Analysis of the Threshold for the Displacement to the Power of Random Sensors

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 $r$ 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 $s$ is established. It is assumed that for a given parameter $s$ two sensors interfere with each other during communication if their distance is less than $s$ (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 $n$ sensors is measured by the sum of the respective displacements to the power of the individual sensors. We define below the concept of $a -$ total displacement.

Definition 1 ( $a -$ total displacement).

Let $a > 0$ be a constant. Suppose the displacement of the $i -$ th sensor is a distance $d_{i}$ . The $a -$ total displacement is defined as the sum $\sum_{i = 1}^{n} d_{i}^{a}$ .

Motivation for this cost metric arises from the fact that the parameter $a$ in the exponents represents various conditions on the region lubrication, friction which affect the sensor movement.

This paper is concerned with the expected minimum $a -$ total displacement of moving $n$ 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 $m \in {1, 2} .$ We consider $n$ mobile sensors are placed independently at random with the uniform distribution in the $m -$ dimensional unit cube $[0, 1]^{m} .$

For the case of $m = 1$ each sensor is occupied with omnidirectional antenna of identical sensing radius $r_{1} > 0.$ Thus, a sensor placed at location $x$ in the unit interval can cover any point at distance at most $r_{1}$ either to the left or right of $x .$ (See Figure 1(a)).

For the case of $m = 2$ each sensor has identical square sensing radius $r_{2} > 0.$

Definition 2 (Square Sensing Radius).

We assume that a sensor located in position $(x_{1}, x_{2})$ where $0 \leq x_{1}, x_{2} \leq 1$ can cover any point in the area delimited by the square with the $4$ vertices $(x_{1} \pm r_{2}, x_{2} \pm r_{2})$ and and call $r_{2}$ the square sensing radius of the sensor. ²²2The 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 ADHOC-NOW 2015.

Figure 1(b) illustrates square sensing radius .

Fig. 1: (a) sensing radius $r_{1}$ on a line. (b) square sensing radius $r_{2} .$

However, in most cases sensing area of a sensor is a circular disk of radius $r_{c}$ 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 $r_{2}$ are obviously valid for circular disk of radius $r_{c}$ equal to $\sqrt{2} r_{2} .$

The sensors are required to move from their current random locations to new positions so as to satisfy the following scheduling requirement.

Definition 3 ( $(r_{m}, s) -$ C&i).

The $(r_{m}, s) -$ coverage & interference problem requires:

Every point in the $m -$ dimensional cube is within the range of a sensor, i.e. $m$ -dimensional unit cube is completely covered.
Each pair of sensors is placed at Euclidean distance greater or equal to $s .$ ³³3It 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 $(r_{m}, s) -$ coverage & interference requirement and the $a -$ total displacement is minimized in expectation.

For the case of unit interval the threshold phenomena around the sensing radius $r_{1} = \frac{1}{2 n}$ and the interference distance $s = \frac{1}{n}$ for the expected minimum $a -$ total displacement of $n$ 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 $r_{2} = \frac{1}{2 \sqrt{n}}$ and the interference distance $s = \frac{1}{\sqrt{n}}$ for the expected minimum $a -$ total displacement of $n$ sensors.

Throughout the paper, we will use the Landau asymptotic notation:

$f (n) = O (g (n))$ if there exists a constant $C_{1} > 0$ and integer $N$ such that $| f (n) | \leq C_{1} | g (n) |$ for all $n > N,$
$f (n) = Ω (g (n))$ if there exists a constant $C_{2} > 0$ and integer $N$ such that $| f (n) | \geq C_{2} | g (n) |$ for all $n > N,$
$f (n) = Θ (g (n))$ if and only if $f (n) = O (g (n))$ and $f (n) = Ω (g (n)),$

1.1 Preliminary Results

In this subsection we distinguish two cases.

1.1.1 Sensors in the unit interval $[0, 1] .$

Observe that in the case when the sensing radius $r_{1} = \frac{1}{2 n}$ and the interference distance $s = \frac{1}{n}$ the only way to achieve $(r_{1}, s) -$ coverage & interference requirement is for the sensors to occupy the equidistant anchor positions $\frac{i}{n} - \frac{1}{2 n}$ , for $i = 1, 2, \dots, n .$ The following exact asymptotic result was proved in Kapelko and Kranakis (2016b).

Theorem 4 (cf. Kapelko and Kranakis (2016b)).

Let $a$ be an even positive natural number. Assume that, $n$ mobile sensors are thrown uniformly and independently at random in the unit interval. The expected sum over all sensors $i$ from $1$ to $n,$ where the contribution of the $i -$ th sensor is its displacement from the current location to the anchor point $\frac{i}{n} - \frac{1}{2 n},$ raised to the $a -$ th power is

The next theorem extends Theorem 4 to all real valued exponent $a > 0.$

Theorem 5.

