## Optimal Emulations by Butterfly-Like Networks

Citations: | 15 - 4 self |

### BibTeX

@MISC{Bhatt_optimalemulations,

author = {Sandeep N. Bhatt and Fan R. K. Chung and Jia-Wei Hong and F. Thomson Leighton and Bojana Obrenić and Arnold L. Rosenberg and Eric J. Schwabe},

title = {Optimal Emulations by Butterfly-Like Networks },

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

1313 |
Introduction to Parallel Algorithms and Architectures
- Leighton
- 1992
(Show Context)
Citation Context ...orem 3, and even Corollaries 1 and 2 do not extend to the hypercube. In fact, both meshes and X-trees are embeddable in hypercubes with simultaneous dilation, congestion, and expansion O(1); [9], cf. =-=[5, 21]-=-. Although embeddings of complete binary trees into hypercubes with better constants appear in [6] and [14], we cannot use those embeddings to prove part (b), since at present we do not know of a cong... |

213 |
Topological properties of hypercubes
- Saad, Schultz
- 1988
(Show Context)
Citation Context ...ddings underlying these results expose X-trees and meshes as the first known graphs that can be embedded very efficiently in the hypercube (simultaneous dilation O(1) and expansion O(1); cf. [5], [8] =-=[26]-=-) but have no efficient embeddings in butterfly-like graphs. An unexpected corollary of our results is that if we focus only on dilation, then---to within constant factors---these graphs cannot be emb... |

207 | The cube connected cycles: a versatile network for parallel computation - Preparata, Vuillemin - 1981 |

132 | A framework for solving VLSI graph layout problems
- Bhatt, Leighton
- 1984
(Show Context)
Citation Context ...he complete binary tree T (lg(n)), with simultaneous dilation O(log(d P i S(n=2 i ))) and expansion O(1). Our proof of Proposition 3.1 employs the following refinement of the notion of separator; cf. =-=[7]-=-. Let k be a positive integer, and let R k (x) be a nondecreasing integer function. The graph G has a k-color recursive bisector of size R k (x) either if jGj ! 2 or if the following holds for every w... |

102 | S.Rao, Packet Routing and Job Shop Scheduling in O(Congestion+Dilation) steps
- Leighton, Maggs
- 1994
(Show Context)
Citation Context ...f resource utilization in an emulation. Less obviously, dilation and congestion combine additively to measure the slowdown incurred by an emulation, as the following result indicates. Proposition 1.1 =-=[20]-=- Say that one can embed the graph G in the graph H, with dilation D and congestion C. Then a processor array based on network H can emulate 1 The networks of interest are defined in Section 1.3. 2 For... |

80 |
Group Action Graphs and Parallel Architectures
- Annexstein, Baumslag, et al.
- 1990
(Show Context)
Citation Context ...eports on an ongoing program of research, dedicated to comparing the respective computational powers of the various networks that have been proposed for use as multicomputer interconnection networks (=-=[3]-=-, [5], [13], [14], [19]). We focus here on one member of the family of butterfly-like networks 1 that have become one of the benchmark architectures for processor arrays; our results have direct impli... |

77 |
Interconnection Networks for Large-Scale Parallel Processing: Theory and Case
- Siegel
- 1990
(Show Context)
Citation Context ...-like and that include the Benes, cube-connected cycles, and multistage cube networks, as well as all bidelta networks (such as the Omega, flip, baseline, and reverse baseline); cf., [4], [18], [23], =-=[27]-=-. In fact, versions of our results hold for an even broader class of networks; cf. Section 5. By dint of the efficient embeddings of butterfly-like graphs in hypercubes reported in [13], our upper bou... |

48 |
Optimal rearrangeable multi-stage connecting networks
- Benes
- 1964
(Show Context)
Citation Context ...ly term butterfly-like and that include the Benes, cube-connected cycles, and multistage cube networks, as well as all bidelta networks (such as the Omega, flip, baseline, and reverse baseline); cf., =-=[4]-=-, [18], [23], [27]. In fact, versions of our results hold for an even broader class of networks; cf. Section 5. By dint of the efficient embeddings of butterfly-like graphs in hypercubes reported in [... |

44 | Workpreserving emulations of fixed-connection networks
- Koch, Leighton, et al.
- 1989
(Show Context)
Citation Context ...that suffers no more slowdown than is incurred by the emulation. Less demanding notions of emulation/simulation, that do not guarantee the automatic-translation property, appear in the literature. In =-=[17]-=-, the range of computations that the emulating network can perform is broadened. In that extended model, butterfly networks can, indeed, emulate meshes and X-trees efficiently. Another approach to com... |

