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60 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
with |t|.
□
On the other hand, it is not difficult to design an algorithm which achieves a computational
complexity that is indeed linear in |t|. This may proceed as follows: since
y t1,...,t n
is contained in w t1,...,t n
=(A T x 1
) t1 (A T x 2
) t2 ···(A T x n
) tn w 0,0,...,0 , it can be computed
by the joint action of t 1 + ...+ t n = |t| matrices of the form A T x i
. It is not
difficult to compute a fixed uniform upper bound on the computational complexity
involved in the action of each of the matrices A T x i
, because only the time instants
that have a total time which does not exceed n(m − 1) can assist in this computation
and their number is finite. In view of the previous theorem this shows that an optimal
algorithm for the computation of y t1,...,t n
has a computational complexity that
increases linearly with the total time |t|. Clearly, similar arguments and results also
hold for the computation of a state vector w t1,...,t n
.
5.3.2 Formulation as a shortest path problem
The problem of finding an optimal algorithm for the computation of y t1,...,t n
from
w 0,...,0 using the recursions (5.3) can be cast into the form of a shortest path problem
(SPP). As it turns out, this SPP will quickly become huge and difficult to solve.
However, it is possible to set up a relaxation of the shortest path problem (RSPP)
which is considerably smaller and easier to solve. In the 2D-case (with full recursions
and uniform costs) the RSPP can be solved analytically and its solution happens to
solve the SPP too. A central role in this approach is played by the notion of stable
patterns, which are shifted along the 2D-grid. The same approach leads to partial
results in the 3D-case (and in higher dimensions). It also underlies the design of the
heuristic methods discussed in Section 5.3.3.
In general, a standard formulation of a shortest path problem requires the specification
of a weighted directed graph G =(V,E,W,v I ,v T ), consisting of a set V of
nodes, a set E ⊆ V × V of edges, a weight function W : E → R, a set of initial nodes
v I ∈ V and a set of terminal nodes v T ∈ V . To compute a shortest path from v I to
v T with smallest total weight, one may apply any classical algorithm (e.g., those of
Dijkstra or Floyd). The set V should correspond to the various ‘states’ in which the
computational procedure can be. It is natural to relate a node v ∈ V in some way to
a set of multidimensional time instants (t 1 ,...,t n ) for which the value of y t1,...,t n
is
either already available or still requires computation. The edges E represent ‘state
transitions’ and they are naturally associated with the recursions in the set (5.3).
The weight function W is used to reflect the computational costs (e.g., the number
of flops (floating point operations)) associated with these recursions.
In setting up a shortest path problem formulation, one may run into the problem
that the number of elements in V becomes infinite, since for n ≥ 2 it may already
happen that one can apply an infinite sequence of recursions without ever arriving at
the specified multidimensional time instant (t 1 ,...,t n ). To avoid this, one may work
backwards from (t 1 ,...,t n ), by figuring out sets of time instants with smaller total
time which may assist in the computation of y t1,...,t n
. Another feature of the problem