Selection, Routing and Sorting on the Star Graph

We consider the problems of selection, routing and sorting on an n-star graph (with n! nodes), an interconnection network which has been proven to possess many special properties. We identify a tree like subgraph (which we call as a \u27(k, l, k) chain network\u27) of the star graph which enables us to design efficient algorithms for the above mentioned problems. We present an algorithm that performs a sequence of n prefix computations in O(n2) time. This algorithm is used as a subroutine in our other algorithms. In addition we offer an efficient deterministic sorting algorithm that runs in O(n3lg n) steps. Though an algorithm with the same time bound has been proposed before, our algorithm is very simple and is based on a different approach. We also show that sorting can be performed on the n star graph in time O(n3) and that selection of a set of uniformly distributed n keys can be performed in O(n2) time with high probability. Finally, we also present a deterministic (non oblivious) routing algorithm that realizes any permutation in O(n3) steps on the n-star graph. There exists an algorithm in the literature that can perform a single prefix computation in O(n lg n) time. The best known previous algorithm for sorting has a run time of O(n3 lg n) and is deterministic. To our knowledge, the problem of selection has not been considered before on the star graph

Properties and algorithms of the (n, k)-star graphs

The (n, k)-star interconnection network was proposed in 1995 as an attractive alternative to the n-star topology in parallel computation. The (n, k )-star has significant advantages over the n-star which itself was proposed as an attractive alternative to the popular hypercube. The major advantage of the (n, k )-star network is its scalability, which makes it more flexible than the n-star as an interconnection network. In this thesis, we will focus on finding graph theoretical properties of the (n, k )-star as well as developing parallel algorithms that run on this network. The basic topological properties of the (n, k )-star are first studied. These are useful since they can be used to develop efficient algorithms on this network. We then study the (n, k )-star network from algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms for basic communication, prefix computation, and sorting, etc. A literature review of the state-of-the-art in relation to the (n, k )-star network as well as some open problems in this area are also provided

Properties and algorithms of the (n, k)-arrangement graphs

The (n, k)-arrangement interconnection topology was first introduced in 1992. The (n, k )-arrangement graph is a class of generalized star graphs. Compared with the well known n-star, the (n, k )-arrangement graph is more flexible in degree and diameter. However, there are few algorithms designed for the (n, k)-arrangement graph up to present. In this thesis, we will focus on finding graph theoretical properties of the (n, k)- arrangement graph and developing parallel algorithms that run on this network. The topological properties of the arrangement graph are first studied. They include the cyclic properties. We then study the problems of communication: broadcasting and routing. Embedding problems are also studied later on. These are very useful to develop efficient algorithms on this network. We then study the (n, k )-arrangement network from the algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms such as prefix sums computation, sorting, merging and basic geometry computation: finding convex hull on the (n, k )-arrangement graph. A literature review of the state-of-the-art in relation to the (n, k)-arrangement network is also provided, as well as some open problems in this area

Online Permutation Routing in Partitioned Optical Passive Star Networks

This paper establishes the state of the art in both deterministic and randomized online permutation routing in the POPS network. Indeed, we show that any permutation can be routed online on a POPS network either with $O(\frac{d}{g}\log g)$ deterministic slots, or, with high probability, with $5c\lceil d/g\rceil+o(d/g)+O(\log\log g)$ randomized slots, where constant $c=\exp (1+e^{-1})\approx 3.927$. When $d=\Theta(g)$, that we claim to be the "interesting" case, the randomized algorithm is exponentially faster than any other algorithm in the literature, both deterministic and randomized ones. This is true in practice as well. Indeed, experiments show that it outperforms its rivals even starting from as small a network as a POPS(2,2), and the gap grows exponentially with the size of the network. We can also show that, under proper hypothesis, no deterministic algorithm can asymptotically match its performance

Diameter of Cayley graphs of permutation groups generated by transposition trees

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $\Gamma$ is bounded as \diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}, where the maximization is over all permutations $\pi$, $c(\pi)$ denotes the number of cycles in $\pi$, and \dist_T is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Observe that evaluating this upper bound requires on the order of $n!$ (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) $n$ computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on $n$ vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value $\beta$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. \diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}.Comment: This is an extension of arXiv:1106.535

Efficient structural outlooks for vertex product networks

In this thesis, a new classification for a large set of interconnection networks, referred to as "Vertex Product Networks" (VPN), is provided and a number of related issues are discussed including the design and evaluation of efficient structural outlooks for algorithm development on this class of networks. The importance of studying the VPN can be attributed to the following two main reasons: first an unlimited number of new networks can be defined under the umbrella of the VPN, and second some known networks can be studied and analysed more deeply. Examples of the VPN include the newly proposed arrangement-star and the existing Optical Transpose Interconnection Systems (OTIS-networks). Over the past two decades many interconnection networks have been proposed in the literature, including the star, hyperstar, hypercube, arrangement, and OTIS-networks. Most existing research on these networks has focused on analysing their topological properties. Consequently, there has been relatively little work devoted to designing efficient parallel algorithms for important parallel applications. In an attempt to fill this gap, this research aims to propose efficient structural outlooks for algorithm development. These structural outlooks are based on grid and pipeline views as popular structures that support a vast body of applications that are encountered in many areas of science and engineering, including matrix computation, divide-and- conquer type of algorithms, sorting, and Fourier transforms. The proposed structural outlooks are applied to the VPN, notably the arrangement-star and OTIS-networks. In this research, we argue that the proposed arrangement-star is a viable candidate as an underlying topology for future high-speed parallel computers. Not only does the arrangement-star bring a solution to the scalability limitations from which the Abstract existing star graph suffers, but it also enables the development of parallel algorithms based on the proposed structural outlooks, such as matrix computation, linear algebra, divide-and-conquer algorithms, sorting, and Fourier transforms. Results from a performance study conducted in this thesis reveal that the proposed arrangement-star supports efficiently applications based on the grid or pipeline structural outlooks. OTIS-networks are another example of the VPN. This type of networks has the important advantage of combining both optical and electronic interconnect technology. A number of studies have recently explored the topological properties of OTIS-networks. Although there has been some work on designing parallel algorithms for image processing and sorting, hardly any work has considered the suitability of these networks for an important class of scientific problems such as matrix computation, sorting, and Fourier transforms. In this study, we present and evaluate two structural outlooks for algorithm development on OTIS-networks. The proposed structural outlooks are general in the sense that no specific factor network or problem domain is assumed. Timing models for measuring the performance of the proposed structural outlooks are provided. Through these models, the performance of various algorithms on OTIS-networks are evaluated and compared with their counterparts on conventional electronic interconnection systems. The obtained results reveal that OTIS-networks are an attractive candidate for future parallel computers due to their superior performance characteristics over networks using traditional electronic interconnects

Distributed Computing on Core-Periphery Networks: Axiom-based Design

Inspired by social networks and complex systems, we propose a core-periphery network architecture that supports fast computation for many distributed algorithms and is robust and efficient in number of links. Rather than providing a concrete network model, we take an axiom-based design approach. We provide three intuitive (and independent) algorithmic axioms and prove that any network that satisfies all axioms enjoys an efficient algorithm for a range of tasks (e.g., MST, sparse matrix multiplication, etc.). We also show the minimality of our axiom set: for networks that satisfy any subset of the axioms, the same efficiency cannot be guaranteed for any deterministic algorithm