J. Kleinberg, R. Rubinfeld, "Short paths in expander graphs," Proc. 37th IEEE Symp. on Foundations of Computer Science, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a constant (depending on and less than the degree) number of pairs. This result is optimal to within constant factors, and has been also has been extended to expander digraphs [7] An immediate consequence of this is an O(log n) approximation for MEDP on such expanders. Kleinberg and Rubinfeld [13] in 1996 had used an earlier result of Broder, Frieze, and Upfal [8] to show that a deterministic online algorithm, the so called bounded greedy algorithm (BGA) gave an O(log n log log n) approximation guarantee. In fact Frieze s result mentioned above implies an O(log n) bound for BGA. In the ....

....network. Thus we obtain: Theorem 7. For UFP on the ring there is a (1 ff) approximation where ff is the approximation factor for the problem on the line. 5 Concluding Remarks We also note that an online O(F log n) approximation for UFP can be obtained by combining the bounded greedy algorithm [13], and the online algorithm of Awerbuch, Azar and Plotkin [2] we defer the details to the full version of the paper. Acknowledgments: We are grateful to Bruce Shepherd for suggesting the unsplittable flow problem on the line and for several discussions. ....

J. M. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pp. 86--95. 1996.

.... LP duality and metric embeddings) are complex and involve non integral flows [57, Chapter 21] However, there are complementary results suggesting that near optimal congestion (up to poly log n factors) can also be achieved with integral short paths and decentralized, on line algorithms (e.g. see [23, 29]. Therefore, a constant expansion factor is thought of as an excellent promise for routing. Random regular graphs are long known to possess constant expansion [43, Chapter 5.6] These, together with explicit constructions [36] have found many applications in networks: 47, 48, 49] for ....

J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proc. 37th Symposium on Foundations of Computer Science (FOCS), pages 86--95. IEEE, 1996.

.... (10) for ISl 2 provided we have e2 rio For Isl 2 we have 421 i and then and so Phase 1 succeeds with respect to A, The same argument applies to B, To ensure these paths are of length O(logn) we can solve a minimum cost maximum flow problem as indicated in Kleinberg and Rubinfeld [11]. Lemma I Throughout the algorithm Il (1 To)n, j = 3, 4, rio 10. Proof: First consider Va. We know from (8) that ra is a (fior 2 ) expander throughout the execution of Phase 2. We can use the strong edge expansion of Pa to prove some vertex expansion and conclude the diameter of Pa is ....

J.Kleinberg and R.Rubinfeld, Short paths in expander graphs, Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, (1996) 86-95.

....Srinivasan [32] where Delta denotes the maximal degree in the graph, ff the expansion of the graph, N is the number of nodes, and M is the number of edges. The latter result is based on multicommodity flow algorithms, which is one of the most common approaches for the MDPP and related problems [28, 22, 23, 17, 33]. 1 The other frequently used approach is based on random walks, which was useful especially for expander graphs [27, 9, 10, 12, 11] Other important results for specific graphs are polylogarithmic and later O(1) approximations for mesh like graphs [5, 1, 18, 19, 16] There are also a few results ....

....p N) lower bound for the N Theta N mesh by Blum, Fiat, Karloff and Rabani. Kleinberg [16] provides an alternative proof. The known deterministic on line algorithms for the MDPP with at most polylogarithmic competitive ratios are for the hex [5] for graphs with strong expansion properties [17] and for hypercubic networks [21] For the expanders and the hypercubic networks the bounded greedy algorithm is shown to have this performance. Most of the afore mentioned randomized algorithms suffer from the drawback that only the expected competitive ratio is good. It may happen that they ....

J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86--95, 1996.

....hypercube, for permanent calls problem. Although there are algorithms with better competitive ratios for other networks (e.g. randomized logarithmically competitive algorithms for line [9] trees [2, 3] and meshes [11] deterministic O(log N) competitive algorithms for expander graphs [10]) we are not aware of any ones for Bene s network and the hypercube. In the next section the crucial notion of i similarity is introduced and some usefull lemmas as well as necessary and sucient condition for existence of a free path in a partialy blocked Bene s network are given. This yields an ....

J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of FOCS, pages 86-95, 1996.

