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Unimodular matrices are often used as (unimodular) time/space-mappings in loop parallelization because they map iteration domains to convex regions. Some methods to automatically parallelize a given loop nest enumerate many different possible mappings and choose the best of these according to some heuristic. This technical report discusses methods...
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... application of a reallocation is shown in figure 3: The carefully chosen reallocation is able to reduce an algorithm's communication requirements. Assume a data-parallel algorithm that uses three dependence vectors, namely d 1 = Mapping the dependence vectors using this mapping, we get the space-components (i.e., all but the first components) of the mapped dependence vectors˜dvectors˜ ...
Citations
... This is accomplished by applying Voronoi's algorithm to enumerate non-isometric perfect forms [24], and then applying the Minkowski reduction algorithm to the obtained perfect lattices. 2) Enumerate all unimodular Z satisfying the bounds in (25). There are specialized algorithms for this task [25]. ...
... 2) Enumerate all unimodular Z satisfying the bounds in (25). There are specialized algorithms for this task [25]. ...
... is non-zero for all the Minkowski extreme lattices L (candidates for the optimum), and thus the bound in(25) is well-defined. A geometrical interpretation is that this inequality bounds the squared lengths sum of the basis vectors in the integer lattice Z N , where the basis vectors are now the rows of Z. Thus, finding the optimal Z can be regarded as searching for basis vectors inside a sphere of a certain radius. ...
This work investigates linear precoding over non-singular linear channels
with additive white Gaussian noise, with lattice-type inputs. The aim is to
maximize the minimum distance of the received lattice points, where the
precoder is subject to an energy constraint. It is shown that the optimal
precoder only produces a finite number of different lattices, namely perfect
lattices, at the receiver. The well-known densest lattice packings are
instances of perfect lattices, however it is analytically shown that the
densest lattices are not always the solution. This is a counter-intuitive
result at first sight, since previous work in the area showed a tight
connection between densest lattices and minimum distance. Since there are only
finitely many different perfect lattices, they can theoretically be enumerated
off-line. A new upper bound on the optimal minimum distance is derived, which
significantly improves upon a previously reported bound. Based on this bound,
we propose an enumeration algorithm that produces a finite codebook of optimal
precoders.
