Sort_trans_expts

# Sort_trans_expts - Convex hulls Preliminaries and...

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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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Convex 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 )-transformable to B ,
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