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2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2005)
Pittsburgh, Pennsylvania, USA
Oct. 23, 2005 to Oct. 25, 2005
ISBN: 0-7695-2468-0
pp: 389-396
Uri Zwick , Tel Aviv University
Raphael Yuster , University of Haifa
<p>Let G = (V, E,w) be a weighted directed graph, where w : \rm E \to {-M, . . . , 0, . . . , M}. We show that G can be preprocessed in O(m^\omega) time, where \omega < 2.376 is the exponent of fast matrix multiplication, such that subsequently, each distance \delta (\upsilon ,\nu ) in the graph, where \upsilon ,\nu \varepsilon V , can be computed exactly in O(n) time. We also present a tradeoff between the processing time and the query answering time. As a very special case, we obtain an O(m^\omega) time algorithm for the Single Source Shortest Paths (SSSP) problem for directed graphs with integer weights of absolute value at most M. For suf?ciently dense graphs, with small enough edge weights, this improves upon the O(m\sqrt n \log M) time algorithm of Goldberg. We note that even the case M = 1, in which all the edge weights are in {-1, 0, +1}, is an interesting case for which no improvement over Goldberg?s O(m\sqrt n ) algorithm was known. Our new Õ(n^\omega) algorithm is faster whenever m > n^{\omega - 1/2} \simeq n^{1.876}.</p>
Uri Zwick, Raphael Yuster, "Answering distance queries in directed graphs using fast matrix multiplication", 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, vol. 00, no. , pp. 389-396, 2005, doi:10.1109/SFCS.2005.20
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