Embeddings into the Pancake Interconnection Network

Article paru en 2002 dans Parallel Processing LettersInternational audienceOwing to its nice properties, the pancake is one of the Cayley graphs that were proposed as alternatives to the hypercube for interconnecting processors in parallel computers. In this paper, we present embeddings of rings, grids and hypercubes into the pancake with constant dilation and congestion. We also extend the results to similar efficient embeddings into the star graph

Embedding Schemes for Interconnection Networks.

Graph embeddings play an important role in interconnection network and VLSI design. Designing efficient embedding strategies for simulating one network by another and determining the number of layers required to build a VLSI chip are just two of the many areas in which graph embeddings are used. In the area of network simulation we develop efficient, small dilation embeddings of a butterfly network into a different size and/or type of butterfly network. The genus of a graph gives an indication of how many layers are required to build a circuit. We have determined the exact genus for the permutation network called the star graph, and have given a lower bound for the genus of the permutation network called the pancake graph. The star graph has been proposed as an alternative to the binary hypercube and, therefore, we compare the genus of the star graph with that of the binary hypercube. Another type of embedding that is helpful in determining the number of layers is a book embedding. We develop upper and lower bounds on the pagenumber of a book embedding of the k-ary hypercube along with an upper bound on the cumulative pagewidth

Fault-tolerance embedding of rings and arrays in star and pancake graphs

The star and pancake graphs are useful interconnection networks for connecting processors in a parallel and distributed computing environment. The star network has been widely studied and is shown to possess attactive features like sublogarithmic diameter, node and edge symmetry and high resilience. The star/pancake interconnection graphs, {dollar}S\sb{n}/P\sb{n}{dollar} of dimension n have n! nodes connected by {dollar}{(n-1).n!\over2}{dollar} edges. Due to their large number of nodes and interconnections, they are prone to failure of one or more nodes/edges; In this thesis, we present methods to embed Hamiltonian paths (H-path) and Hamiltonian cycles (H-cycle) in a star graph {dollar}S\sb{n}{dollar} and pancake graph {dollar}P\sb{n}{dollar} in a faulty environment. Such embeddings are important for solving computational problems, formulated for array and ring topologies, on star and pancake graphs. The models considered include single-processor failure, double-processor failure, and multiple-processor failures. All the models are applied to an H-cycle which is formed by visiting all the ({dollar}{n!\over4!})\ S\sb4/P\sb4{dollar}s in an {dollar}S\sb{n}/P\sb{n}{dollar} in a particular order. Each {dollar}S\sb4/P\sb4{dollar} has an entry node where the cycle/path enters that particular {dollar}S\sb4/P\sb4{dollar} and an exit node where the path leaves it. Distributed algorithms for embedding hamiltonian cycle in the presence of multiple faults, are also presented for both {dollar}S\sb{n}{dollar} and {dollar}P\sb{n}{dollar}

Embedding complete binary trees into star networks

Abstract. Star networks have been proposed as a possible interconnection network for massively parallel computers. In this paper we investigate embeddings of complete binary trees into star networks. Let G and H be two networks represented by simple undirected graphs. An embedding of G into H is an injective mapping f from the vertices of G into the vertices of H. The dilation of the embedding is the maximum distance between f(u), f(v) taken over all edges (u, v) of G. Low dilation embeddings of binary trees into star graphs correspond to efficient simulations of parallel algorithms that use the binary tree topology, on parallel computers interconnected with star networks. First, we give a construction of embeddings of dilation 1 of complete binary trees into n-dimensional star graphs. These trees are subgraphs of star graphs. Their height is fl(n log n), which is asymptotically optimal. Constructions of embeddings of complete binary trees of dilation 28 and 26 + 1, for 8 > 1, into star graphs are then given. The use of larger dilation allows embeddings of trees of greater height into star graphs. For example, the difference of the heights of the trees embedded with dilation 2 and 1 is greater than n/2. All these constructions can be modified to yield embeddings of dilation 1, and 26, for ~ > 1, of complete binary trees into pancake graphs. Our results show that massively parallel computers interconnected with star networks are well suited for efficient simulations of parallel algorithms with complete binary tree topology

ONE BY ONE EMBEDDING THE CROSSED HYPERCUBE INTO PANCAKE GRAPH

Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from vertices of G to the vertices of H. The dilation of embedding is the maximum distance between f(u), f(v) taken over edges (u, v) of G. The Pancake graph is one as viable interconnection scheme for parallel computers, which has been examined by a number of researchers. The Pancake was proposed as alternatives to the hypercube for interconnecting processors in parallel computer. Some good attractive properties of this interconnection network include: vertex symmetry, small degree, a sub-logarithmic diameter, extendability, and high connectivity (robustness), easy routing and regularity of topology, fault tolerance, extensibility and embeddability of others topologies. In this paper, we give a construction of one by one embedding of dilation 5 of crossed hypercube into Pancake graph

A powerful heuristic for telephone gossiping

A refined heuristic for computing schedules for gossiping in the telephone model is presented. The heuristic is fast: for a network with n nodes and m edges, requiring R rounds for gossiping, the running time is O(R n log(n) m) for all tested classes of graphs. This moderate time consumption allows to compute gossiping schedules for networks with more than 10,000 PUs and 100,000 connections. The heuristic is good: in practice the computed schedules never exceed the optimum by more than a few rounds. The heuristic is versatile: it can also be used for broadcasting and more general information dispersion patterns. It can handle both the unit-cost and the linear-cost model. Actually, the heuristic is so good, that for CCC, shuffle-exchange, butterfly de Bruijn, star and pancake networks the constructed gossiping schedules are better than the best theoretically derived ones. For example, for gossiping on a shuffle-exchange network with 2^{13} PUs, the former upper bound was 49 rounds, while our heuristic finds a schedule requiring 31 rounds. Also for broadcasting the heuristic improves on many formerly known results. A second heuristic, works even better for CCC, butterfly, star and pancake networks. For example, with this heuristic we found that gossiping on a pancake network with 7! PUs can be performed in 15 rounds, 2 fewer than achieved by the best theoretical construction. This second heuristic is less versatile than the first, but by refined search techniques it can tackle even larger problems, the main limitation being the storage capacity. Another advantage is that the constructed schedules can be represented concisely