要旨
Given a network $N=(V,E,{c},{l})$, where
$G=(V,E)$, $|V|=n$ and $|E|=m$,
is a directed graph, ${c}(e) > 0$ is the capacity
and ${l}(e) \ge 0$ is the lead time (or delay) for each edge $e\in E$,
the quickest path problem is to find a path for a
given source--destination pair such that the total lead time plus the
inverse of the minimum edge capacity of the path
is minimal. The problem has applications to fast data
transmissions in communication networks. The best previous algorithm for the
single pair quickest path problem runs in time $O(r m+r n \log n)$,
where $r$ is the number of distinct capacities of $N$.
In this paper,
we present algorithms for general, sparse and planar
networks that have significantly lower running times.
For general networks, we show that the time complexity can be
reduced to $O(r^{\ast} m+r^{\ast} n \log n)$,
where $r^{\ast}$ is at most the number of capacities greater than the
capacity of the shortest (with respect to lead time) path in $N$.
For sparse networks, we present an algorithm with time complexity
$O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma})$,
where $\tilde{\gamma}$ is a topological measure of $N$.
Since for sparse networks $\tilde{\gamma}$ ranges from $1$
up to $\Theta(n)$,
this constitutes an improvement over the previously known bound of
$O(r n \log n)$ in all cases that $\tilde{\gamma}=o(n)$.
For planar networks, the complexity becomes $O(n \log n +
n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma})$.
Similar improvements are obtained
for the all--pairs quickest path problem.
We also give the first algorithm for solving the dynamic quickest path problem.
