C.C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426--461, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....set [n] 1, n) ad m edges depends heavily on whether or not edge lengths are allowed to be negative. In fact, if all edge lengths are non negative, then Dijkstras algorithm solves the single source shortest paths problem in near linear time O(m nlogn) 8, 11] and the algorithms of McGeoch [20] ad Karger, Koller, and Phillips [18] solve the all pairs shortest paths problem in time O(nlH I n21ogn) where H is the set of edges that are a shortest path between their endpoints. In the general case, the Bellman Ford algorithm [2, 10] solves the single source shortest paths problem in time ....

....[12, 4.6) and (4.14) This implies that the diameter A(r) is O( logn) n) with high probability. Davis and Prieditis [7] showed that for exponentially distributed r, the expected length of a shortest path is of exactly this order of magnitude. This is also true for r uniformly distributed; see [7, 20]. Furthermore, if (i, j) is the p th shortest edge in the adjacency list of i where p Blogn (for a sufficiently large constant B) then ri,j 7 A(r) with high probability [12, Lemma 4.3] This means that, with high probability, edge (i, j) is irrelevant, that is, is not contained in any ....

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C. C. McGeoch, All-pairs shortest paths and the essential subgraph, Algorithmica 13 (1995) 426-441

....in Department of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. E mail address: zwick math.tau.ac. il the graph and m is the number of edges that participate in shortest paths (Dijkstra [9] Johnson [17] Fredman and Tarjan [12] Karger, Koller and Phillips [18] and McGeoch [20]) The running time of the above algorithm may be as high as Omega Gamma n 3 ) Can the APSP problem be solved in sub cubic time Fredman [11] showed that the APSP problem for weighted directed graphs can be solved non uniformly in O(n 2:5 ) time. More precisely, for every n, there is a ....

C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426--461, 1995.

....this paper appeared in [Zwi98] y Department of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. E mail address: zwick math.tau.ac.il. Work supported in part by THE ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Sciences and Humanities. 1 Phillips [KKP93] and McGeoch [McG95]) For undirected graphs with nonnegative integer edge weights, a running time of O(mn) can be obtained by running a recent single source shortest paths algorithm of Thorup [Tho99] Tho00] from each vertex of the graph. The running time of all the above mentioned algorithms may be as high as n ....

C.C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426-461, 1995.

....case. Two algorithms for the all pairs shortest paths problem with non negative arc costs were proposed that dismiss the idea of iterating over the n possible source vertices and of solving the corresponding single source shortest paths problem in each iteration. Instead, the algorithms of McGeoch [57] and Karger, Koller, and Phillips [49] iterate over the arcs, thereby solving the single source shortest paths problems simultaneously. The running time of their algorithms is O(n H n 2 log n) where H denotes the set of arcs that are a shortest path between their starting point and their ....

....in Lemmata 3.4 and 3.10 are essentially tight, as is indicated by the following results. Davis and Prieditis [17] showed that for r distributed exponentially (with parameter 1) the expected cost of a shortest path is of order (log n) n O(1 n) This is also true for r uniformly distributed; see [17, 57]. In fact, this result holds if the distribution of r is chosen from a fairly general class of distributions, including both exponential and uniform distributions, as was proved (for undirected graphs) by Janson [45] recently. Janson also studied the number of arcs on shortest paths, proving ....

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C. C. McGeoch, All-pairs shortest paths and the essential subgraph, Algorithmica 13 (1995), pp. 426--441

.... also [CLR90] Algorithms for the APSP problem which work on directed graphs with non negative edge weights and whose running times are O(m n n 2 log n) where m is the number of edges participating in shortest paths, were obtained by Karger, Koller and Phillips [KKP93] and by McGeoch [McG95] Karger et al. KKP93] also obtain an Omega Gamma mn) lower bound on any path comparison A preliminary version of this paper appeared in the proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, Burlington, Vermont, 1996, pages 452 461. y Department of Computer ....

C.C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426--461, 1995.

.... Data sets 1 and 2 are the expected costs of Quicksort and Insertion Sort, for whichformulas are known exactly [9] Sets 3 through 6 are from experiments on the FFD and FF rules for bin packing [3] 4] Sets 7 and 8 are from experiments on distances in random graphs having uniform edge weights [11]. The X vectors havevarious ranges and intervals# except for the first two cases, the Y s represent means of several independent trials. Results appear in Figure 2. The left column presents the best analytical bounds known for each. The entries NA for PWD mark cases where this rule was not ....

C. C. McGeoch (1995) "All pairs shortest paths and the essential subgraph," Algorithmica (13), pp. 426--441.

.... Data sets 1 and 2 are the expected costs of Quicksort and Insertion Sort, formulas for which are known exactly [9] Sets 3 through 6 are from experiments on the FFD and FF rules for bin packing [3] 4] Sets 7 and 8 are from experiments on distances in random graphs having uniform edge weights [11]. The X vectors havevarious ranges and intervals# except for the first two cases, the Y s represent means of several independent trials. Results appear in Figure 2. The left column presents the best analytical bounds known for each. The entries NA for PWD mark cases where this rule was not ....

C. C. McGeoch (1995) "All pairs shortest paths and the essential subgraph," Algorithmica (13), pp. 426--441.

.... Data sets 1 and 2 are the expected costs of Quicksort and Insertion Sort, for which formulas are known exactly [9] Sets 3 through 6 are from experiments on the FFD and FF rules for bin packing [3] 4] Sets 7 and 8 are from experiments on distances in random graphs having uniform edge weights [11]. The X vectors have various ranges and intervals; except for the first two cases, the Y s represent means of several independent trials. Results appear in Figure 2. The left column presents the best analytical bounds known for each. The entries NA for PWD mark cases where this rule was not ....

C. C. McGeoch (1995) "All pairs shortest paths and the essential subgraph," Algorithmica (13), pp. 426--441.

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C.C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426--461, 1995.

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C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426--461, 1995.

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C.C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426--461, 1995.

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C.C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426-461, 1995.

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C.C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426--461, 1995.

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