F. T. Leighton and R. L. Rivest. The Markov chain tree theorem, Rep. MIT/LCS/TM-249, Laboratory for Computer Science, MIT, Cambridge, Mass.,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....process is the method to avoid performance degradation in the presence of hot spots. Especially, combining on the route provides a practical method to accomplish the memory reference primitives of strong CRCW PRAM (as well as CREW) models. Several methods have been proposed to accomplish combining [1, 3, 6, 8, 10], which all can be applied to mesh connected computers. These methods can be applied to other interconnection structures as well, but here we deal only with the 2 dimensional and the 3 dimensional mesh structure, since it is simple, regular, modularly extendible, well scalable, and in general it ....

....issues. 2 Virtual leveled network technique In a directed leveled network, each edge connects a node on level i to another node on level i 1. A leveled network is assumed to consist of L levels. Mesh is not directly a leveled network, but it can be seen as a virtual leveled network. The idea [3] is to see the network as one virtual leveled network consisting of d 1 duplicates of the original network. Duplicates are connected together so that they form a snake in which two consecutive duplicates are connected (see Figure 2) by alternating the dimensions one by one. Each processor has to ....

T. Leighton, C.E. Leiserson, and M. Klugerman. Theory of Parallel and VLSI Computation, Lecture notes for 18.435/6.848. Technical Report MIT/LCS/RSS 10, MIT, Laboratory for Computer Science, January 1991.

....p=p 0 fi. Two Markov chains P = fp ij g m i;j=1 and P 0 = fp 0 ij g m i;j=1 are fi close if, for all i and j, p ij and p 0 ij are fi close. As noted by Greenberg and Weiss [5] see also [1, Lemma 3. 2] the following lemma can be proved easily from the Markov chain tree theorem [9, 10]. Lemma 4.1 Let P and P 0 be two m state Markov chains which are fi close. Then a(P ) and a(P 0 ) are fi 2m close. We need a variation of this lemma, where certain corresponding pairs of probabilities p ij and p 0 ij are not known to be fi close, but only in the case that these ....

F. T. Leighton and R. L. Rivest, The Markov chain tree theorem, Report MIT/LCS/TM-249, Laboratory for Computer Science, MIT, Cambridge, MA, 1983.

....V S S The figure shows the sets of nodes U and V . S is a subset of U that has edges to S 0 in V . jS 0 j fijSj if jSj ffjU j. Figure 1: An expander graph It can be shown that classes of (ff; fi) expander graphs with constant ff, fi and degree, but arbitrarily large jU j and jV j exist[13]. We shall simply assume the existence of bipartite graphs in which each node in U has the same degree, each node in V has the same degree and and the graph has the desired expansion properties. We next define a multi Benes network as a particular type of combination of bipartite graphs with the ....

....Figure 2. It appears symmetric about stage k. However, the network need not be symmetric about stage k. Stage 0 Stage 1 Stage k Stage 2k 1 Stage 2k Figure 2: A multi Benes network It is possible to make a multi Benes network such that each node has in degree and out degree d for some d 2 [13]. Clearly, the expansion factor for each expander graph depends upon the actual value of d. We shall assume in our future discussions of multi Benes networks that the indegree and out degree of each node is d where d is some integer greater than 2. Let us construct another network, the ....

F. Tom Leighton, Charles E. Leiserson, and Michael Klugerman. Theory of Parallel and VLSI Computation. Technical Report MIT/LCS/RSS 10, Laboratory for Computer Science, 545 Technology Square, Cambridge MA 02139, January 1991.

....of this network can also be seen as an achievable upper bound. The network topology is based on expander graphs. In Subsection 4.1 we discuss the properties of expander graphs. We borrow the definition of the multi Bene s network and a theorem on multi Bene s networks (Theorem 4. 1) from [14]. We then construct a new graph called the shuffle Bene s network and prove a property of interest (Corollary 4.2) of this new graph. In Subsection 4.2 we use the properties of the shuffle Bene s network to introduce another new graph called the cascaded shuffle Bene s network. We use this graph ....

....U and V are shown to illustrate the expansion property. S and S 0 are subsets of U and V . The dotted lines connecting the smaller ellipses denote that S 0 is the set of neighbors of S. For jSj ffU we have jS 0 j fijSj. In [15] an explicit construction for expander graphs is given. In [14] a proof based on a random graph argument is given to show that classes of (ff; fi) expander graphs with constant ff, fi and degree but arbitrarily large jU j and jV j exist but no explicit construction is given. The argument constructs random graphs in which each node has the same degree and ....