Fix $a > 0.$ Assume that, $n$ mobile sensors are thrown uniformly and independently at random in the unit interval. The expected $a -$ total movement of all $n$ sensors, when the $i -$ th sensor sorted in increasing order moves from their current random location to the equidistant anchor location $\frac{i}{n} - \frac{1}{2 n}$ , for $i = 1, 2, \dots, n$ , respectively, is

Γ(a2+1)2a2(1+a)n1−a2+O(n−a2).\lx@notefootnoteThegammafunction$Γ(a)$isdefinedtobeanextensionofthefactorialtorealnumberarguments.Itisrelatedtothefactorialby$Γ(a2+1)=(a2)!$providedthat$a2∈N.$

(1)

Before proving Theorem 5, we briefly explain the main ideas.

When $a > 0$ and $a$ 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 $Y$

be a random variable with the distribution function

F (x)

and the characteristic function

φ (t) .

Assume that

E [| Y |^{a}] < \infty,

where

a > 0

and

a

is not an even integer. Let

α_{k} = E [Y^{k}],

where

k

is nonnegative integer. Then

where $[\frac{a}{2}]$ is the greatest integer less than or equal to $\frac{a}{2} .$

It turns out that combining estimations when

a

is positive even natural with the representation in Theorem 6, one obtains the desired asymptotic formula (1) for all positive real numbers

a .

It is also worthwhile to mention that, the extension of direct combinatorial method from Kapelko and Kranakis (2016b) leads to exact asymptotic result in Theorem 5 only when $a$

is an odd natural number (see

(Kapelko, , Theorem 2)).

1.1.2 Sensors in the unit square $[0, 1]^{2} .$

Assume that, $n$ mobile sensors with the same square sensing radius $\frac{1}{2 \sqrt{n}}$ are thrown uniformly and independently at random in the unit square $[0, 1]^{2} .$ Observe that to fullfil $(\frac{1}{\sqrt{n}}, \frac{1}{2 \sqrt{n}}) -$ coverage & interference requirement the sensors have to occupy the following positions $(\frac{k}{\sqrt{n}} - \frac{1}{2 \sqrt{n}}, \frac{l}{\sqrt{n}} - \frac{1}{2 \sqrt{n}}),$ where $1 \leq k, l \leq \sqrt{n}$ and $n$ must be the square of a natural number.

It is known that expected $1 -$ total movement in this case is $Θ (\sqrt{ln (n) n}) .$ 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 $n = q^{2}$ for some $q \in N .$ Assume that $n$ random variables $X_{1}, X_{2}, \dots, X_{n}$ are independently uniformly distributed in the unit square $[0, 1]^{2}$ Consider the non-random points $(Z_{i})_{i \leq n}$ evenly distributed as follows: $Z_{i} = (\frac{k}{\sqrt{n}} - \frac{1}{2 \sqrt{n}}, \frac{l}{\sqrt{n}} - \frac{1}{} 2 \sqrt{n}),$ where $1 \leq k, l \leq \sqrt{n},$ $i = k \sqrt{n} + l .$ Then

where the infimum is over all permutations of ${1, 2, \dots, n}$ and where $d$ is the Euclidean distance.

The Euclidean Bipartite Matching Problem to find a permutation $π$ in the set of cardinality $n!$ which minimizes the transportation cost $\sum_{i = 1}^{n} d (X_{i}, Z_{π (i)})$ can be solved $O (n^{2 + ε})$ running time, where $ε$ 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 $a$ provided that $a > 1.$

Theorem 8.

Fix $a > 1.$ Let $n = q^{2}$ for some $q \in N .$ Assume that $n$ random variables $X_{1}, X_{2}, \dots, X_{n}$ are independently uniformly distributed in the unit square $[0, 1]^{2}$ Consider the non-random points $(Z_{i})_{i \leq n}$ evenly distributed as follows:
$Z_{i} = (\frac{k}{\sqrt{n}} - \frac{1}{2 \sqrt{n}}, \frac{l}{\sqrt{n}} - \frac{1}{} 2 \sqrt{n}),$ where $1 \leq k, l \leq \sqrt{n},$ $i = k \sqrt{n} + l .$ Then

E (inf π n \sum i = 1 d^{a} (X_{i}, Z_{π (i)})) = Ω ((ln (n))^{\frac{a}{2}} n^{1 - \frac{a}{2}}),

where the infimum is over all permutations of ${1, 2, \dots, n}$ and where $d$ is the Euclidean distance.