42 | A Unified Theory of Interconnection Network
- Kruskal, Snir
- 1986
(Show Context)
Citation Context ...rm butterfly-like and that include the Benes, cube-connected cycles, and multistage cube networks, as well as all bidelta networks (such as the Omega, flip, baseline, and reverse baseline); cf., [4], =-=[18]-=-, [23], [27]. In fact, versions of our results hold for an even broader class of networks; cf. Section 5. By dint of the efficient embeddings of butterfly-like graphs in hypercubes reported in [13], o... |

36 |
On an estimate of the chromatic class of a p-graph (Russian
- Vizing
- 1964
(Show Context)
Citation Context ...(D) and congestion O(dD). The proof employs two auxiliary notions, edge-coloring of graphs and permutation networks. The following result is traditionally referred to as "Vizing's Theorem". =-=Lemma 3.3 [28]-=- One can color the edges of any maxdegree-d graph H with either d or d + 1 colors, in such a way that no pair of like-colored edges share an endpoint. The importance of Lemma 3.3 is that if one interp... |

30 |
Efficient embeddings of trees in hypercubes
- BHATT, CHUNG, et al.
(Show Context)
Citation Context ...s on an ongoing program of research, dedicated to comparing the respective computational powers of the various networks that have been proposed for use as multicomputer interconnection networks ([3], =-=[5]-=-, [13], [14], [19]). We focus here on one member of the family of butterfly-like networks 1 that have become one of the benchmark architectures for processor arrays; our results have direct implicatio... |

28 |
Embedding of Grids into Optimal hypercubes
- Chan
- 1991
(Show Context)
Citation Context ...urse, Theorem 3, and even Corollaries 1 and 2 do not extend to the hypercube. In fact, both meshes and X-trees are embeddable in hypercubes with simultaneous dilation, congestion, and expansion O(1); =-=[9]-=-, cf. [5, 21]. Although embeddings of complete binary trees into hypercubes with better constants appear in [6] and [14], we cannot use those embeddings to prove part (b), since at present we do not k... |

23 |
X-TREE: A Tree Structured Multiprocessor
- Despain, Patterson
- 1978
(Show Context)
Citation Context ...jssg; the edges of M(s) connect nodes ha; bi and hc; di just when ja \Gamma cj + jb \Gamma dj = 1. All of these networks have been seriously proposed as interconnection networks for processor arrays (=-=[10]-=-, [11], [16]), hence are important candidates for our study. Our results depend on the following structural features of a graph G: 1. The maxdegree of G is the maximum number of edges touching any one... |

18 |
Product-shuffle networks: towards reconciling shuffles and butterflies
- Rosenberg
- 1992
(Show Context)
Citation Context ...e extension to its shuffle-oriented relatives follows from the existence of dilation-O(1) embeddings of the de Bruijn graph into those graphs and of those graphs into the de Bruijn graph. (See, e.g., =-=[25]-=-.) (a), (b)(i): One verifies easily that D(m) contains a copy of T (m \Gamma 1) as a subgraph. Therefore, part (a) is a triviality. Part (b)(i) follows from the embedding in our proof of Proposition 3... |

15 |
The binary tree as an interconnection network: Applications to multiprocessor systems and VLSI
- Horowitz, Zorat
- 1981
(Show Context)
Citation Context ...ges of M(s) connect nodes ha; bi and hc; di just when ja \Gamma cj + jb \Gamma dj = 1. All of these networks have been seriously proposed as interconnection networks for processor arrays ([10], [11], =-=[16]-=-), hence are important candidates for our study. Our results depend on the following structural features of a graph G: 1. The maxdegree of G is the maximum number of edges touching any one of its node... |

15 |
Issues in the study of graph embedding
- Rosenberg
- 1981
(Show Context)
Citation Context ... networks by determining how efficiently each of the networks can emulate the other. Again as is customary, we formalize the notion of emulation in the formal notion of embedding one graph in another =-=[24]-=-. A. Embeddings and Emulations An embedding of one simple undirected graph, G, in another, H, comprises a oneto -one assignment of the nodes of G to nodes of H, plus a routing of each edge of G within... |

14 |
Dilation-2 embeddings of grids into hypercubes
- CHAN
- 1988
(Show Context)
Citation Context ...embeddings underlying these results expose X-trees and meshes as the first known graphs that can be embedded very efficiently in the hypercube (simultaneous dilation O(1) and expansion O(1); cf. [5], =-=[8]-=- [26]) but have no efficient embeddings in butterfly-like graphs. An unexpected corollary of our results is that if we focus only on dilation, then---to within constant factors---these graphs cannot b... |

