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5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 61
is that many computations can be carried out in parallel since their exact order does
not matter, so that many alternatives exist with equal performance. Already for
small values of n, m and |t| this makes that the graph G can become very large. Two
helpful observations for constructing a useful shortest path formulation are: (i) any
sequence of time instants which facilitates the computation of y t1,...,t n
from w 0,...,0
can always be reorganized such that the total time increases monotonically; (ii) the
computation of values at time instants having the same total time can be carried out
in any arbitrary order. Therefore, a node v ∈ V can naturally be associated with a
set of time instants all having the same total time, rather than with individual time
instants. This is formalized in the following definition.
Definition 5.2. For k = 1, 2,..., let T k be the set of all multidimensional time
instants t =(t 1 ,...,t n ) ∈ N n 0 for which |t| = k and max{t 1 ,...,t n }≥m. Let V k be
the power set of T k (i.e., the set of all its subsets). Let V ⋆ be the set of time instants
corresponding to w 0,...,0 (i.e., for which max{t 1 ,...,t n }