R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computing Systems and Sciences, 51:400-403, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....paths problem in a directed graph has complexity #(m # n) if m = O(n # n) 9] Most known shortest path algorithms, including those by Dijkstra, Bellman Ford and Floyd Warshall, satisfy this model. The exceptions to this model are the matrix multiplication based algorithms by Fredman and others [4, 17, 20]. The same lower bound construction shows that any k simple shortest paths algorithm that finds the best candidate path for each possible branch point o# previously chosen paths is also subject to this lower bound, even for k = 2. All known algorithms for the k simple shortest paths fall into ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51(3):400--403, 1995.

....on by Dijkstra [1] Bellman Ford, Floyd Warshall and others (see [2] all work in the comparison addition model with real edge weights. Since then most progress on shortest paths problems has come by assuming integral edge weights. Techniques based on scaling [3 5] integer matrix multiplication [6 10], and fast integer sorting (see [11 15] for recent results) only work with integer edgeweights, and until recently it appeared as though the component hierarchy approach used in [16] and [17] also required integers. We refer the reader to a recent survey paper [18] for more background and ....

R. Seidel, On the all-pairs-shortest-path problem in unweighted undirected graphs, J. Comput. Syst. Sci. 51 (3) (1995) 400-403.

....San Francisco, CA. Authors address: Department of Computer Sciences, The University of Texas at Austin, Austin, TX 78712. Also supported by an MCD Graduate Fellowship. E mail: seth cs.utexas.edu. E mail: vlr cs.utexas.edu. based on scaling [G85b, GT89, G95] and fast matrix multiplication [S95, GM97, AGM97, Tak98, SZ99, Z02] have running times which depend on the magnitude of the integer edge weights, and therefore yield improved algorithms only for suciently small edge weights. In the case of the matrix multiplication based algorithms the critical threshold is rather low: even edge weights sublinear in n can be too ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci., vol. 51 (1995), pp. 400-403.

.... to denote O(f(n)polylog(n) gave an algorithm whose running time is (3 ) 2 ) for APSP on directed graphs whose weights are integers in the set f Gamma1; 0; 1g, where is the best known exponent for matrix multiplication: currently, 2:376 [2] Galil and Margalit [11, 12] and Seidel [24] gave O(n ) algorithms for solving APSP for unweighted undirected graphs whose weights are integers in the set f Gamma1; 0; 1g. Shoshan and Zwick [25] and Zwick [28] achieve the best known bound for APSP with positive integer edge weights less than C: O(Cn ) for undirected graphs [25] ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51(3):400--403, December 1995.

....graphs with small integer weights: they gave an algorithm whose running time is (3 #) 2 ) for APSP on directed graphs whose weights are integers in the set 0, 1 , where # is the best known exponent for matrix multiplication: currently, # 2. 376 [5] Galil and Margalit [16, 17] and Seidel [34] gave ) algorithms for solving APSP for unweighted undirected graphs whose weights are integers in the set 0, 1 . Shoshan and Zwick [35] and Zwick [38] achieve the best known bound for APSP with positive integer edge weights less than C: O(Cn ) for undirected graphs [35] and O(C ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51(3):400--403, December 1995.

....algorithms for the APSP problem for graphs with small integer weights. They obtained an algorithm whose running time is O(n (3 ) 2 ) for solving the APSP problem for directed graphs with edge weights taken from the set f Gamma1; 0; 1g. Galil and Margalit [14] 15] and independently Seidel [22], obtained O(n ) time algorithms for solving the APSP problem for unweighted undirected graphs. Seidel s algorithm is much simpler. The algorithm of Galil and Margalit can be extended to handle small integer weights. In this work we present an improved algorithm for solving the APSP ....

....in O(n 2:5 ) time. Even if this were the case, there would still be a gap between the complexities of the directed and undirected versions of the APSP problem. As mentioned, the APSP for undirected graphs with small integer weights can be solved in O(n ) time, as shown by Seidel [22] and by Galil and Margalit [14] 15] We next show that the gap between the directed and the undirected versions of the APSP problem can be closed if we are willing to settle for approximate shortest paths. We say that a path between two vertices i and j is of stretch 1 ffl if its length is at ....

