Four algorithms are analyzed in the shared and nonshared (distributed) memory models ofparallel computation. The analysis shows that the shared memory model predicts optimality for algorithms and programming styles that cannot be realized on any physical parallel computers. Programs based on these techniques are inferior to programs wrinen in the nonshared memory model. The “unit ” cost charged for a reference to shared memory is argued to be the source of the shared memory model’s inaccuracy. The implications of these observations are discussed. I.

The binary hypercube, or n-cube, has been widely used as the interconnection network in parallel computers. However, the major drawback of the hypercube is the increase in the number of communication links for each node with the increase in the total number of nodes in the system. This paper introduces a new interconnection network for large-scale distributed memory multiprocessors called dual-cube. This network mitigates the problem of increasing number of links in the large-scale hypercube network while keeps most of the topological properties of the hypercube network. We investigate the topological properties of the dualcube, compare them with other hypercube-like networks, and establish the basic routing and broadcasting algorithms for dual-cubes. 1.

The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter while keeping the number of generators limited) are pertinent to networks, "worst" and "average" generators seem a more adequate model for puzzles. We survey a substantial body of recent work by the authors on these subjects. Regarding the "best" case, we show that while the structure of the group is essentially irrelevant if |S| is allowed to exceed (log |G|) 1+c (c > 0), it plays a heavy role when jSj = O(1). In particular, every nonabelian nite simple group has a set of 7 generators giving logarithmic diameter. This cannot happen for groups with an abelian subgroup of bounded index. { Regarding the worst case, we are concerned primarily with permutation groups of degree n and obtain a tight exp((n ln n) 1=2 (1 + o(1))) upper bound. In the average case, the upper bound improves to exp((ln n) 2 (1 + o(1))). As a rst step toward extending this result to simple groups other than An , we establish that almost every pair of elements of a classical simple group G generates G, a result previously proved by J. Dixon for An . In the limited space of this article, we try to illuminate some of the basic underlying techniques.

A dominating set S of a graph G is perfect if each vertex of G is dominated by exactly one vertex in S. We study the existence and construction of PDSs in families of graphs arising from the interconnection networks of parallel computers. These include trees, dags, series-parallel graphs, meshes, tori, hypercubes, cube-connected cycles, cube-connected paths, and de Bruijn graphs. For trees, dags, and series-parallel graphs we give linear time algorithms that determine if a PDS exists, and generate a PDS when one does. For 2- and 3-dimensional meshes, 2-dimensional tori, hypercubes, and cube-connected paths we completely characterize which graphs have a PDS, and the structure of all PDSs. For higher dimensional meshes and tori, cube-connected cycles, and de Bruijn graphs, we show the existence of a PDS in infinitely many cases, but our characterization is not complete. Our results include distance d-domination for arbitrary d. 1 Introduction Suppose G = (V; E) is a graph with vertex se...

. We prove tight bounds for crossing numbers of hypercube and cube connected cycles (CCC) graphs. Key Words: crossing number, cube connected cycles, hypercube, lower bound CR Categories: F.1.2 [Modes of Computations] Parallelism, G.2.2. [Graph Theory] Network problems 1

The binary hypercube, or n-cube, has been widely used as the interconnection network in parallel computers. However, the major drawback of the hypercube is the increase in the number of communication links for each node with the increase in the total number of nodes in the system. This paper introduces a new interconnection network for large-scale parallel computers called dual-cube. This network mitigates the problem of increasing number of links in the large-scale hypercube network while keeps most of the topological properties of the hypercube network. Design of efficient routing algorithms for collective communications is the key issue for any interconnection networks. In this paper, we show that collective communications can be done efficiently in dual-cube. KEY WORDS Interconnection networks, hypercube, collective communication, broadcast and personalized communication 1.

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AbstractÐChordal rings have been proposed in the past as networks that combine the simple routing framework of rings with the lower diameter, wider bisection, and higher resilience of other architectures. Virtually all proposed chordal ring networks are nodesymmetric, i.e., all nodes have the same in/out degree and interconnection pattern. Unfortunately, such regular chordal rings are not scalable. In this paper, periodically regular chordal (PRC) ring networks are proposed as a compromise for combining low node degree with small diameter. By varying the PRC ring parameters, one can obtain architectures with significantly different characteristics (e.g., from linear to logarithmic diameter), while maintaining an elegant framework for computation and communication. In particular, a very simple and efficient routing algorithm works for the entire spectrum of PRC rings thus obtained. This flexibility has important implications for key system attributes such as architectural scalability, software portability, and fault tolerance. Our discussion is centered on unidirectional PRC rings with in/out-degree of 2. We explore the basic structure, topological properties, optimization of parameters, VLSI layout, and scalability of such networks, develop packet and wormhole routing algorithms for them, and briefly compare them to competing fixed-degree architectures such as symmetric chordal rings, meshes, tori, and cube-connected cycles. Index TermsÐChordal rings, fault tolerance, greedy routing, hierarchical parallel architectures, interconnection networks, routing algorithms, skip links. 1

this paper, we study functions called list homomorphisms, which represent a particular pattern of parallelism.

A new, scalable interconnection topology called the Spanning Multichannel Linked Hypercube (SMLH) is proposed. This proposed network is very suitable to massively parallel systems and is highly amenable to optical implementation. The SMLH uses the hypercube topology as a basic building block and connects such building blocks using two-dimensional multichannel links (similar to spanning buses). In doing so, the SMLH combines positive features of both the hypercube (small diameter, high connectivity, symmetry, simple routing, and fault tolerance) and the spanning bus hypercube (SBH) (constant node degree, scalability, and ease of physical implementation), while at the same time circumventing their disadvantages. The SMLH topology supports many communication patterns found in different classes of computation, such as bus-based, mesh-based, and tree-based problems, as well as hypercube-based problems. A very attractive feature of the SMLH network is its ability to support a large n...