Before proving Theorem 8 we recall Jensen’s inequality for expectations. If $f$ is a convex function, then

f (E [X]) \leq E [f (X)]

(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

T^{(b)} = inf π \in S_{n} n \sum i = 1 d^{b} (X_{i}, Z_{π (i)}), 1 \leq b < \infty

where permutations $S_{n}$ is the set of all permutations of the numbers $1, 2, \dots, n .$
Fix $a > 1.$ Let $π^{⋆} \in S_{n}$ be the permutation which gives us

T^{(a)} = n \sum i = 1 d^{a} (X_{i}, Z_{π^{⋆} (i)}) .

(3)

Applying discrete Hölder inequality we get

n \sum i = 1 d (X_{i}, Z_{π^{⋆} (i)}) \leq {(n \sum i = 1 d^{a} (X_{i}, Z_{π^{⋆} (i)}))}^{\frac{1}{a}} {(n \sum i = 1 1)}^{\frac{a - 1}{a}} .

Hence

{(n \sum i = 1 d (X_{i}, Z_{π^{⋆} (i)}))}^{a} \leq n \sum i = 1 d^{a} (X_{i}, Z_{π^{⋆} (i)}) n^{a - 1} .

(4)

Observe that

{(T^{(1)})}^{a} = {(inf π \in S_{n} n \sum i = 1 d (X_{i}, Z_{π (i)}))}^{a} \leq {(n \sum i = 1 d (X_{i}, Z_{π^{⋆} (i)}))}^{a} .

(5)

Combining together equations (5), (4) and (3) we obtain

{(T^{(1)})}^{a} \leq T^{(a)} n^{a - 1} .

Passing to the expectations and using Jensen inequality ( see (2)) for $X := T^{(1)}$ and $f (x) = x^{a}$ we get the following estimation

(E {(T^{(1)})}^{a}) \leq E (T^{(a)} n^{a - 1}) .

(6)

Putting together Theorem 7 and inequality (6) we obtain

E (T^{(a)}) \geq n^{1 - a} {(Θ (\sqrt{ln (n) n}))}^{a} = Θ ((ln (n))^{\frac{a}{2}} n^{1 - \frac{a}{2}}) .

Therefore

E (inf π n \sum i = 1 d^{a} (X_{i}, Z_{π (i)})) = Ω ((ln (n))^{\frac{a}{2}} n^{1 - \frac{a}{2}}) .

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 $a -$ 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 $r_{1}$ is close to $\frac{1}{2 n}$ and the square sensing radius $r_{2}$ is close to $\frac{1}{2 \sqrt{n}}$ 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 $r_{1}$ is close to $\frac{1}{2 n},$ i.e. $r_{1} = \frac{1 + ϵ}{2 n}$ and the square sensing radius $r_{2}$ is close to $\frac{1}{2 \sqrt{n}},$ i.e. $r_{2} = \frac{1 + ϵ}{2 ⌊ \sqrt{n} ⌋}$ for both coverage and interference, provided that $ϵ$ is arbitrary small constant independent on $n .$

Compared to the coverage problem, the complex $(r_{m}, s) - C & I$ 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 $n$ sensors are initially deployed on the $[0, \infty)$ according to the arrival times of the Poisson process with arrival rate $λ > 0$ 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 $a > 0$ be a constant. Fix $m \in {1, 2} .$ Let $ϵ, δ > 0$ be arbitrary small constants independent on $n .$

Assume that $n$ mobile sensors with identical sensing range are placed independently at random in the with the uniform distribution in the unit $m -$ dimensional cube $[0, 1]^{m} .$

Table $1$ summarizes our main contribution in one dimension.

Sensing

radius

r_{1}

Interference

distance

s

Expected minimum total

displacement for

(r_{1}, s) - C & I

requirement

Theorem

r_{1} = \frac{1}{2 n}

s = \frac{1}{n}

\frac{Γ (\frac{a}{2} + 1)}{2^{\frac{a}{2}} (1 + a)} n^{1 - \frac{a}{2}} + O (n^{- \frac{a}{2}}),

a > 0

r_{1} = \frac{1 + ϵ}{2 n},

ϵ > 0

s = \frac{1 - δ}{n},

δ > 0

O (n^{1 - a}),

a > 0

Table 1: The expected minimum

a -

total movement of

n

random sensors in the unit interval

[0, 1]

as a function of the sensing radius and interference value.

As the sensing radius $r_{1}$ increases from $\frac{1}{2 n}$ to $\frac{1 + ϵ}{2 n}$ and the interference distance $s$ decreases from $\frac{1}{n}$ to $\frac{1 - δ}{n} .$
It is the sharp decline (the threshold) from $Θ (n^{\frac{a}{2}} n^{1 - a})$ to $O (n^{1 - a})$ in the expected minimum $a -$ total displacement for all powers $a > 0.$

Table $2$ summarizes our main contribution in two dimension.