11 |
Parallel computations using mesh of trees
- Leighton
- 1982
(Show Context)
Citation Context ...rogram of research, dedicated to comparing the respective computational powers of the various networks that have been proposed for use as multicomputer interconnection networks ([3], [5], [13], [14], =-=[19]-=-). We focus here on one member of the family of butterfly-like networks 1 that have become one of the benchmark architectures for processor arrays; our results have direct implications for all butterf... |

10 | Optimal rearrangeable multistage connecting networks - Beneˇs - 1964 |

7 |
Optimal embeddings of butterfly-like graphs in the hypercube
- Greenberg, Heath, et al.
- 1990
(Show Context)
Citation Context ...an ongoing program of research, dedicated to comparing the respective computational powers of the various networks that have been proposed for use as multicomputer interconnection networks ([3], [5], =-=[13]-=-, [14], [19]). We focus here on one member of the family of butterfly-like networks 1 that have become one of the benchmark architectures for processor arrays; our results have direct implications for... |

6 |
Fault tolerance in hypercube-derivative networks
- Annexstein
- 1989
(Show Context)
Citation Context ...ic shifts of node labels, and therefore can be accomplished with dilation O(D) and overall congestion O(dD). For a fixed c, the second step can be accomplished with dilation O(D) and congestion O(dD) =-=[2]-=-, yielding overall congestion O(dD 2 ), as 0scs2D. The analog of Theorem 2 for shuffle-oriented networks appears in [22]. Rather than embedding the bucket tree into the complete binary tree (Propositi... |

4 |
A comparative study of X-tree, pyramid, and related machines
- Aggarwal
- 1984
(Show Context)
Citation Context ..., indeed, emulate meshes and X-trees efficiently. Another approach to comparing the power of networks involves implementing and analyzing specific algorithms on the networks. This approach is used in =-=[1]-=-, to compare the networks we have studied here, and in [12], to compare the hypercube with some of its bounded-degree derivatives. Our Host Networks. We restrict attention here to embeddings into butt... |

4 |
A note on pipelining mesh-connected multiprocessor for nite element problems by nested dissection
- GANNON
- 1980
(Show Context)
Citation Context ...the edges of M(s) connect nodes ha; bi and hc; di just when ja \Gamma cj + jb \Gamma dj = 1. All of these networks have been seriously proposed as interconnection networks for processor arrays ([10], =-=[11]-=-, [16]), hence are important candidates for our study. Our results depend on the following structural features of a graph G: 1. The maxdegree of G is the maximum number of edges touching any one of it... |

3 |
Embedding the polytomic tree into the n-cube, ˘ Casopis pro Pĕstování Matematiky 98
- Havel, Liebl
- 1973
(Show Context)
Citation Context ...ddable in hypercubes with simultaneous dilation, congestion, and expansion O(1); [9], cf. [5, 21]. Although embeddings of complete binary trees into hypercubes with better constants appear in [6] and =-=[14]-=-, we cannot use those embeddings to prove part (b), since at present we do not know of a congestion-control theorem for hypercubes, that would be equivalent to our Proposition 3.2. Actually, it is not... |

3 |
Graphs that are almost binary trees
- Rosenberg, Hong
- 1982
(Show Context)
Citation Context ... is now an immediate consequence of Lemmas 4.6 and 4.7, and by Lemma 4.2 it follows for G. 2 4.2 Applications to X-Trees and Meshes Corollaries 1 and 2 now follow from the following Lemmas. Lemma 4.8 =-=[15]-=- \Sigma(X (h)) = \Omega\Gamma h) =\Omega\Gamma268 jX (h)j), and \Phi(X (h)) = 4 (under the natural embedding). Lemma 4.9 (e.g., [15]) \Sigma(M(s)) = \Omega\Gamma s) = \Omega\Gamma q jM(s)j), and \Phi(... |

2 |
An approach to emulating separable graphs
- Obrenić
- 1993
(Show Context)
Citation Context ...ol congestion in hypercube embeddings leaves us with only one way to utilize the hypercube: embedding indirectly via intermediate butterflies. In place of the butterfly, we could use the cell tree of =-=[22]-=-, which has sufficient congestion-control properties. In the case of hypercube embeddings, this would change only our constants (improving the expansion and worsening the slowdown); in the case of shu... |

1 | Topological comparison of perfect shuffle and hypercube. Typescript, The Technion
- Ginosar, Egozi
- 1987
(Show Context)
Citation Context ...approach to comparing the power of networks involves implementing and analyzing specific algorithms on the networks. This approach is used in [1], to compare the networks we have studied here, and in =-=[12]-=-, to compare the hypercube with some of its bounded-degree derivatives. Our Host Networks. We restrict attention here to embeddings into butterfly graphs because focussing on one specific network stru... |