Thorup recently showed that single-source shortest-paths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0,…,2w−1} can be solved in O(n+m) time and space on a unit-cost random-access machine with a word length of w bits. His algorithm works by traversing a so-called component tree. Two new related results are provided here. First, and most importantly, Thorup’s approach is generalized from undirected to directed networks. The resulting time bound, O(n + m log w), is the best deterministic linear- space bound known for sparse networks unless w is superpolynomial in log n. As an application, all-pairs shortest-paths problems in directed networks with n vertices, m edges, and edge weights in {−2w,…,2w} can be solved in O(nm + n2 log log n) time and O(n + m) space (not counting the output space). Second, it is shown that the component tree for an undirected network can be constructed in deterministic linear time and space with a simple algorithm, to be contrasted with a complicated and impractical solution suggested by Thorup. Another contribution of the present paper is a greatly simplified view of the principles underlying algorithms based on component trees.
CITATION STYLE
Hagerup, T. (2000). Improved shortest paths on the word RAM. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1853, pp. 61–72). Springer Verlag. https://doi.org/10.1007/3-540-45022-x_7
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