....provably good routability and fault tolerance properties; the reader is referred to [45] Let Delta = Delta(G) denote the maximum degree of the vertices of G. Given an instance (G; T ) of the ucufp, let ff (T ) denote the optimum value of the LP relaxation studied above. Then, a nice result of [39] shows the following: for any constant ffl 0, there exists a feasible solution to the LP relaxation with: ffl objective function value at least ff (T ) 1 ffl) and such that ffl all flow paths have length at most d 0 = O( Delta 2 fi Gamma2 log 3 n) We can easily add the linear ....

....of O(log a= log log a) Similar improvements are easily seen for other ranges of y also. Thus, routing along short paths is very beneficial in keeping the congestion low. Indeed, specific useful classes of graphs such as expanders have been shown to be rich in the structure of such short paths [48, 39]. Recalling the graph theoretic parameters such as Delta and fi from Section 5, the work of [39] can essentially be used to ensure that a = O( Delta 2 fi Gamma2 log 3 n) So if, for instance, Delta and fi Gamma1 are O(polylog(n) then a also can be bounded by O(polylog(n) Hence, if ....

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J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proc. IEEE Symposium on Foundations of Computer Science, pages 86--95, 1996.

....UFP (that is, all edges have a capacity one) Baveja and Srinivasan [3] gave an algorithm with an O( 2 2 log 3 n) approximation ratio, where is the maximum degree, is the expansion, and n is the number of nodes in the network. Using the techniques of Kleinberg and Rubinfeld [9] and results of Leighton and Rao [16] this can be decreased to O( 2 2 log 2 n) Recently, Kolman and Scheideler [13] improved this ratio to O( 2 1 log n) Using a new parameter called the ow number F of a network, we improve the ratio further to O(F ) with F = O( 1 log n) For ....

J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86-95, 1996.

.... the maximal degree in the graph and the edge expansion of the graph (we stress that may be a function of n, e.g. log 1 n) for n node butter y graphs) The result is based 1 on multicommodity ow algorithms, which is one of the most common approaches for the MDPP and related problems [32, 26, 27, 20, 37]. The other frequently used approach is based on random walks, which was useful especially for expander graphs [31, 11, 12, 14, 13] Other important results for speci c graphs are polylogarithmic and later O(1) approximations for mesh like graphs [5, 1, 21, 22, 19] There are also a few results ....

....p n) lower bound for the p n p n mesh by Blum, Fiat, Karlo and Rabani. Kleinberg [19] provides an alternative proof. The known deterministic on line algorithms for the MDPP with at most polylogarithmic competitive ratios are for the hex [5] for graphs with strong expansion properties [20] and for hypercubic networks [24] Combination of the techniques of Kleinberg and Rubinfeld [20] and the results of Leighton and Rao [26] see Lemma 1.2 of this paper) yields a competitive ratio of O( 1 log n) for general graphs for the special case that the sequence of requests forms a ....

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J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proc. of the 37th IEEE Symposium on Foundations of Computer Science, pages 86-95, 1996.

.... s i ; t i 2 T . If the graph is directed and the directed paths are requested, the problem is called max directed vertex edge disjoint paths (DVDP DEDP) All the above versions were proved to be NP complete [5, 6] A lot of effort has been made in design polynomial time approximation algorithms [7, 8, 9, 10]. Most of them are about special classes of graphs, e.g. expander graphs [10] densely embedded graphs [8] high diameter planar networks [9] and high bandwidth models [1] For general graphs, the best known performance ratios are O( p jE 0 j) for EDP and O( p jV 0 j) for VDP, where jE 0 j ....

.... problem is called max directed vertex edge disjoint paths (DVDP DEDP) All the above versions were proved to be NP complete [5, 6] A lot of effort has been made in design polynomial time approximation algorithms [7, 8, 9, 10] Most of them are about special classes of graphs, e.g. expander graphs [10], densely embedded graphs [8] high diameter planar networks [9] and high bandwidth models [1] For general graphs, the best known performance ratios are O( p jE 0 j) for EDP and O( p jV 0 j) for VDP, where jE 0 j and jV 0 j are the numbers of edges and vertices appear in the paths in an ....

J. Kleinberg and R. Rubinfeld, "Short Paths in Expander Graphs", Proc. 37th IEEE Symposium on Foundations of Computer Science, pp. 86-95, 1996.

