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In-Depth Information
vertices. If a node
v
i
is a leaf, its weight defaults to
w
i
= 1
. If the node
v
i
has
k
children {v
i,1
, v
i,2
, … v
i,j
,.., v
i,k
}, its weight is calculated by using the
formula:
k
¦
w
(
v
)
1
C
w
(
v
)
i
i
,
j
j
1
Where
C
is a constant
(0 < C < 1)
, and
w
i,j
is the weight assigned to
the
j
th
child of vertex
v
i
. Constant
C
is a scalar that determines the
difference of local region sizes based on the number of descendants of
these vertices. In other words, the larger
C
's value, the bigger the
difference of local regions
P(v)
of vertices with more descendants and
vertices with fewer descendants.
After the calculation of local regions, we calculate the position of
nodes. Within a non-self-intersecting closed polygon, we position the label
at the centroid of the polygon. The vertices are initially numbered in order
of their occurrence along the polygon's perimeter clockwise, and the
vertex (
x
n
, y
n
) is assumed to be the same as (
x
0
, y
0
). The centroid of a non-
self-intersecting closed polygon defined by n vertices (
x
0
, y
0
), (
x
1
, y
1
), ...,
(
x
n-1
, y
n-1
) is the point (
C
x
, C
y
), where:
and:
where A is the polygon's signed area and it is defined by the formula
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