A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....result here is the first constant factor approximation for any class of graphs other than trees, when one does not require Omega Gammaqui n) parallel copies of each edge. A different approach is taken in papers of Peleg and Upfal [19] and Broder, Frieze, and Upfal [7] see also Broder et al. [6]) Here the underlying graph G is assumed to have strong expansion properties; in this case one can prove that any set of terminal pairs of at most a given size must be realizable in G, and that corresponding paths can be found in (randomized) polynomial time. The results in [7] are strong enough ....

A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.

....interests [1] 2] 3] 4] This paper introduces a distributed algorithm for constructing expander networks, which are suitable for peer to peer networking, without using any globally known server. Interesting properties and algorithms have been discovered for random regular graphs [5] [6], 7] 8] In particular, it has been found that random regular graphs are expected to have big eigenvalue gaps [9] with high probability, and thus are good expanders. In this paper, we form expander graphs by constructing a class of regular graphs which we call H graphs. An H graph is a ....

A. M. Frieze and L. Zhao, "Optimal construction of edge-disjoint paths in random regular graphs," in Proceedings of the Tenth Annual ACMSIAM Symposium on Discrete Algorithms,Jan. 17--19 1999, pp. 346-- 355.

.... if paths are required to be shortest paths connecting the respective nodes [24] which is another suggestion that a minimal adaptive routing might be weaker than a fully adaptive routing even for hypercube, which possess a very large number of paths of optimal length) Broder, Frieze and Upfal [4, 5] proved that in expander graphs such collections of edge disjoint paths always exist, provided the size of W is at most cn log n for some positive constant c that depend only on expanding factor of the graph. However, the algorithm that finds such paths is extremely complicated and the constant c ....

A. Z. Broder, A. M. Frieze, S. Suen, and E. Upfal, "Optimal construction of edge-disjoint paths in random graphs", in Proceeding of the 5th ACM-SIAM Symposium on Discrete Algorithms, pp. 603-612, 1994.

....result here is the first constant factor approximation for any class of graphs other than trees, when one does not require Omega Gammaqui n) parallel copies of each edge. A different approach is taken in papers of Peleg and Upfal [19] and Broder, Frieze, and Upfal [7] see also Broder et al. [6]) Here the underlying graph G is assumed to have strong expansion properties; in this case one can prove that any set of terminal pairs of at most a given size must be realizable in G, and that corresponding paths can be found in (randomized) polynomial time. The results in [7] are strong enough ....

A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.

....is similar to that of Aumann and Rabani, but improves their bound by a factor of log n. See Section 2 for more on the comparison of the algorithms. Finally, a different line of work related to the construction of disjoint paths can be found in papers of Broder, Frieze, Peleg, Suen, and Upfal [5, 6, 17]. Here the underlying graph G is assumed to have strong expansion properties; in this case one can prove that any set of terminal pairs of at most a given size must be realizable in G. The goal then is to find the paths in (randomized) polynomial time. In this paper we deal only with planar ....

A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.

....in their own right, as they provide tools for constructing disjoint paths in more general settings. These techniques are surveyed in Section 2.2. Finally, a different line of work related to the construction of disjoint paths can be found in papers of Broder, Frieze, Peleg, Suen, and Upfal [5, 6, 18]. Here the underlying graph G is assumed to have strong expansion properties; in this case one can prove that any set of terminal pairs of at most a given size must be realizable in G. The goal then is to find the paths in (randomized) polynomial time. In this paper we deal only with planar ....

A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.

....for the circuits. It then uses the Lov asz Local Lemma to show that routes can be selected from the candidates to resolve the remaining congestion. Recently, Broder, Frieze, and Upfal have used similar ideas to prove better existential results [8] In related work, Broder, Frieze, Suen, and Upfal [5, 6] achieve results to within a constant factor of optimal for both node disjoint and edgedisjoint paths on a certain set of random graphs (with nonconstant degree) This result is an nice demonstration of the utility of using random walks in this setting. 1.2 Our Results. In this paper, we describe ....