Square

sensing

radius

r_{2}

Interference

distance

s

Expected minimum total

displacement for

(r_{2}, s) - C & I

requirement

Theorem

r_{2} = \frac{1}{2 \sqrt{n}}

s = \frac{1}{\sqrt{n}}

Θ (\sqrt{ln (n) n})

a = 1

Ω ((ln (n))^{\frac{a}{2}} n^{1 - \frac{a}{2}})

a > 1

r_{2} = \frac{1 + ϵ}{2 ⌊ \sqrt{n} ⌋},

ϵ > 0

s = \frac{1 - δ}{⌊ \sqrt{n} ⌋},

δ > 0

O (n^{1 - \frac{a}{2}})

a > 0

Table 2: The expected minimum

a -

total movement of

n

random sensors in the unit square

[0, 1]^{2}

as a function of the square sensing radius and interference value.

As the square sensing radius $r_{2}$ increases from $\frac{1}{2 \sqrt{n}}$ to $\frac{1 + ϵ}{2 ⌊ \sqrt{n} ⌋}$ and the interference distance $s$ decreases from $\frac{1}{\sqrt{n}}$ to $\frac{1 - δ}{⌊ \sqrt{n} ⌋} .$
It is the sharp decline (the threshold) from $Ω ((ln (n))^{\frac{a}{2}} n^{1 - \frac{a}{2}})$ to $O (n^{1 - \frac{a}{2}})$ in the expected minimum $a -$ total displacement for all powers $a \geq 1.$

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 $[0, 1] .$ Let $X_{(l)}$ be the position of the $l -$ th sensor after sorting the initial random locations of $n$ sensors with respect to the origin of the interval $[0, 1],$ i.e. the $l -$ th order statistics of the uniform distribution on the unit interval. It is known that the random variable $X_{(l)}$ obeys the Beta distribution with parameters $l, n + 1 - l$ (see Arnold et al. (2008)).

Assume that $c, d$ are positive integers. The Beta distribution (see of Mathematical Functions ) with parameters $c, d$ is the continuous distribution on $[0, 1]$

with probability density function

f_{c, d} (t)

given by

f_{c, d} (t) = c (\frac{c + d - 1}{c}) t^{c - 1} (1 - t)^{d - 1}, when 0 \leq t \leq 1.

(7)

The cumulative distribution function of the Beta distribution with parameters

c, d

I_{z} (c, d) = c (\frac{c + d - 1}{c}) \int_{0}^{z} t^{c - 1} (1 - t)^{d - 1} d t for 0 \leq z \leq 1.

(8)

denotes the incomplete Beta function.

Moreover, the incomplete Beta function with is related to the binomial distribution by

1 - I_{z} (c, d) = c - 1 \sum j = 0 (\frac{c + d - 1}{j}) z^{j} (1 - z)^{c + d - 1 - j}

(9)

(see (of Mathematical Functions, , Identity 8.17.5) for $c = m,$ $d = n - m + 1$ and $x = z$ ), as well as the binomial identity

c + d - 1 \sum j = 0 (\frac{c + d - 1}{j}) z^{j} (1 - z)^{c + d - 1} = 1.

(10)

The following inequality which relates binomial and Poisson distribution was discovered by Yu. V. Prohorov (see

(LeCam, 1965, Theorem 2), Prohorov (1953)).

(\frac{n}{j}) x^{j} (1 - x)^{n - j} \leq {(\frac{n}{m})}^{\frac{1}{2}} e^{- n x} \frac{(n x)^{j}}{j!},

(11)

where $m$ is integer which satisfies $n (1 - x) - 1 < m \leq n (1 - x) .$

We will also use the classical Stirling’s approximation for factorial (see (Feller, 1968, page 54))

\sqrt{2 π} N^{N + \frac{1}{2}} e^{- N + \frac{1}{12 N + 1}} < N! < \sqrt{2 π} N^{N + \frac{1}{} 2} e^{- N + \frac{1}{12 N}} .

(12)

We use the following notation

| x |^{+} = max {x, 0}

(13)

for positive parts of $x \in R .$

We are now ready to give some useful properties of Beta distribution in the following sequences of lemmas.

Lemma 9.

Let $a > 0.$ Assume that random variable $B e t a (l, n - l + 1)$ obeys the Beta distribution with parameters $l, n - l + 1$ provided that $l, n$ are positive integers and $l \leq n .$ Then

Pr [B e t a (n, 1) < 1 - \frac{1}{n^{\frac{a}{1 + a}}}] < \frac{1}{e^{n^{\frac{1}{1 + a}}}} .