....results obtained by Leighton and Rao [13] using the multi commodity ow approach. The case (n) 1= log n) improves the 2 bound of Theorem 1 of [13] the case (n) O(1) improves the bound of Theorem 2 in [13] and makes the result constructive. We also note that Kleinberg and Rubinfeld [10] have recently used our result in their analysis of a greedy algorithm for nding short disjoint paths on expanders. We next show constructively that K = n= log n) 2 pairs of vertices can be connected by edge disjoint paths, provided the graph has suciently strong expansion. Theorem 2 Suppose ....

J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In 37th Annual Symposium on Foundations of Computer Science, Burlington, Vermont, 14-16 Oct. 1996. IEEE.

.... S) j Sj 1 j Sj 0 r 2 rj Sj ( S) S) Therefore, Phase 1 succeeds with respect to X; X. The same argument applies to Y; Y . To ensure these paths are of length O(log n) we can solve a minimum cost maximum ow problem as indicated in Kleinberg and Rubinfeld [13]. 2.3.2 On the size of V a Lemma 2 Throughout GenPaths we have jV a j (1 0 )n where 0 = 0 10 : Proof: It follows from (9) that a is a ( 0 r=2) expander throughout the execution of Phase 1. It follows from Lemma 1 that the diameter of a is always at most = d2 log 1 0=2 n ....

J. Kleinberg and R. Rubinfeld, Short paths in expander graphs, Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, (1996) 86-95.

....Let L be a suitable chosen parameter (a multiple of the diameter in the case of hypercubic and expander graphs) Given a request, reject it if there is no free path of length at most L between its terminal nodes. Otherwise accept it and for routing use any such path. Kleinberg and Rubinfeld [17] showed that this algorithm achieves polylogarithmic competitive ratio on bounded degree graphs with strong expansion properties. This paper extends this result by proving that the algorithm achieves polylogarithmic competitive ratio on hypercubic networks as well, namely O(log N) on the Bene s ....

....diameter d tree. Awerbuch et al. [6] mention a deterministic p N) lower bound for N N mesh, by Blum, Fiat, Karlo and Rabani. Kleinberg [16] provides an alternative proof. The known deterministic on line algorithms for the MDPP, with polylogarithmic competitive ratios, are for expander graphs [17] and for the hex [6] There was also an e ort to design algorithms that work well on any network topology. The deterministic lower bound N) for the line shows that there is no hope for deterministic algorithms with reasonable competitive ratio. Bartal, Fiat and Leonardi [7] prove this e ort to ....

J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86-95, 1996.

....3 N) by Srinivasan [32] where denotes the maximal degree in the graph, the expansion of the graph, N is the number of nodes, and M is the number of edges. The latter result is based on multicommodity ow algorithms, which is one of the most common approaches for the MDPP and related problems [28, 22, 23, 17, 33]. 1 The other frequently used approach is based on random walks, which was useful especially for expander graphs [27, 9, 10, 12, 11] Other important results for speci c graphs are polylogarithmic and later O(1) approximations for mesh like graphs [5, 1, 18, 19, 16] There are also a few results ....

....deterministic p N) lower bound for the N N mesh by Blum, Fiat, Karlo and Rabani. Kleinberg [16] provides an alternative proof. The known deterministic on line algorithms for the MDPP with at most polylogarithmic competitive ratios are for the hex [5] for graphs with strong expansion properties [17] and for hypercubic networks [21] For the expanders and the hypercubic networks the bounded greedy algorithm is shown to have this performance. Most of the afore mentioned randomized algorithms su er from the drawback that only the expected competitive ratio is good. It may happen that they ....

J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86-95, 1996.

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J. Kleinberg, R. Rubinfeld, "Short paths in expander graphs," Proc. 37th IEEE Symp. on Foundations of Computer Science, 1996.

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J. Kleinberg and R. Rubenfield, Short paths in expander graphs, in Proceedings of the 37th Symposium Foundations of Computer Science, 1996, pp. 86--95.

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J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86--95, 1996.

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J. Kleinberg and R. Rubinfeld, Short paths in expander graphs, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, 1996, pp. 86--95.

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J. M. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86-95, 1996.

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J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86-95, 1996. 39

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J. M. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86--95, 1996.

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J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 86--95, 1996.

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J. Kleinberg and R. Rubinfeld, Short paths in expander graphs, Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, (1996) 86-95.

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J. Kleinberg and R. Rubenfield. Short paths in expander graphs. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pages 86--95, 1996.

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J. Kleinberg and R. Rubenfield. Short paths in expander graphs. In Proc. 37th Symp. Foundations of Computer Science, pages 86--95, 1996.

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