....[2] That is, you can embed a polylogarithmically smaller version of their network into a constantdegree expander. This exercise, however, leads to ever more cumbersome solutions and performance measures. finding disjoint paths. In this section, we heavily cite without proof previous results in [12, 9, 5] as well as technical lemmas that we prove in section 3. 2 Constant Degree Expander Graphs. In this section, we give an outline of a method that gives disjoint paths in constant degree expander graphs with sufficient expansion. We proceed with some definitions. Definition 1: A graph G = V; E) ....

A. Broder, A. Frieze, S. Suen, and E. Upfal. Optimal construction of edge-disjoint paths in random graphs. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 603--612, 1994.

....result here is the first constant factor approximation for any class of graphs other than trees, when one does not require Omega Gammaqui n) parallel copies of each edge. A different approach is taken in papers of Peleg and Upfal [21] and Broder, Frieze, and Upfal [8] see also Broder et al. [7]) Here the underlying graph G is assumed to have strong expansion properties; in this case one can prove that any set of terminal pairs of at most a given size must be realizable in G, and that corresponding paths can be found in (randomized) polynomial time. The results in [8] are strong enough ....

A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.

....proof [3] 12] via the local lemma) that in a sufficiently strong expander K = n= log n) 2 ) is achievable. Random Graphs Random graphs are well known to be excellent expanders and so it is perhaps not surprising that they very highly routable . Broder, Frieze, Suen and Upfal [4] and Frieze and Zhao [9] see Theorems 7,8) show that they are K routable where K is within a constant factor of a simple lower bound, something that has not yet been achieved for arbitrary expander graphs. Low Congestion Path Sets One way of generalising the problem is to bound the number of ....

....than all the edges available. In the case of bounded degree expanders, this absolute upper bound on k is O(n= log n) The results mentioned above use only a vanishing fraction of the set of edges of the graph, thus are far from reaching this upper bound. In contrast, Broder, Frieze, Suen and Upfal [4] and Frieze and Zhao [9] show that for G n;m and G r reg the absolute upper bound is achievable within a constant factor, and present algorithms that construct the required paths in polynomial time. 10 Theorem 7 Let m = m(n) be such that d = 2m=n (1 o(1) log n. Then, as n 1, with ....

A.Z.Broder, A.M.Frieze, S.Suen and E.Upfal, Optimal construction of edge disjoint paths in random graphs, Proceedings of 4th Annual Symposium on Discrete Algorithms, (1994) 603-612.

....more edges than all the edges available. In the case of bounded degree expanders, this absolute upper bound on is O(n= log n) The results mentioned so far use only a vanishing fraction of the set of edges of the graph, thus are far from reaching this upper bound. Broder, Frieze, Suen and Upfal [8] have proved that for the model of random graphs, G n;m , the absolute upper bound is achievable to within a constant factor, but only if the average degree is at least ln n. In this work, we show that this result holds when the minimum degree is a large enough constant. Without significant loss ....

....to b i , for each i = 1; 2; Furthermore, there is an O(n 3 ) time randomized algorithm for constructing these paths. This result is the best possible up to constant factors. The median distance between pairs of vertices in G is Omega Gamma 16 r n) The need for fi 1 is discussed in [8]. The analogous problem of finding vertex disjoint paths in random graphs is dealt with in [9] 2 Preliminaries The paper contains a few unspecified absolute constants of which ff above is the first. Exact values could be given but it is easier for us and the reader if we simply give the ....

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A. Z. Broder, A. M. Frieze, S. Suen, and E. Upfal, Optimal Construction of EdgeDisjoint Paths in Random Graphs, to appear.

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A. Z. Broder, A. M. Frieze, S. Suen, and E. Upfal, Optimal Construction of EdgeDisjoint Paths in Random Graphs, SIAM Journal on Computing 28, 541-574 (1999) 603-612.

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A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.

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A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM SODA, 1994, pp. 603--612.

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A. Broder, A. Frieze, S. Suen, E. Upfal, "Optimal construction of edge-disjoint paths in random graphs," Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, 1994, pp. 603--612.

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