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ω
1
∈
Γ
1
ω
2
∈
Γ
2
(
p
,
ω
1
)
≡
h
(
p
,
ω
2
)
,
where
and
(6)
(
,
ω
)
=
|
ω
|≥|
|
(
,
ω
)
=
|
ω
|≥|
|
such that
N
p
0 with
Q
1
,and
N
p
0 with
Q
2
.
1
1
2
2
Definition 12.
Suppose that the prerequisite for skipping to the node labeled eq.(5)
in question is satisfied as in Definition 11, and that
v
=====
⇒
T
u
\
x
/
¯
(
p
,
ω
1
)
≡
h
(
p
,
ω
2
)
T
1
:
T
2
)
(
q
,
γ
)
≡
(
q
,
γ
)
(
and FIRST
live
(
q
,
γ
)=
FIRST
live
(
q
,
γ
)=
{
ε
}
¯
,where
γ
=
ε
or
N
(
q
,
γ
)=
0 with 1
≤
∈
(
Σ
∪{
λ
}
)
∗
,
|
γ
|≤|
Q
1
|
,and ¯
γ
=
ε
or
N
(
q
,
γ
)=
¯
0 with 1
≤|
γ
|≤|
¯
Q
2
|
,forsome
x
∈
Δ
∗
with
u
u
,
v
=
hv
,
q
∈
F
1
,
q
∈
F
2
.
x
0
)
∈
Σ
∗
such that
Now find a
shortest
string
σ
(
v
0
=====
⇒
T
u
0
\
x
0
/
(
,
ω
)
≡
(
,
ω
)
T
1
:
T
2
)
(
,
γ
)
≡
(
,
γ
)
,
¯
p
h
p
q
q
1
2
(
σ
(
u
0
===
⇒
T
1
x
0
)
/
σ
(
v
0
===
⇒
T
2
x
0
)
/
¯
(
,
ω
)
(
,
ζ
)
(
,
ω
)
(
,
ζ
)
p
q
and
p
q
1
2
¯
∈
Δ
∗
, and check whether it is successful
|
ζ
|≥|
γ
|
|
ζ
|≥|
γ
|
¯
,
with
and
,forsome
u
0
v
0
=
or not to have
u
0
hv
0
. Then the skipping to the node in question is said to be
applicable
if the above checking is successful for every possible
(
q
,
γ
)
≡
(
q
,
γ
)
¯
as
,
γα
)
≡
(
γβ
)
above. A node labeled
is defined to be a
skipping-end
from the
node in question, and an edge label between them is defined to be
u
0
\
(
q
q
,
¯
v
0
.
When skipping is applicable to the node in question, we expand it to have
skipping-ends in
unchecked
status, then we turn the node in question to be
skip-
ping
. The step of developing the comparison tree in this way is named
skipping
to
the node labeled eq.(5) with respect to eq.(6). Whenever a new node is added to the
comparison tree, we must apply skipping again to the node which has been already
applied skipping, because it is possible that some new skipping-ends will be added
to it afterward.
x
0
/
When the skipping has been applied to the node in question, the number of
skipping-ends of it is at most
|
F
1
|
(
|
Q
1
|
+
1
)
×|
F
2
|
(
|
Q
2
|
+
1
)
.
4.4
The Whole Algorithm
The whole algorithm is shown in Fig. 1. Here, the next node to be visited is cho-
sen as the “smallest” of the
unchecked
or
skipping
nodes, where the
size
of a node
labeled
(
p
,
α
)
≡
h
(
p
,
β
)
is the pair
(
Max
{|
α
|,|
β
|},
Min
{|
α
|,|
β
|}
)
, under lexico-
graphic ordering.
Example 1.
Let us apply our algorithm to the following pair of droct's:
T
1
=(
{
p
0
,
p
1
,
p
2
,
p
3
}, {
A
}, {
a
,
b
,
c
}, {
a
,
b
},
μ
1
,
p
0
, {
p
3
}
)
and
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