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F. Tom Leighton, Charles E. Leiserson, and Michael Klugerman. Theory of Parallel and VLSI Computation. Technical Report MIT/LCS/RSS 10, Laboratory for Computer Science, 545 Technology Square, Cambridge MA 02139, January 1991.

....PRAM simulation algorithms, and were based on sorting and the combining queues method, respectively. In paper [12] we also presented a third CRCW simulation method: an improved levelled network routing strategy for coated meshes. The virtual levelled routing strategy was proposed for meshes in [10], and our improvement is simply based on moving the turning points from the sides to the center of the mesh. The known proofs for these routing algorithms do not take the effect of overloading properly into consideration. Therefore, in this paper, we show that overloading can be used to ....

....virtual levelled network technique In a directed levelled network, each edge connects a node on level i to another node on level i 1. A levelled network is assumed to consist of L levels. Mesh is not directly a levelled network, but it can be seen as a virtual levelled network. The idea [10] is to see the network as one virtual levelled network consisting of d 1 duplicates of the original network. Duplicates are connected together, so that they form a snake, in which two consecutive duplicates are connected (see Figure 3) by alternating the dimensions one by one. Each processor has ....

T. Leighton, C.E. Leiserson, and M. Klugerman. Theory of Parallel and VLSI Computation, Lecture notes for 18.435/6.848. Technical Report MIT/LCS/RSS 10, MIT, Laboratory for Computer Science, January 1991.

....process is the method to avoid performance degradation in the presence of hot spots. Especially, combining on the route provides a practical method to accomplish the memory reference primitives of strong CRCW PRAM (as well as CREW) models. Several methods have been proposed to accomplish combining [1, 3, 6, 8, 10], which all can be applied to mesh connected computers. These methods can be applied to other interconnection structures as well, but here we deal only with the 2 dimensional and the 3 dimensional mesh structure, since it is simple, regular, modularly extendible, well scalable, and in general it ....

....issues. 2 Virtual leveled network technique In a directed leveled network, each edge connects a node on level i to another node on level i 1. A leveled network is assumed to consist of L levels. Mesh is not directly a leveled network, but it can be seen as a virtual leveled network. The idea [3] is to see the network as one virtual leveled network consisting of d 1 duplicates of the original network. Duplicates are connected together so that they form a snake in which two consecutive duplicates are connected (see Figure 2) by alternating the dimensions one by one. Each processor has to ....

T. Leighton, C.E. Leiserson, and M. Klugerman. Theory of Parallel and VLSI Computation, Lecture notes for 18.435/6.848. Technical Report MIT/LCS/RSS 10, MIT, Laboratory for Computer Science, January 1991.

....w Gamma 1 using a log N universal hash function. The leveled network algorithm can be adapted to meshes, since a d dimensional d p N Theta d p N Theta : Theta d p N processor mesh can be seen as consisting of 2 d overlapping leveled networks of depth L = d( d p N Gamma 1) [48]. One leveled network is constructed for each direction combination e.g. for the 2D mesh the combinations are ( GammaX; GammaY ) GammaX; Y ) X; GammaY ) and ( X; Y ) Construction guarantees that there are at least one leveled network for each packet, so that the network can route the ....

....1) and there are many ways, how packets can be routed on each network. We call this the overlapping leveled network routing strategy R oln to separate it from the virtual leveled network routing strategy R vln , which is discussed next. 5.1. 2 Virtual leveled network The idea behind R vln [48] is to see the network as one virtual leveled network consisting of d 1 duplicates of the original network. Duplicates are connected together, so that they form a snake, in which two consecutive duplicates are connected (see Figure 7) by alternating the dimensions one by one. Each processor now ....

T. Leighton, C.E. Leiserson, and M. Klugerman. Theory of Parallel and VLSI Computation, Lecture notes for 18.435/6.848. Technical Report MIT/LCS/RSS 10, MIT, Laboratory for Computer Science, January 1991.

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F. T. Leighton and R. L. Rivest. The Markov chain tree theorem, Rep. MIT/LCS/TM-249, Laboratory for Computer Science, MIT, Cambridge, Mass.,

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Leighton, F. T., and Rivest, R. L., The Markov Chain Tree Theorem, Rep. MIT/LCS, TM--249, Laboratory of Computer Science, MIT, Cambridge, Mass., 1983.

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T. Leighton, C.E. Leiserson, and M. Klugerman. Theory of Parallel and VLSI Computation, Lecture notes for 18.435/6.848. Technical Report MIT/LCS/RSS 10, MIT, Laboratory for Computer Science, January 1991.

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