Analysis of the Threshold for Energy Consumption in Displacement of Random Sensors

. 2022 Nov 14;22(22):8789. doi: 10.3390/s22228789

Algorithm 2

C V_{1} (n, r_{1}, s)

for the

(r_{1}, s)

-coverage and interference requirement on

[0, 1]

when

r_{1} = \frac{1 + ϵ}{2 n},

s = \frac{1 - δ}{n}

provided that

ϵ > 0

and

1 > δ > 0

are fixed and independent of

n

Require: The initial locations of n mobile sensors with the identical sensing radius $r_{1} = \frac{1 + ϵ}{2 n}$ , placed uniformly and independently at random on the unit interval $[0, 1] .$

Ensure: The final positions of sensors the to satisfy the $(r_{1}, s)$ -coverage and interference requirement on the interval $[0, 1] .$

Initialization: Apply Algorithm

M V (n, ρ, s)

for

ρ : = \frac{1 + \frac{ϵ}{2}}{n},

s : = \frac{1 - δ}{n}

and the random sensors

X_{1}, X_{2}, \dots, X_{n} .

Let

Y_{1}, Y_{2}, \dots, Y_{n}

be the location of n sensors

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

after Algorithm

M V (n, ρ, s);

1:
switch ()
2:
case A $(Y_{n} \geq 1 - r_{1})$
3:
do nothing;
4:
case B $(Y_{n} \in (1 - \frac{2}{n^{\frac{a}{a + 1}}}, 1 - r_{1}))$
5:
move the sensor $Y_{n}$ to the new position $1 - r_{1},$ $i : = n - 1;$
6:
while $Y_{i + 1} - Y_{i} > 2 r_{1}$ do
7:
move the sensor $Y_{i}$ to the new position $1 - r_{1} - (n - i) 2 r_{1},$ $i : = i - 1$ ;
8:
end while
9:
case C $(Y_{n} \leq 1 - \frac{2}{n^{\frac{a}{a + 1}}})$
10:
for $i = 1$ to n do
11:
move the sensor $Y_{i}$ to the position $(\frac{i}{n} - \frac{1}{2 n});$
12:
end for
13:
end switch