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要旨:
We consider the problem of preprocessing an $n$-vertex digraph with
real edge weights so that subsequent queries for the shortest path or distance
between any two vertices can be efficiently answered.
We give algorithms
that depend on the {\em treewidth} of the input graph. When the
treewidth is a constant, our algorithms can answer distance queries in
$O(\alpha(n))$ time after $O(n)$ preprocessing. This improves upon
previously known results for the same problem.
We also give a
dynamic algorithm which, after a change in an edge weight, updates the
data structure in time $O(n^\beta)$, for any constant $0 < \beta < 1$.
Furthermore, an algorithm of independent interest is given:
computing a shortest path tree, or finding a negative cycle in linear
time.
