Proximity Planar triangulations

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ProximityPlanar triangulations
Problem definitions, 1 TRIANGULATION INSTANCE
Set S p1, p2, ..., pN of N points in the
plane. QUESTION Join the points in S with
nonintersecting straight line segments so that
every region internal to the convex hull of S is
a triangle.
A triangulation for a set S is not necessarily
unique. As a planar graph, a triangulation on N
vertices has ? 3N - 6 edges.
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ProximityPlanar triangulations
Problem definitions, 2 CONSTRAINED
TRIANGULATION INSTANCE Set S p1, p2, ...,
pN of N points in the plane, and set E pi,
pj, of nonintersecting edges on S. QUESTION
Join the points in S with nonintersecting
straight line segments so that every region
internal to the convex hull of S is a triangle
and every edge of E is in the triangulation.
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ProximityPlanar triangulations
Problem definitions, 3 POLYGON TRIANGULATION INSTA
NCE Polygon P p1, p2, ..., pN of N points
in the plane. QUESTION Triangulate the interior
of P. As a polygon, P has a set of edges by
definition, hence POLYGON TRIANGULATION is a
special case of CONSTRAINED TRIANGULATION. We
encountered this problem in 1. Kirkpatricks
triangle refinement method for point location 2.
Hertels convex polyhedra intersection algorithm
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ProximityPlanar triangulations
Optimizations Beyond simply triangulating a set
S, some problems involve optimizations of the
triangulation. For example 1. Minimize total
length of triangulation edges 2. Minimize
difference among internal angles (i.e.,
triangles that are more or less equilateral)
A
A
For example, A and B are both valid
triangulations, but B is preferred to A under
both optimization criteria. Complexity Time
complexity depends on the problem and
optimization.
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ProximityPlanar triangulations
Greedy triangulation A greedy algorithm chooses
the optimum choice available at each step of an
algorithm and never undoes anything done
earlier. Suppose the problem is TRIANGULATION,
with the desired optimization of minimizing the
total edge length. We will examine some greedy
algorithms for this problem. If the objective is
to minimize total edge length, the greedy
approach is to add at each stage the shortest
possible edge that does not intersect any
existing edge.
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ProximityPlanar triangulations
Simple greedy algorithm 1 Generate all ( ) ?
O(N2) possible edges. store in P. / O(N2)
/ 2 Sort P on increasing edge length. / O(N2
log N2) / 3 Initialize triangulation T to
?. 4 while P ? ? / O(N2) / 5 e shortest
(first) edge in P 6 P P - e 7 keep
TRUE 8 for every edge t ? T / O(N), by Eulers
formula / 9 if e ? t ? ? 10 keep
FALSE 11 endif 12 endfor 13 if keep
TRUE 14 T T ? e 15 endif 16 endwhile Analy
sis O(N2) ? O(N) ? O(N3) (An aside Preparata
says sorting requires O(N2 log N) time, rather
than O(N2 log N2).)
N 2