A.Z. Broder, A.M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, Volume 623, pages 308-317, July 1992, Springer-Verlag. 17  Home/Search   Document Not in Database   Summary   Related Articles   Check

....physically exchanged between the processors, which would be too costly for our purposes, but instead they are moved by manipulating the corresponding tree rings, with a cost logarithmic in the number of nodes being moved. BALANCE is based on a balancing strategy introduced by Broder et al. in [BFSU92], which makes use of a special kind of expander de ned below. De nition 6 ( BFSU92] An undirected graph G = V; E) is an (a; b) extrovert graph, for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbours in V S than in ....

....but instead they are moved by manipulating the corresponding tree rings, with a cost logarithmic in the number of nodes being moved. BALANCE is based on a balancing strategy introduced by Broder et al. in [BFSU92] which makes use of a special kind of expander de ned below. De nition 6 ([BFSU92]) An undirected graph G = V; E) is an (a; b) extrovert graph, for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbours in V S than in S. The existence of regular extrovert graphs of constant degree is proved through ....

[Article contains additional citation context not shown here]

A.Z. Broder, A.M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, Volume 623, pages 308-317, July 1992, Springer-Verlag. 17

....23] hypercubes [20, 33] and meshes [17, 28] Another class of networks on which load balancing has been studied is the class of expanders. Peleg and Upfal [31] pioneered this study by identifying certain small degree expanders as being suitable for load balancing. Their work has been extended in [9, 18, 32]. These algorithms either use strong expanders to approximately balance the network, or the AKS sorting network [3] to perfectly balance the network. Thus, they do not work on networks of arbitrary topology. Also, these algorithms work by setting up fixed paths through the network on which load is ....

A. Broder, A. M. Frieze, E. Shamir, and E. Upfal. Near-- perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming, pages 308--317, 1992.

....which would be too costly for our purposes, but instead they are virtually moved by manipulating the corresponding tree rings, with a cost logarithmic in the number of nodes being moved. The algorithm for the BALANCE substep is based on a balancing strategy introduced by Broder et al. in [BFSU92], which makes use of a special kind of expander defined below. Definition 5 [BFSU92] A graph G = V; E) is called (a; b) extrovert , for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbors in V Gamma S than in S. The ....

....by manipulating the corresponding tree rings, with a cost logarithmic in the number of nodes being moved. The algorithm for the BALANCE substep is based on a balancing strategy introduced by Broder et al. in [BFSU92] which makes use of a special kind of expander defined below. Definition 5 [BFSU92]. A graph G = V; E) is called (a; b) extrovert , for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbors in V Gamma S than in S. The existence of extrovert graphs can be proved through the probabilistic method ....

[Article contains additional citation context not shown here]

A.Z. Broder, A.M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming, pages 308--317, July 1992.

....by an undirected graph, whose nodes correspond to PEs and edges correspond to communication links. 1. 1 Dimension Exchange Algorithms There are many data distribution methods that achieve a balanced token distribution by gathering and making use of a certain amount of global information [1, 2, 4, 5, 11, 16]. Such methods are often unsatisfactory, in that they do not take into account the practical limitations of the parallel architecture, or result in algorithms that are unnecessarily complex. One method that requires no such global information is the so called dimension exchange method, which is ....

....For each colour c 2 f0; 1; 1g, 9 let E c = e 0 ; e 1 ; e n 1 ) be the ordering of the arcs coloured c in the observer tour, where e 0 is the outgoing arc at s coloured c. Each node has one outgoing arc in E c . The c ordering starting at s is the ordering (s[0] s[1]; s[n 1] of the nodes of T , where s = s[0] and e i is the outgoing arc at s[i] coloured c for all i 2 f0; 1; n 1g. For two nodes v and w of a tree T , if v = s[i] and w = s[j] in the c ordering starting at some node s, then we say the c distance between v and w is dist c (v; w) ....

A. Z. Broder, A. M. Frieze, E. Shamir, and E. Upfal, Near-perfect token distribution. Random Structures Algorithms, 5(4):559-572, 1994.

....78] hypercubes [72, 101] and meshes [68, 93] Another class of networks on which load balancing has been studied is the class of expanders. Peleg and Upfal [97] pioneered this study by identifying certain small degree expanders as being suitable for load balancing. Their work was extended in [34, 69, 98]. These algorithms either use strong expanders to approximately balance the network, or the AKS sorting network [6] to perfectly balance the network. Thus, they do not work on networks of arbitrary topology. Also, these algorithms work by transferring load along fixed paths in the network and, ....

A. Broder, A. M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. Random Structures and Algorithms, pages 559--572, 5 1994. 201

....the stage. A feature of the balancing procedure is to move only pointers to trees between the TRs without physically moving the nodes, which will be too costly for our purposes. The algorithm we present is essentially a PRAM implementation of a variant of the balancing algorithm by Broder et al. [BFSU92]. We need the following definition. Definition 1 ( BFSU92] A graph G = V; E) is called (a; b) extrovert, for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbors in V Gamma S than in S. 10 We identify the PRAM ....

....only pointers to trees between the TRs without physically moving the nodes, which will be too costly for our purposes. The algorithm we present is essentially a PRAM implementation of a variant of the balancing algorithm by Broder et al. BFSU92] We need the following definition. Definition 1 ([BFSU92]) A graph G = V; E) is called (a; b) extrovert, for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbors in V Gamma S than in S. 10 We identify the PRAM processors with the nodes of a d degree (a; b) extrovert graph, ....

[Article contains additional citation context not shown here]

A.Z. Broder, A.M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proc. of the 19th Int. Colloquium on Automata, Languages and Programming, pages 308--317, July 1992.

