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5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 69
Figure 5.8: Two different viewpoints of Figure 5.7(b)
a solution also constitutes a solution to the SPP itself, because these stable patterns
have convex configurations.
For the case n = 3 and m = 3, the stable pattern N is depicted in Figure 5.10:
all the points on the blue layer can be computed once all the points of the green
layer are known. Here N is a stable pattern which allows for shifts along each of the
three time axes. However, N is not minimal in this situation. Smaller stable patterns
which allow for shifts in just one of the time axes are depicted in Figure 5.11: a green
layer denotes the stable pattern from Figure 5.10 from which 4 points, denoted by the
black dots, are removed. The patterns in the Figures 5.11 a, b, and c, respectively,
allow for shifts along the t 1 , t 2 , and t 3 time axis. Although these three patterns are
smaller than the non-minimal stable pattern N in Figure 5.10, they are not minimal
either.
When three (instead of four) points in one of the corners of N in Figure 5.10 are
deleted, a stable pattern remains which still allows for shifts in all three directions of
the time axes. This demonstrates the complexity of the shortest path problem and
its relaxation.
5.3.3 Some heuristic procedures for the shortest path problem
From the discussion in the previous section it is clear that the size and the complexity
of the SPP increases rapidly with n. However, optimal solution of the SPP is not a
goal by itself; it serves to facilitate the efficient computation of state vectors w t1,...,t n
.