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Workpreserving emulations of fixedconnection networks
 21st ACM Symp. on Theory of Computing
, 1989
"... Abstract. In this paper, we study the problem of emulating T G steps of an N Gnode guest network, G, on an N Hnode host network, H. We call an emulation workpreserving if the time required by the host, T H,isO(T GN G/N H), because then both the guest and host networks perform the same total work ..."
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Cited by 47 (16 self)
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Abstract. In this paper, we study the problem of emulating T G steps of an N Gnode guest network, G, on an N Hnode host network, H. We call an emulation workpreserving if the time required by the host, T H,isO(T GN G/N H), because then both the guest and host networks perform the same total work (i.e., processortime product), �(T GN G), to within a constant factor. We say that an emulation occurs in realtime if T H � O(T G), because then the host emulates the guest with constant slowdown. In addition to describing several workpreserving and realtime emulations, we also provide a general model in which lower bounds can be proved. Some of the more interesting and diverse consequences of this work include: (1) a proof that a linear array can emulate a (much larger) butterfly in a workpreserving fashion, but that a butterfly cannot emulate an expander (of any size) in a workpreserving fashion, (2) a proof that a butterfly can emulate a shuffleexchange network in a realtime workpreserving fashion, and vice versa, (3) a proof that a butterfly can emulate a mesh (or an array of higher, but fixed, dimension) in a realtime workpreserving fashion, even though any O(1)to1 embedding of an Nnode mesh in an Nnode butterfly has dilation �(log N), and
On the Fault Tolerance of Some Popular BoundedDegree Networks
 SIAM Journal on Computing
, 1992
"... In this paper, we analyze the ability of several boundeddegree networks that are commonly used for parallel computation to tolerate faults. Among other things, we show that an Nnode butterfly containing N 1\Gammaffl worstcase faults (for any constant ffl ? 0) can emulate a faultfree butterfly ..."
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Cited by 46 (8 self)
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In this paper, we analyze the ability of several boundeddegree networks that are commonly used for parallel computation to tolerate faults. Among other things, we show that an Nnode butterfly containing N 1\Gammaffl worstcase faults (for any constant ffl ? 0) can emulate a faultfree butterfly of the same size with only constant slowdown. Similar results are proved for the shuffleexchange graph. Hence, these networks become the first connected boundeddegree networks known to be able to sustain more than a constant number of worstcase faults without suffering more than a constantfactor slowdown in performance. We also show that an Nnode butterfly whose nodes fail with some constant probability p can emulate a faultfree version of itself with a slowdown of 2 O(log N) , which is a very slowly increasing function of N . The proofs of these results combine the technique of redundant computation with new algorithms for (packet) routing around faults in hypercubic networks. Tech...
Cycles in Networks
, 1993
"... We study the presence of cycles and long paths in graphs that have been proposed as interconnection networks for parallel architectures. The study surveys and complements known results. 1 Introduction This paper is devoted to studying embeddings of the simplest possible guest graphs, the path P ..."
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We study the presence of cycles and long paths in graphs that have been proposed as interconnection networks for parallel architectures. The study surveys and complements known results. 1 Introduction This paper is devoted to studying embeddings of the simplest possible guest graphs, the path PN and the cycle CN , in graphs that have been proposed as interconnection networks for parallel architectures. In addition to their intrinsic interest, in terms of the development of algorithms on parallel architectures, these two guest graphs are important because of the fact that many structurally richer graphs can be constructed from paths and cycles by various product constructions. A few of the results we present are original; several appear in the literature and are duly cited; many belong to the folklore of the field. Indeed this paper is motivated by a desire to find a single repository for this important, yet scattered material. Before proceeding further, we define formally the ...
Emulating Direct Products by IndexShuffle Graphs
, 1998
"... In the theoretical framework of graph embedding and network emulations, we show that the indexshuffle graph (a boundeddegreehypercubelike interconnection network, recently introduced by [ Baumslag and Obreni c (1997): IndexShuffle Graphs, :::]) efficiently approximates the hypercube in general c ..."
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In the theoretical framework of graph embedding and network emulations, we show that the indexshuffle graph (a boundeddegreehypercubelike interconnection network, recently introduced by [ Baumslag and Obreni c (1997): IndexShuffle Graphs, :::]) efficiently approximates the hypercube in general computations, by emulating the directproduct structure of the hypercube. In the direct product G = G 1 # G 2 #####G k let any factor G i be an instance of any of the three following graphs: cycle, complete binary tree, X tree. Given an N node indexshuffle graph Yn , where N = 2 n , and any collection of 2 ` copies of G, such that: jG i j#2 n i ; for i = 1;:::k,where ` + P k i=1 n i # n and 2 dlog 2 ke # #max 1#i#k n i # # n; Yn emulates any factor G i in all copies of G in this collection with slowdown O#log k + log n i #=O#log log N#. As a consequenceof these and previous results, the indexshuffle graph emerges as a uniquely "universal" boundeddegree hypercube substitute. Th...
Theory of Computing Systems © 2001 SpringerVerlag New York Inc. On the Bisection Width and Expansion of Butterfly Networks ∗
"... Abstract. This paper proves tight bounds on the bisection width and expansion of butterfly networks with and without wraparound. We show that the bisection width of an ninput butterfly network is 2 ( √ 2 − 1)n + o(n) ≈ 0.82n without wraparound, and n with wraparound. The former result is surprisi ..."
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Abstract. This paper proves tight bounds on the bisection width and expansion of butterfly networks with and without wraparound. We show that the bisection width of an ninput butterfly network is 2 ( √ 2 − 1)n + o(n) ≈ 0.82n without wraparound, and n with wraparound. The former result is surprising, since it contradicts the prior “folklore ” belief that the bisection width is n. We also show that every set of k nodes has at least (k/(2 log k))(1−o(1)) neighbors in a butterfly without wraparound, and at least (k / log k)(1 − o(1)) neighbors in a butterfly with wraparound, if k is o ( √ n) and o(n), respectively.