....physically exchanged between the processors, which would be too costly for our purposes, but instead they are moved by manipulating the corresponding tree rings, with a cost logarithmic in the number of nodes being moved. BALANCE is based on a balancing strategy introduced by Broder et al. in [BFSU92], which makes use of a special kind of expander de ned below. De nition 6 ( BFSU92] A graph G = V; E) is an (a; b) extrovert graph, for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbours in V S than in S. The ....

....but instead they are moved by manipulating the corresponding tree rings, with a cost logarithmic in the number of nodes being moved. BALANCE is based on a balancing strategy introduced by Broder et al. in [BFSU92] which makes use of a special kind of expander de ned below. De nition 6 ([BFSU92]) A graph G = V; E) is an (a; b) extrovert graph, for some a; b with 0 a; b 1, if for any set S V , with jSj ajV j, at least bjSj of its vertices have strictly more neighbours in V S than in S. The existence of extrovert graphs of constant degree is proved through the probabilistic ....

[Article contains additional citation context not shown here]

A.Z. Broder, A.M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming,Lecture Notes in Computer Science, Volume 623, pages 308-317, July 1992, Springer-Verlag.

....networks M with p processors and total load N = p, the same authors show in [8] that for all k 2, T ad (M ; p; k; 0) O(p) and T ad (M ; p; k; 0) Omega (k diam(M) with diam(M) denoting the diameter of M . Matching upper bounds are shown in [8] by Herley [2] and by Broder et al. [1] for certain classes K of expander related, bounded degree, low diameter networks, i.e. T ad (M ; p; k; O(1) O(k log p) for all M 2 K. Plaxton [9] investigates the TD problem on the d dimensional hypercube Q d . He shows that for k 2 Delta N p , T ad (Q d ; N; k; 0) Omega k ....

A. Z. Broder, A. M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proceedings of the 19th ICALP, pages 308--317, 1992.

....are redistributed among the nodes. We concentrate on the stage of distributing tasks among the processors, which is often called load balancing. The problem of load balancing has been studied extensively for various kinds of networks, including hypercubes [6, 11, 12, 13] meshes [8] and expanders [2, 5, 9, 10]. There was also work on arbitrary network topologies [1, 3, 4] We present a new simple and easy to implement algorithm on the hypercube with a competitive performance for the case of relatively small initial distribution of tasks among the processors. Problem and model of computation. The ....

A. Broader, A.M. Frieze, E. Shamir, and E. Upfal, Near-perfect Token Distribution, Random Structures & Algorithms 5 (1994).

....3. Extrovert graphs. Given a graph G = V; E) and a subset X of V , an element of X is said to be extrovert if at least half of its neighbors are outside X. A family of graphs is called extrovert if all linear sized subsets contain a constant fraction of extrovert nodes. Such graphs have been used [11] to solve the token distribution problem. Theorem 3 shows that the average degree of the nodes of a linear sized induced subgraph of a k regular Ramanujan graph is upper bounded by roughly 1 p k Gamma 1, which is less than k=2 for k 7. This shows that Ramanujan graphs of degree at least 7 ....

A. Z. Broder, A. M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In 19th International Colloquium on Automata, Languages, and Programming, pages 308--317. Springer Verlag, W. Berlin, 1992.

....22] hypercubes [19, 33] and meshes [16, 28] Another class of networks on which load balancing has been studied is the class of expanders. Peleg and Upfal [31] pioneered this study by identifying certain smalldegree expanders as being suitable for load balancing. Their work has been extended in [9, 17, 32]. These algorithms either use strong expanders to approximately balance the network, or the AKS sorting network [3] to perfectly balance the network. Thus, they do not work on networks of arbitrary topology. Also, these algorithms work by setting up fixed paths through the network on which load ....

A. Broder, A. M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming, pages 308--317, July 1992.

....[4, 22] hypercubes [19, 32] and meshes [16, 27] Expanders form a broad class of networks on which load balancing has been well studied. Peleg and Upfal [30] pioneered this study by identifying certain small degree expanders as being suitable for load balancing. Their work has been extended in [9, 17, 31]. The expander network required by their algorithms is complex since it contains the AKS sorting network [3] as a subgraph. Thus, they do not work on networks of arbitrary topology. Also, these algorithms work by setting up fixed paths through the network on which load is moved and therefore fail ....

A. Broder, A. M. Frieze, E. Shamir, and E. Upfal. Near--perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming, pages 308--317, 1992.

....23] hypercubes [20, 34] and meshes [17, 29] Another class of networks on which load balancing has been studied is the class of expanders. Peleg and Upfal [32] pioneered this study by identifying certain small degree expanders as being suitable for load balancing. Their work has been extended in [9, 18, 33]. These algorithms either use strong expanders to approximately balance the network, or the AKS sorting network [3] to perfectly balance the network. Thus, they do not work on networks of arbitrary topology. Also, these algorithms work by setting up fixed paths through the network on which load is ....

A. Broder, A. M. Frieze, E. Shamir, and E. Upfal. Near-perfect token distribution. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming, pages 308--317, July 1992.

....stores finally at most d( P v2V ) v) j V j)e tokens, that is, the network is weakly balanced. Tokens may be moved only along the edges of the graph. A relocation of a token between two adjacent nodes is called a transition. Variations of the above problem have been studied by many authors [2, 4, 5, 7, 9, 10, 11]. The weak balancing was considered by Meyer auf der Heide et al. 7] Most of the related research was concerned with the minimum time needed to achieve a balanced distribution, assuming that one transition takes a unit of time. In this paper we are interested in another natural measure of token ....

Broder, A.Z, and Frieze, A.M. and Shamir, E. and Upfal, E.: " Near-perfect Token Distribution," Proceedings, 19th International Colloquium on Automata, Languages and Programming, Springer Lecture Notes on Computer Science 623, 1992, pp. 308--317.

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