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5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 71
Figure 5.10: A non-minimal stable pattern for n = 3 and m = 3 allowing for shifts in
three time directions
compared for a number of situations involving full recursions and uniform costs.
The heuristic methods under investigation are the following.
(i) The linear method starts by computing the values of y t1,...,t n
at all the points on
and below the hyperplane which corresponds to the total time n(m − 1), thereby just
covering the area (hypercube) of initial values. Then this stable pattern of values N
(see Proposition 5.7) is shifted step by step, with each step involving a single shift
in one of the axis directions, until the requested location is reached. The sequence
of shifts may depend on the multidimensional time instant at the requested location,
but this does not affect the total amount of points encountered in this procedure.
(ii) The diagonal method proceeds by first computing y t1,...,t n
for all the time instants
with constant total time |t| = 1, then all those with total time |t| = 2 and so on, until
the requested location is reached.
(iii) The equalizing method proceeds by computing a value y t1,...,t n
by employing that
recursion of the system (5.3) which reduces the largest possible coordinate of the
time instant y t1,...,t n
(which is larger than 2d − 2, to exclude negative time indices).
It uses a lexicographic ordering in case there is more than one largest coordinate.
The path between the initial state w 0,...,0 and the requested location is determined
in a backwards fashion, i.e., starting from the requested location. This method has
the effect that the calculated points are first directed towards the diagonal and then
to the origin.
(iv) The axis method uses the same methodology as the equalizing method with the
difference that it applies that recursion of the system (5.3) which reduces the smallest
possible coordinate of the time instant y t1,...,t n
which is larger than 2d − 2. This