Proof.

First of all observe that (see (7) for $c := n$ and $d := 1.$ )

	$Pr [B e t a (n, 1) < 1 - \frac{1}{n^{\frac{a}{1 + a}}}]$	$= \int_{0}^{1 - \frac{1}{n^{\frac{a}{1 + a}}}} f_{n, 1} (t) d t = {(1 - \frac{1}{n^{\frac{a}{1 + a}}})}^{n}$
		$= {⎛ ⎜ ⎜ ⎝ {(1 - \frac{1}{n^{\frac{a}{1 + a}}})}^{n^{\frac{a}{a + 1}}} ⎞ ⎟ ⎟ ⎠}^{n^{\frac{1}{a + 1}}} .$		(14)

Using (2) and the basic inequality $(1 - x)^{1 / x} < e^{- 1}$ when $x > 0$ we derive easily

Pr [B e t a (n, 1) < 1 - \frac{1}{n^{\frac{a}{1 + a}}}] < \frac{1}{e^{n^{\frac{1}{1 + a}}}} .

This proves Lemma 9. ∎

Lemma 10.

Let $a > 0$ be a constant. Fix $γ > 0$ independent on $n .$ Let $ρ = \frac{1 + γ}{n} .$ Assume that random variable $B e t a (l, n - l + 1)$ obeys the Beta distribution with parameters $l, n - l + 1$ provided that $l, n$ are positive integers and $l \leq n .$ Then

\Large∀l∈{1,2,…,n}E[(|Beta(l,n−l+1)−ρl|+)a]=O(1na),

(15)

n \sum l = 1 \frac{n}{l} E [{(| B e t a (l, n - l + 1) - ρ l |^{+})}^{a}] = O (n^{1 - a}) .

(16)

Lemma 11.

Let $a > 0$ be a constant. Fix $δ > 0$ independent on $n .$ Let $s = \frac{1 - δ}{n} .$ Assume that random variable $B e t a (l, n - l + 1)$ obeys the Beta distribution with parameters $l, n - l + 1$ provided that $l, n$ are positive integers and $l \leq n .$ Then

n \sum l = 1 \frac{n}{l} E [{(| s l - B e t a (l, n - l + 1) |^{+})}^{a}] = O (n^{1 - a}) .

(17)

The following lemma will simplify the upper bound estimations in Section 3 and Section 4.

Lemma 12.

Fix $a > 0.$ Assume that the sensor movement $M$ is the finite sum of movements $M_{i}$ for $i = 1, 2, \dots, l,$ i.e. $M = \sum_{i = 1}^{l} M_{i} .$ Then

E [M^{a}] \leq C_{a, l} l \sum i = 1 E [M_{i}^{a}],

where $C_{a, l}$ is some constant which depend only on fixed $a$ and $l .$

Proof.

Firstly we recall two elementary inequalities.

Fix $a \geq 1.$ Let $x, y \geq 0.$ Then

(x + y)^{a} \leq 2^{a - 1} (x^{a} + y^{a}) .

(18)

Notice that, Inequality (18) is the consequence of the fact that $f (x) = x^{a}$ is convex over $R_{+}$ for $a \geq 1.$

Fix $a \in (0, 1) .$ Let $x, y \geq 0.$ Then

(x + y)^{a} \leq x^{a} + y^{a} .

(19)

Combining together Inequality (18) and Inequality (19) for the sum $\sum_{i = 1}^{l} M_{i}$ and passing to the expectations we derive

E [M^{a}] \leq C_{a, l} l \sum i = 1 E [M_{i}^{a}] .

This is enough to prove Lemma 12. ∎

3 Sensors in 1D

In this section, we study the expected $a -$ total displacement to achieve coverage & interference requirement when $n$ mobile sensors are thrown independently at random with the uniform distribution in the unit interval $[0, 1] .$

3.1 Analysis of Algorithm 1

Fix $a > 0.$ Let $γ, δ > 0$ be arbitrary small independent on $n$ and let $ρ = \frac{1 + γ}{n},$ $s = \frac{1 - δ}{n} .$

This subsection is concerned with reallocating of the $n$ random sensors within the unit interval to achieve only the following property:

The distance between consecutive sensors is greater than or equal to $s$ and less than or equal to $ρ .$
The first leftmost sensor is in the distance less than or equal to $\frac{ρ}{2}$ from the origin.

We present basic and energy efficient algorithm $M V (n, ρ, s)$ (see Algorithm 1). Theorem 13 states that the expected $a -$ total displacement of algorithm $M V (n, ρ, s)$ is in $O (n^{1 - a})$ when $ρ = \frac{1 + γ}{n}$ and $s = \frac{1 - δ}{n} .$ Algorithm 1

is very simple but the asymptotic analysis is not totally trivial.