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R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400--403, 1995.

....algorithms for the APSP problem for graphs with small integer edge weights. They obtained an algorithm whose running time is O(n (3 ) 2 ) 1 for solving the APSP problem for directed graphs with edge weights taken from the set f1; 0; 1g. Galil and Margalit [GM97a] GM97b] and Seidel [Sei95], obtained O(n ) time algorithms for solving the APSP problem for unweighted undirected graphs. Seidel s algorithm is much simpler. The algorithm of Galil and Margalit has the advantage that it can be extended to handle small integer weights. The running time of their algorithm, when used ....

....in O(n 2:5 ) time. Even if this were the case, there would still be a gap between the complexities of the directed and undirected versions of the APSP problem. As mentioned, the APSP for undirected graphs with small integer weights can be solved in O(n ) time, as shown by Seidel [Sei95] and by Galil and Margalit [GM97a] GM97b] See also Shoshan and Zwick [SZ99] We next show that the gap between the directed and the undirected versions of the APSP problem can be closed if we are willing to settle for approximate shortest paths. We say that a path between two vertices i and j ....

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R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400-403, 1995. 26

.... which makes O(n 2:5 ) comparisons and additions (the best implementations of Fredman s algorithm [F76, Tak92] are only marginally better than n 3 ) There is a large body of work on fast APSP algorithms for dense graphs that restrict the range of edge weights to relatively small integers (see [S95, GM97, SZ99] for undirected graphs and [AGM97, Z98] for directed graphs) We have focussed on shortest path algorithms for general graphs. There are several faster algorithms for restricted inputs, e.g. Fre91, KS93, HK 97, FR01, Tho01] for planar graphs and [SR94, EH97, EH99, HS99, CKT00] for certain ....

....edge lengths. No adaptations of scaling algorithms [G95, G85b, GT89] to the reals or the comparison addition model are known; the RAM based priority queue algorithms [AM 90, Tho96, Ram96, Ram97, CGS97, Hag00] require word sized integers, as do the matrix multiplication based algorithms in [S95, GM97, SZ99, AGM97, Z98]. One interesting aspect of our SSSP algorithm is that for certain ranges of m, n, and r (e.g. m = O(n) r = 2 2 o( m;n) its comparison addition complexity is unknown: it depends upon the decision tree complexity of the minimum spanning tree problem see [PR00] To date the best upper ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Sys. Sci., vol. 51 (1995), 400-403.

.... small integer weights: they gave an algorithm whose running time is O(n (3 ) 2 ) 1 for APSP on directed graphs whose weights are integers in the set f1; 0; 1g, where is the best known exponent for matrix multiplication: currently, 2:376 [4] Galil and Margalit [11, 12] and Seidel [22] gave O(n ) algorithms for solving APSP for unweighted undirected graphs whose weights are integers in the set f1; 0; 1g. Shoshan and Zwick [23] and Zwick [25] achieve the best known bound for APSP with positive integer edge weights less than C: O(Cn ) for undirected graphs [23] ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51(3):400-403, December 1995.

....algorithm which makes O(n 2:5 ) comparisons and additions. The best implementations of Fredman s algorithm [F76, Tak92] are only marginally better than n 3 . There is a large body of work on fast APSP algorithms for dense graphs that restrict the range of values allowed for edge weights (see [S95, GM97] for undirected graphs and [AGM97, Z98] for directed graphs) In this paper we study undirected shortest paths problems. Since these problems are only interesting on positive edge lengths 1 , Dijkstra s algorithm may be used with its usual time bounds. Until recently no SSSP techniques speci c ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Sys. Sci., vol. 51 (1995), 400-403.

....for each agent, it is sufficient to solve the all pairs shortest paths (APSP) problem. No faster exact method is known. The APSP problem can be solved by various algorithms in time O(nm n 2 log n) 13, 19] O(n 3 ) 12] or more quickly using fast matrix multiplication techniques [2, 11, 25, 26]. Faster specialized algorithms are known for graph classes such as interval graphs [3, 9, 23] and chordal graphs [8, 17] and the APSP problem can be solved in averagecase in time O(n 2 log n) for various types of random graph [10, 16, 20, 22] Because these results are slow, specialized, # ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51(3):400-- 403, December 1995.

