to the position The displacement to the power is
Since we upper bound the displacement to the power as follows:
Let us recall that is the th order statistic of the uniform distribution on the unit interval, i.e., the position of the th sensor in the interval We know that the random variable
(see (Arnold et al., 2008, Formula 2.5.21, page 33)).
Applying this for and and passing to the expectations we deduce that
Next we make an important observation that extends our estimation to general specification of phase in Algorithm 1.
Let for some We assume that phase is divided into phases as follows.
For Algorithm 1 moves cumulatively the sequence of all sensors
(the sensors
)
into one chosen direction left to right or right to left.
The movement direction of the sequence of all sensors is opposite to the movement direction of the sequence of all sensors for
Let be the expected movement to the power in the considered phases of Algorithm 1. Observe that
(20) 
Now we are ready to discuss the cost of the proposed algorithm. Let be the expected total displacement of Algorithm 1 at the steps () . Combining together (20) and the observation that Algorithm 1 is the sequence of the two phases and we get the following upper bound
(21) 
Observe that the expected costs: and
can appear in the double sum (21) at most times. Hence
Finally, using Equation (16) in Lemma 10 and Equation (17) in Lemma 11 we conclude that
This is enough to prove the desired upper bound in the first case.
Case 2: The steps () of Algorithm (1)
Observe that, after the steps () the sensor has to be at the position provided Hence for each sensor we upper bound the movement to the power by
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