⁵⁵5We 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.

n

mobile sensors placed randomly and independently with the uniform distribution on the unit interval

[0, 1] .

0: The final positions of the sensors such that:

The distance between consecutive sensors is greater than or equal to $s$ and less than or equal to $ρ .$
The first leftmost sensor is in the distance less than or equal to $\frac{ρ}{2}$ from the origin.

Initialization: Sort the initial locations of sensors with respect to the origin of the interval, the locations after sorting

X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)};

X_{0} = 0;

2: for

i = 1

n

3: if

X_{(i)} - X_{(i - 1)} < s

then

4: move left to right the sensor

X_{(i)}

at the new position

min (s + X_{(i - 1)}, 1);

5: else if

X_{(i)} - X_{(i - 1)} > ρ

then

6: move right to left the sensor

X_{(i)}

at the new position

ρ + X_{(i - 1)};

7: else

8: do nothing;

9: end if

10: end for

11: if

X_{(1)} > \frac{1}{2} ρ

then

12:

z := X_{(1)} - \frac{1}{2} ρ;

13: for

i = 1

n

14: move right to left the sensor

X_{(i)}

at the new position

X_{(i)} - z;

15: end for

16: end if

Algorithm 1

M V (n, ρ, s)

Moving sensors in the

[0, 1] .

Theorem 13.

Let $a > 0$ be a constant. Fix $γ, δ > 0$ independent on $n .$ Assume that $n$ mobile sensors are thrown uniformly and independently at random in the unit interval. Then Algorithm 1 for $ρ = \frac{1 + γ}{n}$ and $s = \frac{1 - δ}{n}$ reallocate the random sensors within the unit interval so that:

The distance between consecutive sensors is greater than or equal to $s$ and less than or equal to $ρ .$
The first leftmost sensor is in the distance less than or equal to $\frac{ρ}{2}$ from the origin.
The expected $a -$ total displacement in $O (n^{1 - a}) .$ ⁶⁶6This theorem is valid regardless of the sensing radius.

Before starting the proof of Theorem 13, we briefly discuss one technical issue in the steps (2-4) of Algorithm 1. Let $Y_{1}, Y_{2}, \dots, Y_{n}$ be the locations of $n$ sensors after Algorithm 1. It is possible there exists $l_{0} \in N_{+}$ with the following property $Y_{i} < 1$ for all $i = 1, 2, \dots, l_{0}$ and $Y_{i} = 1$ for all $i = l_{0} + 1, l_{0} + 2, \dots, n .$ Then to avoid interference to achieve the property that the distance between consecutive sensors is greater than or equal to $s$ , we have to deactivate some set of sensors. Namely,

if $1 - Y_{l_{0}} < s$ then for all $i = l_{0} + 1, l_{0} + 2, \dots, n$ the sensors $Y_{i}$ will not sense any longer,
if $1 - Y_{l_{0}} \geq s$ then for all $i = l_{0} + 2, l_{0} + 3, \dots, n$ the sensors $Y_{i}$ will not sense any longer.

We are now ready to give the proof of Theorem 13.

Proof.

(Theorem 13) Let $ρ = \frac{1 + γ}{n}$ and $s = \frac{1 - δ}{n} .$ There are two cases to consider.

Case 1: The steps ( $1$ - $11$ ) of Algorithm (1)

We observe that Algorithm 1 is the sequence of the two phases: $A$ and $B .$ During phase $A,$ Algorithm 1 moves the sensors $X_{(i + 1)}, X_{(i + 2)}, \dots X_{(i + p)}$ at the new positions. Then in phase $B,$ Algorithm 1 leaves the sensors $X_{(i + p + 1)}, X_{(i + p + 2)}, \dots$ $X_{(i + p + k)}$ at the same positions.

Consider the phase $A$ as specified above. Let $p = p_{1} + p_{2}$ for some $p_{1}, p_{2} \in N_{+} .$

The sensors $X_{(i + 1)}, X_{(i + 2)}, \dots X_{(i + p_{1})}$ move right to left. Observe that the sensors $X_{(i + 1)}, X_{(i + 2)}, \dots X_{(i + p_{1})}$ have to move cumulatively, namely for $l = 1, 2, \dots, p_{1}$ the sensor $X_{(i + l)}$ move right to left to the position $X_{(i)} + ρ l .$ The displacement to the power $a$ is

