Exact Shortest Paths with Rational Weights on the Word RAM

Abstract

Exact computation of shortest paths in weighted graphs has been traditionally studied in one of two settings. First, one can assume that the edge weights are real numbers and all the performed operations on reals (typically comparisons and additions) take constant time. Classical Dijkstra's and Bellman-Ford algorithms have been described in this setting. More efficient exact shortest paths algorithms have been obtained for integer-weighted graphs. Integrality assumption not only enables faster algorithms but also allows implementing the aforementioned algorithms in a much more realistic word RAM model where only arithmetic operations on $O(\log{n})$-bit integers are performed in constant time. On the word RAM one can as efficiently exactly encode even \emph{rational-weighted} instances with $O(\log{n})$-bit numerators and denominators. However, the known exact real-weighted shortest paths algorithms, run on such a rational input, can easily encounter intermediate values of $\Theta(n)$ bits if represented exactly. This leads to a factor-$\Omega(n)$ slowdown on the word RAM. At the same time, the scaling algorithms suited for integer weights do not produce exact solutions for rational inputs without dramatically increasing their accuracy. In this paper, we design randomized exact single-source shortest paths algorithms for rational-weighted graphs on the word RAM. Most importantly, in the non-negative case, we obtain a near-linear time algorithm matching Dijkstra's algorithm running time up to polylogarithmic factors. In presence of negative weights, we give an $\tilde{O}(n^{2.5})$-time algorithm breaking through the best known strongly polynomial bound attained by Bellman-Ford for sufficiently dense graphs.