Interconnection Networks with Hypercubic Skeletons

The hypercubic family of interconnection networks, encompassing the hypercube and its derivatives and variants, has a wide range of applications in parallel processing. Various problems in general complex networks can be addressed by choosing a hypercubic network as a skeleton. In this paper, we provide insight into why hypercubic networks are suitable as network skeletons and discuss a mapping scheme to take advantage of the symmetry of such networks for developing efficient algorithms

Differential Geometry of Group Lattices

In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset S. There is a distinguished subclass of "bicovariant" Cayley graphs with the property that ad(S)S is contained in S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The first order calculi extend to higher orders and then allow to introduce further differential geometric structures. Furthermore, we explore the properties of "discrete" vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner product with forms is defined. The Lie-Cartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analogue of the lattice gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with so-called Schreier diagrams are obtained.Comment: 51 pages, 11 figure

Combinatorial Design and Analysis of Optimal Multiple Bus Systems for Parallel Algorithms.

This dissertation develops a formal and systematic methodology for designing optimal, synchronous multiple bus systems (MBSs) realizing given (classes of) parallel algorithms. Our approach utilizes graph and group theoretic concepts to develop the necessary model and procedural tools. By partitioning the vertex set of the graphical representation CFG of the algorithm, we extract a set of interconnection functions that represents the interprocessor communication requirement of the algorithm. We prove that the optimal partitioning problem is NP-Hard. However, we show how to obtain polynomial time solutions by exploiting certain regularities present in many well-behaved parallel algorithms. The extracted set of interconnection functions is represented by an edge colored, directed graph called interconnection function graph (IFG). We show that the problem of constructing an optimal MBS to realize an IFG is NP-Hard. We show important special cases where polynomial time solutions exist. In particular, we prove that polynomial time solutions exist when the IFG is vertex symmetric. This is the case of interest for the vast majority of important interconnection function sets, whether extracted from algorithms or correspond to existing interconnection networks. We show that an IFG is vertex symmetric if and only if it is the Cayley color graph of a finite group $\Gamma$ and its generating set $\Delta.$ Using this property, we present a particular scheme to construct a symmetric $MBS\ M(\Gamma,\Delta)$ with minimum number of buses as well as minimum number of interfaces realizing a vertex symmetric IFG. We demonstrate several advantages of the optimal $MBS\ M(\Gamma,\Delta)$ in terms of its symmetry, number of ports per processor, number of neighbors per processor, and the diameter. We also investigate the fault tolerant capabilities and performance degradation of $M(\Gamma,\Delta)$ in the case of a single bus failure, single driver failure, single receiver failure, and single processor failure. Further, we address the problem of designing an optimal MBS realizing a class of algorithms when the number of buses and/or processors in the target MBS are specified. The optimality criteria are maximizing the speed and minimizing the number of interfaces

Comments on "a new family of Cayley graph interconnection networks of constant degree four"

Vadapalli and Srimani [2] have proposed a new family of Cayley graph interconnection networks of constant degree four. Our comments show that their proposed graph is not new but is the same as the wrap-around butterfly graph. The structural kinship of the proposed graph with the de Bruijn graph is also discussed. © 1997 IEEE.published_or_final_versio

Analysis of Topological Structure and Fault Tolerance of Interconnection Networks

信息社会的基础是计算机互连网络,信息交换的关键是通信算法,寻找具有对称性等良好性质的互连网络是实现各种有效的通信算法和协议的前提.自从S.B.Akers与B.Krishnamurthy$^{[6]}$倡导把$Cayley$图(群图)作为对称互连网络模型之后,网络设计者和图论学者利用各种技巧提出并研究了一系列互连网络模型,如超立方体,De\Bruijn和$Kautz$网络,洗牌交换网络(Shuffle-Exchange\Network),星图(Star\Graph),蝴蝶网络(Butterfly\Network),蜂窝网络(Honeycomb\Network),金字塔网络($...The base of information society is interconnection network, and communication algorithm is the key of information exchange. Seeking network topologies with good properties such as symmetry is indispensable to realize all kinds of effective communication algorithms and protocols. Since S.B. Akers and B. Krishnamurthy advocated using$Cayley$$graph$($group$$graph$) as the models of symmetric inter...学位：理学博士院系专业：数学系_基础数学学号：B20022300