Mapping pyramid algorithms into hypercubes (original) (raw)

Dilation d embedding of a hyper-pyramid into a hypercube

1989

A p(lc, d) hyper-pyramid is a level structure of X: Boolean cubes wh.ere the cube at level i is of dimension id, and a node at level i-1 connects to every node in a d dimensional Boolean subcube at level i, except for the leaf level ir,. Hype]:-pyramids contain pyramids as proper ,. subgraphs. We sh~ow that a P(k, d) hyper-pyramid can be embedded in a Boolean cube with minimal expansion and dilation d. The congestion is bounded from above by and from beIow by 1 + [@I. For @(k, 2) hyper-pyramids we present a dilation 2 and congestion 2 embedd.ing. As a corollary a complete n-ary tree can be embedded in a Boolean cube with dilation m42, Poh 4) a.nd expansion 2kr10g+l+1/*. We' also discuss multi.ple pyramid embeddings. *This work was done while the author was with the Department of Computer Science, Yale University. tThc author is a.1s.a with Thinking Machines Corp.,

Extended Hypercube: A Hierarchical Interconnection Network of Hypercubes

Parallel and Distributed Systems, …, 1992

A new interconnection topology-the extended hypercube-consisting of an interconnection network of k-cubes is discussed. The extended hypercube is a hierarchical, expansive, recursive structure with a constant predefined building block. The extended ...

Pyramid mappings onto hypercubes for computer vision: Connection machine comparative study

Concurrency: Practice and Experience, 1993

The paper presents a comparative analysis for algorithms that map pplamids onto hypercubes. The analysis is based on some important performance measures h m graph theory and actual results from a Connection Machine system CM-2 conEaining 16K processors. Connection Machine results are presented for pyramid algorithms that compute the perimeter of objects, apply 2-dimensional convolution, and segment images.

A parallel algorithm for an efficient mapping of grids in hypercubes

IEEE Transactions on Parallel and Distributed Systems, 1993

Abstruct-This paper parallelizes the embedding strategy for mapping any two-dimensional grid into its optimal hypercube with minimal dilation. The parallelization allows each hypercube node to independently determine, in1 constant time, which grid node it will simulate and the communication paths it will take to reach the hypercube nodes which simdlate its grid-neighbors. The paths between grid-neighbors are chosen in such a way as to curb the congestion at each hypercube node and across each hypercube edge. Explicity, the node congestion for our embedding is at most 6 (one above optimal), while the edge congestion is at most 5.

Embedding linear array onto the tree-hypercube network

Graph embedding or graph mapping is an important problem in interconnection networks. A good mapping is said to exist when adjacent processors in the guest network are mapped to reasonably close processors in the host network (i.e. small dilation) and when the paths between adjacent processors in the guest network are chosen in such a way that the congestion at each host node and across each host edge is moderately small (i.e. small nodeand edge-congestion). In the case of mapping guest networks onto smaller hosts, the processors of the host have to be assigned to about the same number of processes from the guest (i.e. small load-factor). In this paper an approach for embedding linear array onto Tree-Hypercubes networks is proposed.

KH-map: A new way of representing the hypercube structure

Journal of Systems Architecture, 1998

The hypercube structure is an attractive and powerful topology for interconnecting processing elements in a multiprocessor system since it allows simple deadlock-free routing and broadcasting. In a hypercube architecture, one has to visualize multidimensional objects to develop efficient algorithms or to analyze system behavior. However, human beings are habituated to at most three-dimensional objects. In this paper, we propose a novel two-dimensional representation of the hypercube structure. The proposed representation is used to design routing and broadcasting algorithms. Finally, a comparison of the proposed representation with the conventional hypercube representation is elucidated. its fault tolerance and modularity [2,3]. There are simple and efficient algorithms for node-to-node communication, broadcasting, and other forms of communication in the hypercube [4]. Several variations of hypercube topology have been developed which further improve some of its properties. These include the twisted cube [5], cube connected cycles [6], bridged hypercube [7], generalized hypercubes [8], and binary orthogonal multiprocessors [9]. Several commercially available machines including Intel, N-Cube, and Thinking Machine as well as the Caltech Cosmic Cub have their communication network configured as a 1383-7621/0165-6074/98/$19.00

Performance analysis of pyramid mapping algorithms for the hypercube

1993

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Efficient VLSI Layout of Grid Pyramid Networks

Reducing the VLSI layout area of on-chip networks can result in lower costs and better performance. Those layouts that are more compact can result in shorter wires and therefore the signal propagation through the wires will take place in less time. The grid-pyramid network is a generalized pyramid network based on a general 2D Grid structure (such as mesh, torus, hypermesh or WK-recursive mesh). Such pyramid networks form a wide class of interconnection networks that possess rich topological properties. In this paper, we study these topologies from the VLSI-layout efficiency point of view. Also, we investigated on the layout of RTCC-pyramid networks that we believe can be considered in the class of Grid-pyramid networks.

A comparative performance analysis of n-cubes and star graphs

Proceedings 20th IEEE International Parallel & Distributed Processing Symposium, 2006

Many theoretical-based comparison studies, relying on the graph theoretical viewpoints with using structural and algorithmic properties, have been conducted for the hypercube and the star graph. None of these studies, however, considered real working conditions and implementation limits. We have compared the performance of the star and hypercube networks for different message length and virtual channels and considered two implementation constraints, namely the constant bisection bandwidth and constant node pin-out. We use two accurate analytical models already proposed for the star graph and hypercube and implement the parameter changes imposed by technological implementation constraints. The comparison results reveal that the star graph has a better performance compared to the equivalent hypercube under light traffic loads while the opposite conclusion is reached for heavy traffic loads. The hypercube with more channels compared to its equivalent star graph saturates later showing that it can bear heavier traffic loads.