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...ed, imply an O(n 2 log n + mn) time bound for any graph with m edges; the first term has been lowered to O(n 2 log log n) and O(n 2 α(m, n)) for directed and undirected graphs respectively, by Pettie =-=[28]-=- and Pettie and Ramanchandran [29].) ∗ School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (tmchan@uwaterloo.ca). This work has been supported by NSERC. A preliminary...

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...lable, in various settings. For the best available 0(a) (b) A Bs nW 2nW . . . n2W Fig. 1. Reductions of static problems to incremental or decremental SSSP problems results see [10], [25], [13], [18], =-=[17]-=-. Karger et al. [15] show that any pathcomparison algorithm for the problem must have a running time of Ω(mn). The reduction of Theorem 1 shows that if there is an incremental or decremental SSSP algo...

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...n several applications. However, the complexity of the fastest known algorithm for the APSP problem in a graph with m edges and n vertices with real non-negative edge weights is O(mn + n 2 log log n) =-=[14]-=-. For graphs with m =Θ(n 2 ), this algorithm has a running time of Θ(n 3 ) which matches ∗ This work was done while the author was at the Max-Planck-Institut für Informatik, Saarbrücken, Germany. Teli...

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...re additions are granted for free. A result by Karger et al. [11] shows that any APSP-algorithm (on directed graphs) that is based on path-comparison takes time Ω(n 3 ) on some graph of n 2 edges. In =-=[22]-=- Pettie shows that any algorithm needs Ω(m·n+n 2 ·log n) time to compute APSP on directed graphs if it uses an hierarchybased approach as in [10, 23, 26]. By hierarchy-based approach we mean that one ...

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...2 1 It can also be sped up using an integer sorting algorithm in conjunction with Thorup’s reduction [Tho00] from priority queues to sorting. 2 Hagerup actually proved P = O(min{mlog log C, mn}); see =-=[Pet04]-=- for the O(mlog n) bound.s1400 SETH PETTIE AND VIJAYA RAMACHANDRAN where C is the maximum edge weight and M = O(m + n log log n). After the initial publication of our results [PR02a], Pettie [Pet04, P...

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...nce, 66123 Saarbrücken, Germany. ‡the work was done while the author was a masters student in the department of computer science at I.I.T. Delhi, India. 1 real edge-weights, the best known algorithm =-=[14]-=- for this problem requires O(mn + n2 log log n) time. However, for graphs with m = Θ(n2), this algorithm has a running time of Θ(n3) which matches that of the old and classical algorithm of Floyd and ...

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...The fastest known all-pairs shortest paths algorithm has only marginally subcubic running time [5]. For sparser graphs, repeated applications of Dijkstra’s algorithm yield O(mn+n 2 logn) time. Pettie =-=[12]-=- gave a slightly improved bound of O(mn + n 2 loglogn). Even for sparse graphs, these algorithms are too slow for many ∗ Department of Mathematics and Computer Science, University of Southern Denmark....

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...s, we can compute SSSP in O(m+n log log n) time [26], or even in O(m) time if the graph is undirected [23]. In the case of real weights, we can solve APSP in O(mn+n2 log log n) time for sparse graphs =-=[20]-=- and in O(n3√log log n/log n) time for dense graphs [22, 27]. However, the above improvements are “only” by logarithmic factors, and in our results for distance sensitivity oracles, we are aiming at p...