A note upon minimal path problem

Dynamic programming is optimal for certain sequential decision processes

“…Moravek [4] used a dimensionality argument to obtain a bound on the number of comparisons needed to prove optimality in any ordinary shortest path problem on an acyclic diagraph. The extension of his result to all loop-free monotone sequential decision processes does not seem straight-forward.…”

Shortest Paths in Nearly Conservative Digraphs

“…As D * is a simple digraph, paths can be given by only listing the sequence of its vertices. We can calculate APSP in D * in time O(n 3 ) by the method of Morávek [7] (see also in [3]) if we run this famous algorithm from all possible sources s. It gives distance function dD * (where if t is not reachable from s, then we write dD * (s, t) = +∞). The total running time is still O(2 k 1 · n 4 ).…”

Constrained decompositions of integer matrices and their applications to intensity modulated radiation therapy

“…To conclude this section we explain how to combine both constraints. In this case we have to add the new constraints (18) and (19) to the model (BOT-IDC'), while keeping the constraints (13) and (14). This corresponds to adding new arcs in the network.…”

“…The optimal beam-on-time is exactly the negative of the length of a shortest path from vertex 0 to vertex N + 2 in the network. As the network D is essentially acyclic we can adapt the standard dynamic programming algorithm due to Morávek [14] to compute an optimal potential in time O(|A|) = O (M 2 N). However, as we show here, an optimal potential can be computed in time O(MN).…”