Star varietal cube: A New Large Scale Parallel Interconnection Network

This paper proposes a new interconnection network topology, called the Star varietalcube SVC(n,m), for large scale multicomputer systems. We take advantage of the hierarchical structure of the Star graph network and the Varietal hypercube to obtain an efficient method for constructing the new topology. The Star graph of dimension n and a Varietal hypercube of dimension m are used as building blocks. The resulting network has most of the desirable properties of the Star and Varietal hypercube including recursive structure, partionability, strong connectivity. The diameter of the Star varietal hypercube is about two third of the diameter of the Star-cube. The average distance of the proposed topology is also smaller than that of the Star-cube

I/O embedding and broadcasting in star interconnection networks

The issues of communication between a host or central controller and processors, in large interconnection networks are very important and have been studied in the past by several researchers. There is a plethora of problems that arise when processors are asked to exchange information on parallel computers on which processors are interconnected according to a specific topology. In robust networks, it is desirable at times to send (receive) data/control information to (from) all the processors in minimal time. This type of communication is commonly referred to as broadcasting. To speed up broadcasting in a given network without modifying its topology, certain processors called stations can be specified to act as relay agents. In this thesis, broadcasting issues in a star-based interconnection network are studied. The model adopted assumes all-port communication and wormhole switching mechanism. Initially, the problem treated is one of finding the minimum number of stations required to cover all the nodes in the star graph with i-adjacency. We consider 1-, 2-, and 3-adjacencies and determine the upper bound on the number of stations required to cover the nodes for each case. After deriving the number of stations, two algorithms are designed to broadcast the messages first from the host to stations, and then from stations to remaining nodes; In addition, a Binary-based Algorithm is designed to allow routing in the network by directly working on the binary labels assigned to the star graph. No look-up table is consulted during routing and minimum number of bits are used to represent a node label. At the end, the thesis sheds light on another algorithm for routing using parallel paths in the star network

Neighbourhood Broadcasting in Hypercubes

International audienceIn the broadcasting problem, one node needs to broadcast a message to all other nodes in a network. If nodes can only communicate with one neighbor at a time, broadcasting takes at least $\lceil \log_2 N \rceil$ rounds in a network of $N$ nodes. In the neighborhood broadcasting problem, the node that is broadcasting needs to inform only its neighbors. In a binary hypercube with $N$ nodes, each node has $\log_2 N$ neighbors, so neighborhood broadcasting takes at least $\lceil \log_2 \log_2 (N+1) \rceil$ rounds. In this paper, we present asymptotically optimal neighborhood broadcast protocols for binary hypercubes

Conflict-free star-access in parallel memory systems

We study conflict-free data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same memory module. The mapping algorithm should: (i) be fast in terms of data access (possibly mapping each node in constant time); (ii) minimize the required number of memory modules for accessing any instance in G of the given template type; and (iii) guarantee load balancing on the modules. In this paper, we consider conflict-free access to star templates. i.e., to any node of G along with all of its neighbors. Such a template type arises in many classical algorithms like breadth-first search in a graph, message broadcasting in networks, and nearest neighbor based approximation in numerical computation. We consider the star-template access problem on two specific host graphs-tori and hypercubes-that are also popular interconnection network topologies. The proposed conflict-free mappings on these graphs are fast, use an optimal or provably good number of memory modules, and guarantee load balancing. (C) 2006 Elsevier Inc. All rights reserved

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))

Neighbourhood Broadcasting in Hypercubes

International audienceIn the broadcasting problem, one node needs to broadcast a message to all other nodes in a network. If nodes can only communicate with one neighbor at a time, broadcasting takes at least $\lceil \log_2 N \rceil$ rounds in a network of $N$ nodes. In the neighborhood broadcasting problem, the node that is broadcasting needs to inform only its neighbors. In a binary hypercube with $N$ nodes, each node has $\log_2 N$ neighbors, so neighborhood broadcasting takes at least $\lceil \log_2 \log_2 (N+1) \rceil$ rounds. In this paper, we present asymptotically optimal neighborhood broadcast protocols for binary hypercubes

Improved upper bounds and lower bounds on broadcast function

Given a graph G=(V,E) and an originator vertex v, broadcasting is an information disseminating process of transmitting a message from vertex v to all vertices of graph G as quickly as possible. A graph G on n vertices is called broadcast graph if the broadcasting from any vertex in the graph can be accomplished in \lceil log n\rceil time. A broadcast graph with the minimum number of edges is called minimum broadcast graph. The number of edges in a minimum broadcast graph on n vertices is denoted by B(n). A long sequence of papers present different techniques to construct broadcast graphs and to obtain upper bounds on B(n). In this thesis, we study the compounding and the vertex addition broadcast graph constructions, which improve the upper bound on B(n). We also present the first nontrivial general lower bound on B(n)

Embeddings in hypercubes

One important aspect of efficient use of a hypercube computer to solve a given problem is the assignment of subtasks to processors in such a way that the communication overhead is low. The subtasks and their inter-communication requirements can be modeled by a graph, and the assignment of subtasks to processors viewed as an embedding of the task graph into the graph of the hypercube network. We survey the known results concerning such embeddings, including expansion/dilation tradeoffs for general graphs, embeddings of meshes and trees, packings of multiple copies of a graph, the complexity of finding good embeddings, and critical graphs which are minimal with respect to some property. In addition, we describe several open problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27512/1/0000556.pd