Information Technology Reference
In-Depth Information
1.
The prototype of heap is a pyramid, and its anti-prototype a flat collection of
grains
.
1
2
1
2
(
,
,
)
The pyramid is that in the figure (a), with its vertex the point
1
, height
1
3
. The flat collection of grains can be supposed to be
like in figure (b), with a small height
h
=
1, and volumen
V
=
ε
, and volume=area of the base
×
ε
=
ε
.
An instance of a heap is in figure (c).
(a)
(b)
(c)
Let
N
a sufficiently big number and let us design by
S
N
,asetof
p
grains, with which we will state 'S(p) is a heap'. That is, the universe of dis-
course in
X
(
p
)
,
p
≤
)
≤
∗
S
=
{
S
(
p
)
;
p
≤
N
}
, endowed with the partial order '
S
(
p
(
q
)
⇔
p
≤
q
'.
(
)
The heap is undoubtedly constituted by the collection
S
p
of grains, but it is
(
)
recognized that
S
is a heap by comparing it with a prototype like
P
.Such
comparison is a matter of perceptive similarity. Thus, and provided the number
of p can be estimated,
a. Let it be
p
1, a coefficient perceptively established with
which we compare the statement '
h
is a heap' with the statements '
F
is a
heap' and '
P
is a heap'. Clearly,
λ
(
h
)
,0
<
λ
(
h
)
≤
comes from a perceptive comparison
with the volumes of F and P. It will be supposed that if
h
1
has
p
1
grains, and
h
2
has
p
2
grains,
p
1
≤
λ
(
h
)
p
2
,itis
λ
(
h
1
)
≤
λ
(
h
2
)
. For instance, for
h
in figure
1
3
, that represents that the volume of P is around
three times de volume of
h
in the figure (c).
b. Let it be
(c) it could be taken
λ
(
h
)=
ϕ
:
[
0
,
1
]
→
[
0
,
1
]
, continuous non-decreasing, and such that
ϕ
(
0
)=
0,
ϕ
(
1
)=
1 (an order-automorphism of the ordered unit interval). With
ϕ
,
the degree up to which 'S(p) is a heap' could be defined by
p
N
)
,
μ
h
(
S
(
p
)) =
λ
(
h
)
ϕ
(
q
N
⇒
ϕ
(
q
n
)
)
≤
∗
S
p
N
≤
p
N
)
≤
ϕ
(
since
S
(
p
(
q
)
⇔
p
≤
q
⇔
, and provided
λ
also increases with the number of grains, is
)
≤
∗
S
S
(
p
(
q
)
⇒
μ
h
(
S
(
p
))
≤
μ
h
(
S
(
q
))
.