$T_{1} = p_{1} \sum l = 1 {({∣ ∣ X_{(i + l)} - X_{(i)} - ρ l ∣ ∣}_{(i + l)}^{+})}^{a} .$
The sensors $X_{(i + p_{1} + 1)}, X_{(i + p_{1} + 2)}, \dots X_{(i + p_{1} + p_{2})}$ move left to right. Notice that the sensors $X_{(i + p_{1} + 1)}, X_{(i + p_{1} + 2)}, \dots X_{(i + p_{1} + p_{2})}$ have to move cumulatively, namely for $l = 1, 2, \dots, p_{2}$ the sensors $X_{(i + p_{1} + l)}$ move left to right to the position $X_{(i)} + ρ p_{1} + s l .$ The displacement to the power $a$ is

$T_{2} = p_{2} \sum l = 1 {({∣ ∣ X_{(i)} + ρ p_{1} + s l - X_{(i + p_{1} + l)} ∣ ∣}_{(i)}^{+})}^{a} .$

Since $X_{(i)} + ρ p_{1} < X_{(i + p_{1})}$ we upper bound the displacement to the power $a$ as follows:

$T_{2}$ $\leq p_{2} \sum l = 1 {({∣ ∣ X_{(i + p_{1})} + s l - X_{(i + p_{1} + l)} ∣ ∣}_{(i + p_{1})}^{+})}^{a}$

$= p_{2} \sum l = 1 {({∣ ∣ s l - (X_{(i + p_{1} + l)} - X_{(i + p_{1})}) ∣ ∣}^{+})}^{a} .$

Fig. 2: The positions of mobile sensors specified by

1.

and

2.

in the phase

B

of Algorithm 1.

Let us recall that $X_{(j)}$ is the $j -$ th order statistic of the uniform distribution on the unit interval, i.e., the position of the $j -$ th sensor in the interval $[0, 1] .$ We know that the random variable

X_{(j + l)} - X_{(j)} has the B e t a (l, n - l + 1) % distribution .

(see (Arnold et al., 2008, Formula 2.5.21, page 33)).

Applying this for $j := i$ and $j := i + p_{1}$ and passing to the expectations we deduce that

	$E (T_{1})$	$= p_{1} \sum l = 1 E {({∣ ∣ X_{(i + l)} - X_{(i)} - ρ l ∣ ∣}_{(i + l)}^{+})}^{a}$
		$= p_{1} \sum l = 1 E {({\| B e t a (l, n - l + 1) - ρ l \|}^{+})}^{a},$

	$E (T_{2})$	$\leq p_{2} \sum l = 1 E {({∣ ∣ s l - (X_{(i + p_{1} + l)} - X_{(i + p_{1})}) ∣ ∣}^{+})}^{a}$
		$= p_{2} \sum l = 1 E {({\| s l - B e t a (l, n - l + 1) \|}^{+})}^{a} .$

Next we make an important observation that extends our estimation to general specification of phase $A$ in Algorithm 1. Let $p = p_{1} + p_{2} + \dots p_{m}$ for some $p_{1}, p_{2}, \dots p_{m} \in N_{+} .$ We assume that phase $A$ is divided into $m$ phases as follows. For $j = 1, 2, \dots, m$ Algorithm 1 moves cumulatively the sequence of all $p_{j}$ sensors (the sensors
$X_{(p_{1} + p_{2} + \dots p_{j - 1} + 1)}, X_{(p_{1} + p_{2} + \dots p_{j - 1} + 2)}, \dots, X_{(p_{1} + p_{2} + \dots p_{j - 1} + p_{j})}$ ) into one chosen direction left to right or right to left. The movement direction of the sequence of all $p_{j}$ sensors is opposite to the movement direction of the sequence of all $p_{j + 1}$ sensors for $j = 1, 2, \dots, m - 1.$ Let $E (T_{A})$ be the expected movement to the power $a$ in the considered phases $A$ of Algorithm 1. Observe that

	$E (T_{A}) \leq$	$max 0 < p_{1} + p_{2} + \dots p_{m} \leq p m \sum j = 1 p_{j} \sum l = 1 E {(\| B e t a (l, n - l + 1) - ρ l \|^{+})}^{a}$
		$+ max 0 < p_{1} + p_{2} + \dots p_{m} \leq p m \sum j = 1 p_{j} \sum l = 1 E {(\| s l - B e t a (l, n - l + 1) \|^{+})}^{a} .$		(20)

Now we are ready to discuss the cost of the proposed algorithm. Let $E (T)$ be the expected $a -$ total displacement of Algorithm 1 at the steps ( $1$ - $11$ ) . Combining together (20) and the observation that Algorithm 1 is the sequence of the two phases $A$ and $B$ we get the following upper bound

	$E (T) \leq$
	$+$			(21)

Observe that the expected costs: $E {(| B e t a (l, n - l + 1) - ρ l |^{+})}^{a}$ and
$E {(| s l - B e t a (l, n - l + 1) |^{+})}^{a}$ can appear in the double sum (21) at most $\frac{n}{l}$ times. Hence

	$E (T)$	$\leq n \sum l = 1 \frac{n}{l} E {(\| B e t a (l, n - l + 1) - ρ l \|^{+})}^{a}$
		$+ n \sum l = 1 \frac{n}{l} E {(\| s l - B e t a (l, n - l + 1) \|^{+})}^{a} .$

Finally, using Equation (16) in Lemma 10 and Equation (17) in Lemma 11 we conclude that

E (T) = O (n^{1 - a}) .

