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Abstract:
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 sourcedestination 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 singlepair quickest path problem runs in time $O(r m+r n \log n)$, where $r$ is the number of distinct capacities of $N$ \cite{ROS}. 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 allpairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem.
Bibliographic citation for this report: [plain text] [BIB] [BibTeX] [Refer]
Or copy and paste:
Dimitrios Kagaris,
Grammati E. Pantziou,
Spyros Tragoudas, and
Christos D. Zaroliagis,
"Quickest Paths: Faster Algorithms and Dynamization."
Dartmouth Computer Science Technical Report PCSTR94204,
1994.
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