An Extension of the Nested Partitions Method
EBERT BREA
Function Integer
Double

Uniform(¯a1,¯b1, ¯a2,¯b2, n)
Given:
¯a1 ∈−Z:
a

lower

integer

bound

of

the

first

interval

uniform

random

distribution;
¯b1 ∈−Z:
an

upper

integer

bound

of

the

first

interval

uniform

random

distribution;
¯a2 ∈−Z:
a

lower

integer

bound

of

the

second

interval

uniform

random

distribution;
¯b2 ∈−Z:
an

upper

integer

bound

of

the

second

interval

uniform

random

distribution;
n ∈−N:
a

nth

seed

from

an

available

pseudorandom

number

generator

set;
u(n)
∈−

(0, 1):

a

uniformly

distributed

random

number

between

0

and

1

from

the

nth
index

seed,

namely,

{u(s)}I(S)

s=I(1)
,

which

depends

on

the

sth

random

seed

I(s);
Output: a uniformly distributed random
integer number belonging

to

two disjunct

intervals,
namely:

[¯a1,¯b1]

or

[¯a2,¯b2]

;
Function
w(¯a1,¯b1, ¯a2,¯b2, n):
Calculate:
θ ∈−R and

x ∈−R,

namely:
θ =
1
+

¯b1 −−

¯a1
2
+

¯b1 +

¯b2 −−

¯a1 −−

¯a2
;
x =
¯a1 +

(2

+

¯b1 +

¯b2 −−

¯a1 −−

¯a2)

u(n);
if θ ≤−u(n)
< 1

then
Let x ←−x +
2 −−

¯b1 −−

1;
return ⌊x⌋;
Figure 23.
Double integer uniform distribution function
Function depth
D(k, n, m, σ(k), Θ)
Given:
The
iteration

counter,

k;
The
number

of

real

components,

n;
The
number

of

integer

components,

m;
The
kth

promising

region

σ(k)

⊆−Θ,

which

is

then

defined

by
lσ(i) ≤−x(i) ≤−u(iσ ),
∀i ∈−

{1,

.

.

.

, n},
¯lσ(i) ≤−
y(i) ≤−u¯(iσ ),

∀i ∈−

{1,

.

.

.

, m};
the
bounded

feasible

region

Θ,

namely:
l(i) ≤−x(i) ≤−u(i),
∀i ∈−

{1,

.

.

.

, n},
¯l(i) ≤−
y(i) ≤−u¯(i),

∀i ∈−

{1,

.

.

.

, m};
Declare:
d =
(d(1),

.

.

.

, d(n)
|
{z
}
n
;
(n+1),

.

.

.

, d¯(n+m)
|
{z
}
m
)t ∈−Rn ×−Zm;
Output: an updated depth vector d;
Function D(k, n, m, σ(k), Θ):
switch k do
case k =
0

do
for j ←−
1

to n do
d(j) ←−u(j) −−
l(j);
for j ←−n +
1

to n +

m do
(j) ←−u¯(j) −−
¯l(j);
other wise do
for j ←−
1

to n do
d(j) ←−u(jσ ) −−
lσ(j);
for j ←−n +
1

to n +

m do
(j) ←−u¯(jσ ) −−
¯lσ(j);
return d
Figure 24.
Depth function
Revista
TEKHNÉ

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
139