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Convex hulls
Preliminaries 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
(
τ
(
N
)) time, we say that
A
is
τ
(
N
)transformable to
B
, written
A
α
τ
(
N
)
B
Transformability is not necessarily symmetric.
If problems
A
and
B
are mutually transformable, they are
equivalent
.
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View Full DocumentConvex hulls
Preliminaries 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
τ
(
N
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 Summer '11
 Mukherjee
 Algorithms, Sort

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