Convex hulls Preliminaries and definitions

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Convex hullsPreliminaries and definitions
Transformation of problems We would like to
establish lower bounds for performance
measures (time and space) for problems (not
algorithms!). One reason to avoid a futile
search for an algorithm faster than the
theoretical lower bound for the
problem. Generally, proving a lower (or upper)
bound is difficult, because it is a proof about
all algorithms to solve the problem. There is a
technique to do so, useful in some
cases transformation of problems. Suppose there
are two problems A and B, which are related so
that every instance of A can be solved as
follows 1. Convert the instance of A to a
suitable instance of B. 2. Solve problem
B. 3. Convert the answer to B to a correct answer
to the original instance of A. We say A has
been transformed to B. If the transformation
steps 1 and 3 can be performed in a total
of O(?(N)) time, we say that A is
?(N)-transformable to B, written A ??(N)
B Transformability is not necessarily
symmetric. If problems A and B are mutually
transformable, they are equivalent.
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Convex hullsPreliminaries and definitions
Lower and upper bounds via transformation Under
the assumption that the transformation preserves
the order of the instance size, the
transformation has two properties Property 1
(Lower bound via transformation). If problem A
is known to require at least T(N) time and A is
?(N)-transformable to B, then B requires at least
T(N) - O(?(N)) time. Otherwise, you could use
the transformation to solve A too fast. Property
2 (Upper bound via transformation). If problem B
can be solved in T(N) time and A is
?(N)-transformable to B, then A requires at most
T(N) O(?(N)) time. You can always solve A via
the transformation.
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Convex hullsPreliminaries and definitions
A known problem Note that the technique assumes
that there is a problem with a known lower bound
(to use property 1). We have such a problem
SORTING requires ?(N log N). Lower bound for
CONVEX HULL Can we get a lower bound for time for
CONVEX HULL? Recall that that the constructed
hull H(S) must be given in order. This suggest a
connection to SORTING.
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Convex hullsPreliminaries and definitions
Theorem. SORTING is linear-time transformable
to CONVEX HULL SORTING ?O(N) CONVEX
HULL. Therefore, CONVEX HULL requires at least
?(N log N) time. Proof. We show the
transformation. Suppose the instance of SORTING
is the set of N real numbers x1, x2, ,
xN. Transform that set to an instance of CONVEX
HULL by converting each real number xi to the
point (xi,xi2). O(N) All of these points lie on
the parabola y x2.
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Convex hullsPreliminaries and definitions
Lower bound for CONVEX HULL The convex hull of
the converted instance will consist of a list of
points, sorted by abscissa. At most one pass
through the list will find the smallest and the
answer to SORTING can be read off from there in
one pass. Takes O(N) time.
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Convex hullsPreliminaries and definitions
Lower bound for CONVEX HULL Note that the O(N)
transformation is dominated by the O(N log
N) complexity of the problem. As a consequence of
this transformation, we know that CONVEX HULL
requires at least ?(N log N) time.
Can we find an algorithm for CONVEX HULL that
requires O(N log N) time, which will be the best
we can do? We have already shown that such an
algorithm exists. Thus, the Graham Scan
Algorithm .
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Convex hullsExtreme points algorithm
Extreme points A point p of a convex set S is an
extreme point if no two points a, b ? S exist
such that p lines on the open segment ab.
Extreme points and the convex hull The set E of
extreme points of S (E ? S) is the smallest
subset of S such that H(E) H(S). E is the set
of vertices of H(S). This suggests an algorithm
for CONVEX HULL 1. Find the extreme points E of
S. 2. Order the points E so that they form a
convex polygon.
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Convex hullsExtreme points algorithm
Determining if a point is an extreme point If we
could determine whether a given point p ? S was
an extreme point in S, then we could find E by
testing each point in S. Theorem. A point p
fails to be an extreme point of a plane
convex set S iff it lies in some triangle whose
vertices are in S but is not itself one of the
vertices of the triangle.
p ? E
p ? E
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Convex hullsExtreme points algorithm
Determining if a point is an extreme point There
are O(N3) triangles determined by the N points of
S. Point enclosure in a triangle can be performed
in O(1) time. To determine if p is an extreme
point, test it for inclusion in each of the O(N3)
triangles. If all fail, p ? E. Doing so for each
point p ? S requires O(N4) time. (Determining
extreme edges has O(N3 ) algorithmsee ORourke
p.67)
p ? E
p ? E
for each i do for each j not equal to i do
for each k not equal to i or j do
for each l not equal to i, j or k do
if pl is in triangle (pi,pj,pk) then pl is
non-extreme
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Extreme Edge Algorithm for each i do for each
j not equal to i do for each k not equal
to i or j do if pk is not left
or on (pi,pj) then is (pi,pj)
non-extreme edge The algorithm runs in O(n3)
time. Actually, the best way to find the extreme
points and extreme edges will be to first find
the convex Hull and then by a linear scan find
the extreme points and edges. This takes O(nlogn)
time and it can be shown that this is also the
lower bound. However, there is no direct
constructive way to find the extreme points
taking this amount of time.