SESSION NOVEL ALGORIHMS AND APPLICATIONS + ...

48 Int'l Conf. Foundations of Computer Science | FCS'11 |
2 The The related woks
Recently, Recently, the the the generation generation of of random random geometric objects
objects
objects
and and specially specially simple simple polygons polygons polygons has has received received some some attention attention by
by
researchers. For For example, example, Epstein Epstein studied studied studied the the uniformly
uniformly
random random generation generation of of triangulations
triangulations [8]. ]. Zhu et al. presented
on on an an algorithm algorithm for for generating generating x
x-monotone monotone polygons on a
given given set set of of vertices vertices uniformly uniformly at at random
random [3]. A heuristic for
the the generation generation of of simple simple polygo
polygons ns was was investigated by
OO'Rourke
Rourke and Virmani [6]. ]. Auer and and Held presented the
the
following following heuristic heuristic algorithms:
algorithms:
• Steady Growth Growth: : an incremental algorithm adding one
point after after the other other whose time time complexity in in the
the
the
worst case and and in in the the best best case case is
is
, respectively.
and
• Space Partitionin Partitioning: g: which is a divide divide and and conquer
conquer
algorithm and it it has time time complexity complexity of
of .
• Permute & & Reject Reject: : which creates random
•
permutations (polygons) (polygons) and surveys surveys whether whether it it is
is
corresponding with a simple polygon polygon or or not, not, until until a
a
simple polygon polygon is encountered. encountered. Its Its complexity complexity is
is
.
2-opt opt Moves Moves: : which by starting from a a completely
completely
random polygon polygon and replacing its intersected edges edges edges
encounters a a a simple simple simple polygon polygon and and its its its complexity complexity complexity complexity is
is
[2].
3 Preliminaries
Let be a set of random random vertices, there exists simple
polygons on
in total, such that every every polygon is is generated
with with probability . it is s supposed that no no three points points are
are
linear. linear. A A simple simple simple polygon, polygon, polygon, is is a a limited limited plane plane by by by a a limited limited limited set set set of
of
of
line line segments segments that that form form a a simple simple closed closed closed curve. curve. curve. In In In other other other words,
words,
words,
a a simple polygon on , is a a polygon whose edges edges don't
intersect one another except except on on ve vertices .
A A convex convex polygon, polygon, is is is a a a simple simple simple polygon polygon which which for for both both both of
of
of
vertices and from polygon, the the line line segment segment lies inside
or on its border, i.e.,
. The convex convex hull of a finite
set set of of points points on on the the plane plane (
( ) is is the smallest smallest co convex
polygon that encloses . The smallest polygon means that
there there is no polygon such that (Figure2).
Let be the number of of all all convex convex convex hull hull layers layers on on the the set set of
of
points . . Every layer is is defined defined by
by ( and ).
By By the the supposition supposition of of numbering numbering numbering layers layers layers from from from the the most
most
most
internal internal to to to the the most most external external convex convex hull hull layers,
layers, layers, is
supposed supposed to to be be the the the number number of of the the most most most internal internal convex convex hull
hull
layer, i.e., .
Vertex sees vertex if and doesn't lie out of .
In this case
is visible from or in other words words has
visibility visibility toward toward . The The polar sorting, sorting, is is the sorting of of a a set set of
of
vertices vertices around around a a given vertex vertex acco
according rding to polar angle. The
star star-shaped shaped shaped polygon, polygon, is is is a a a polygon polygon polygon which which is is is visible visible visible at at least
least
fro from m m an an interior interior vertex. vertex. In In the the next next section, section, the the proposed
proposed
heuristic heuristic algorithm algorithm is is is posed posed posed to to to generate generate generate random random random sunflower
sunflower
sunflower
polygon.
4
Fig.
2. The convex hull of of set of of points
The proposed algorithm
In In this this section, section, an an an algorithm algorithm is is posed posed to generate generate generate the
the
random sunflower sunflower polygon polygon with with time time complexity
complexity
in which is the number of vertices.
To To generate generate a a random random sunflower sunflower sunflower polygon polygon on on a a a vertex vertex vertex set set it it it is
is
is
performed in in this this way that first, According According to Graham greedy
algorithm, the assumption assumption is is obtained. obtained. The The Graham
algorithm for the the generation generation of
of performs performs in this way:
• First, the vertex with lowest -coordinate coordinate which is
the the rightmost rightmost point, point, has has selected selected from from the the point point set
set
and is called ! &.
• All All the the remaining remaining points points are are sorted sorted around around
! &
according to to polar polar angle
angle ! ' * ' ! 1, .
• A stack is constructed and ! & and ! are pushed to it.
• By starting from point point ! to ! +, , for every vertices
the following case is investigated:
• ! is is pushed pushed in in the the stack stack if it is is left left of of two two tops tops of of the
the
stack and is incremented counter "; ; otherwise, the
top of stack stack is removed if ! is is right of of two tops of
of
the stack (Figure 3).
Algorithm Algorithm: : Graham Scan
Find rightmost lowest point; lable it ! &.
Sort all other other points angulary about ! &.
Stack # ! ' ! & # ! (''
! (, - . indexes top.
" # $
While " % do
If ! strictly left of ! (, ! ( Then
Push ! )' and set " / " 0
Else
Pop
End If
End While hile
Fig Fig. 3. The Graham scan algorithm
Time complexity of Graham Graham algorithm is is . After
determining , while is is opposite of zero, Graham
algori algorithm thm is recalled for any remaining point set . Let the