This is enough to prove the desired upper bound in the first case.

Case 2: The steps ( $12$ - $17$ ) of Algorithm (1)

Observe that, after the steps ( $1$ - $11$ ) the sensor $X_{1}$ has to be at the position $P_{1}$ provided $0 \leq P_{1} \leq ρ = \frac{1 + γ}{n} .$ Hence for each sensor we upper bound the movement to the power $a$ by $($

Analysis of the Threshold for the Displacement to the Power of Random Sensors

Authors

On the Robot Assisted Movement in Wireless Mobile Sensor Networks

Sampling and Reconstruction of Spatial Fields using Mobile Sensors

A New Sufficient Condition for 1-Coverage to Imply Connectivity

Mobile Sensor Networks: Bounds on Capacity and Complexity of Realizability

On Detection of Critical Events in a Finite Forest using Randomly Deployed Wireless Sensors

Probabilistic Gathering Of Agents With Simple Sensors

Visualizing Sensor Network Coverage with Location Uncertainty

1 Introduction

Definition 1 ( $a -$ total displacement).

Definition 2 (Square Sensing Radius).

Definition 3 ( $(r_{m}, s) -$ C&i).

1.1 Preliminary Results

1.1.1 Sensors in the unit interval $[0, 1] .$

Theorem 4 (cf. Kapelko and Kranakis (2016b)).

Theorem 5.

Theorem 6 (cf. Von Bahr (1965), Ushakov (2011)).

1.1.2 Sensors in the unit square $[0, 1]^{2} .$

Theorem 7 (cf. Talagrand (2014), Chapter 4.3).

Theorem 8.

Proof.

1.2 Related Work

1.3 Contribution and Outline of the Paper

2 Preliminaries

Lemma 9.

Proof.

Lemma 10.

Lemma 11.

Lemma 12.

Proof.

3 Sensors in 1D

3.1 Analysis of Algorithm 1

Theorem 13.

Proof.

	$T_{2}$	$\leq p_{2} \sum l = 1 {({∣ ∣ X_{(i + p_{1})} + s l - X_{(i + p_{1} + l)} ∣ ∣}_{(i + p_{1})}^{+})}^{a}$
		$= p_{2} \sum l = 1 {({∣ ∣ s l - (X_{(i + p_{1} + l)} - X_{(i + p_{1})}) ∣ ∣}^{+})}^{a} .$

Analysis of the Threshold for the Displacement to the Power of Random Sensors

Authors

Related Research

On the Robot Assisted Movement in Wireless Mobile Sensor Networks

Sampling and Reconstruction of Spatial Fields using Mobile Sensors

A New Sufficient Condition for 1-Coverage to Imply Connectivity

Mobile Sensor Networks: Bounds on Capacity and Complexity of Realizability

On Detection of Critical Events in a Finite Forest using Randomly Deployed Wireless Sensors

Probabilistic Gathering Of Agents With Simple Sensors

Visualizing Sensor Network Coverage with Location Uncertainty

This week in AI

1 Introduction

Definition 1 (a−total displacement).

Definition 2 (Square Sensing Radius).

Definition 3 ((rm,s)−C&i).

1.1 Preliminary Results

1.1.1 Sensors in the unit interval [0,1].

Theorem 4 (cf. Kapelko and Kranakis (2016b)).

Theorem 5.

Theorem 6 (cf. Von Bahr (1965), Ushakov (2011)).

1.1.2 Sensors in the unit square [0,1]2.

Theorem 7 (cf. Talagrand (2014), Chapter 4.3).

Theorem 8.

Proof.

1.2 Related Work

1.3 Contribution and Outline of the Paper

2 Preliminaries

Lemma 9.

Proof.

Lemma 10.

Lemma 11.

Lemma 12.

Proof.

3 Sensors in 1D

3.1 Analysis of Algorithm 1

Theorem 13.

Proof.

Definition 1 ( $a -$ total displacement).

Definition 3 ( $(r_{m}, s) -$ C&i).

1.1.1 Sensors in the unit interval $[0, 1] .$

1.1.2 Sensors in the unit square $[0, 1]^{2} .$