....for the APSP problem on general graphs without negative length cycles is designed by Floyd [8] and it requires O(n 3 ) time. As to the algorithms of solving the APSL problem, the most popular method is offered by Johnson [11] and it can be run in O(nm n 2 log n) time. Recently, Seidel [16] showed that the APSL problem on unweighted graphs (i.e. all the edges on the graph have the same weight) can be solved in O(M(n) log n) time with O(n 2 ) space. M(n) denotes the time complexity of multiplying two n Theta n matrices for small integers, and the time required is currently known ....

R. Seidel, On the all-pairs-shortest-path problem in unweighted undirected graphs, J. Comput. System Sci., 51 (1995) 400-403.

....the problem can actually be solved in O(mn) time. In the last decade, it was shown that fast algorithms for algebraic matrix multiplication can be used to obtain faster algorithms for solving the APSP problem in dense graphs with small integer edge weights. Galil and Margalit [8] 9] and Seidel [11], obtained O(n ) time algorithms for solving the APSP problem for unweighted undirected graphs. Seidel s algorithm is simpler. The algorithm of Galil and Margalit can be extended to handle small integer weights. The currently best upper bound on , the exponent of matrix multiplication, is ....

....were found, we can find a concise representation of shortest paths between all pairs of vertices in the graph using only three witnessed Boolean matrix products, in the unweighted case, or three witnessed distance products in the weighted case. For the unweighted case, this was shown by Seidel [11]. We extend his technique to the weighted case. Let A and B two n Theta n matrices . An n Theta n matrix W is said to be a matrix of witnesses for the distance product A B if for every 1 i; j n we have 1 w ij n and (A B) ij = a i;w ij b w ij ;j . As shown in Zwick [16] a matrix of ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci., 51:400-- 403, 1995.

....then be called a shortest lightest path. We decided against it as we did not want to redefine the term shortest paths. 1 of the graph are integers of small absolute value, then fast matrix multiplication algorithms can be used to speed up the operation of algorithms for the APSP problem. Seidel [Sei95] and Galil and Margalit [GM97a] GM97b] give O(n ) algorithms for solving the APSP problem for undirected graphs with small integer weights, where 2:376 is the exponent of square matrix multiplication. The algorithm of Seidel [Sei95] works only for unweighted graphs. For directed ....

....operation of algorithms for the APSP problem. Seidel [Sei95] and Galil and Margalit [GM97a] GM97b] give O(n ) algorithms for solving the APSP problem for undirected graphs with small integer weights, where 2:376 is the exponent of square matrix multiplication. The algorithm of Seidel [Sei95] works only for unweighted graphs. For directed graphs with small integer weights, Alon, Galil and Margalit [AGM97] give an algorithm whose complexity is O(n (3 ) 2 ) Note that (3 ) 2 2:688. The author [Zwi98] recently improved this result and obtained an algorithm that solves the ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400--403, 1995.

....multiplication. It is fairly easy to see that solving the APSP problem exactly, even on unweighted graphs, is at least as hard as Boolean matrix multiplication. Recent works, by Alon, Galil and Margalit [AGM97] Alon, Galil, Margalit and Naor [AGMN92] Galil and Margalit [GM93] GM97] and Seidel [Sei95] have shown that if matrix multiplication can be performed in O(M(n) time, then the APSP problem for unweighted directed graphs can be solved in O( p n 3 M(n) time and the APSP problem for unweighted undirected graphs can be solved in O(M(n) time ( O(f) means O(f polylog n) The ....

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400--403, 1995.

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R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computing Systems and Sciences, 51:400-403, 1995.

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Seidel, R. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences 51, 3 (Dec. 1995), 400--403.

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R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400--403, 1995.

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R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400--403, 1995.

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R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400--403, 1995.

No context found.

R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51:400-403, 1995.

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Seidel, R. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences 51, 3 (Dec. 1995), 400--403.

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R. Seidel, On the all-pairs-shortest-path problem in unweighted undirected graphs, J. Comput. System Sci. 51 #1995# 400#403.

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