This book provides the most basic problems, concepts, and well-established results from the topological structure and analysis of interconnection networks in the graph-theoretic language. It covers the basic principles and methods of network design, several well-known networks such as hypercubes, de Bruijn digraphs, Kautz digraphs, double loop, and other networks, and the newest parameters to measure performance of fault-tolerant networks such as Menger number, Rabin number, fault-tolerant diameter, wide-diameter, restricted connectivity, and (l,w)-dominating number. Audience: The book is suitable for those readers who are working on or intend to start research in design analysis of the topological structure of interconnection networks, particularly undergraduates and postgraduates specializing in computer science and applied mathematics.

Topological Structure and Analysis of Interconnection Networks

The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations.
The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved.
In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

A Survey of Combinatorial Gray Codes

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'. This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

This survey provides a comprehensive and unified analysis of symmetry in a wide variety of Cayley graphs of permutation groups. These include the star graph, bubble-sort graph, modified bubble-sort graph, complete-transposition graph, prefix-reversal graph, alternating-group graph, binary and base-b (b ≥ 3) hypercube, cube connected cycles, bisectional graph, folded hypercube and binary orthogonal graph. In addition, we also define a variety of generalizations of the hypercube and orthogonal graphs. The types of symmetry analyzed include vertex and edge transitivity, distance regularity and distance transitivity. Since these notions of symmetry depend on the shortest paths, as a by product we also describe the shortest path routing algorithms for these graphs. We present a number of open problems related to the networks described in this paper.

Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey

Due to recent developments of parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years and is still stimulated by the new technologies of communication networks such as optic fibers. There are many advantages in using Cayley (di)graphs as models for interconnection networks. This work first surveys some classes of Cayley graphs which are well studied as models of interconnection networks. Results and problems related to routings in networks are then presented, with emphasis on loads of nodes and links in routings.

Cayley graphs and interconnection networks

The authors develop an algebraic framework that exposes the structural kinship among the deBruijn, shujCfle-exchange, butterjy, and cube-connected cycles networks and illustrate algorithmic benefits that ensue from the exposed relationships. The framework builds on two alge- braically specified genres of graphs: A group action graph (GAG, for short) is given by a set V of vertices and a set H of permutations of V: For each v E V and each r EII, there is an arc labeled r from vertex v to vertex v. A Cayley graph is a GAG (V, H), where V is the group Gr(H) generated by H and where each r 6 H acts on each g 6 Gr(H) by right multiplication. The graphs (Gr(H), H) and (V, H) are called associated graphs. It is shown that every GAG is a quotient graph of its associated Cayley graph. By applying such general results, the authors determine the following: The butterfly network (a Cayley graph) and the deBruijn network (a GAG) are associated graphs. The cube-connected cycles network (a Cayley graph) and the shuffle-exchange network (a GAG) are associated graphs. The order-n instance of both the butterfly and the cube-connected cycles share the same underlying group, but have slightly different generator sets II. By analyzing these algebraic results, it is delimited, for any Cayley graph G and associated GAG 7-/, a family of "leveled" algorithms which run as efficiently on T/ as they do on (the much larger) G. Further analysis of the results yields new, surprisingly efficient simulations by the shuffle-oriented networks (the shuffle-exchange and deBruijn networks) of like-sized butterfly-oriented networks (the butterfly and cube-connected cycles networks): An N-vertex butterfly-oriented network can be simulated by the smallest shuffle-oriented network that is big enough to hold it with slowdown O(log log N). This simulation is exponentially faster than the anticipated logarithmic slowdown. The mappings that underlie the simulation can be computed in linear time; and they afford one an algorithmic tech- nique for translating any program developed for a butterfly-oriented architecture into an equivalent program for a shuffle-oriented architecture, the latter program incurring only the indicated slowdown factor.

Group action graphs and parallel architectures

A unified framework for finding efficient permutation routes on parallel networks in an off-line setting is presented. If the underlying graph of a parallel network contains an appropriate “approximate” product structure then our method guarantees the existence of non-blocking near-optimal permutation routes. The routes in question can be determined in polynomial time. Furthermore, our results are extended to finding permutation routes among the remaining “live” nodes in a faulty network.

A unified approach to off-line permutation routing on parallel networks

In this paper we present a general strategy for finding efficient permutation routes in parallel networks. Among the popular parallel networks to which the strategy applies are mesh networks, hypercube networks, hypercube-derivative networks, ring networks, and star networks. The routes produced are generally congestion-free and take a number of routing steps that is within a small constant factor of the diameter of the network. Our basic strategy is derived from an algorithm that finds (in polynomial time) efficient permutation routes for aproduct network, G×H, given efficient permutation routes forG andH. We investigate the use of this algorithm for routingmultiple permutations and extend its applicability to a wide class of graphs, including several families ofCayley graphs. Finally, we show that our approach can be used to find efficient permutation routes among the remaining live nodes infaulty networks.

A unified framework for off-line permutation routing in parallel networks

We present bounds on two combinatorial properties of Cayley graphs in terms relating to the structure of their underlying group, Included in this work is a presentation of lower bounds on the diameter of Cayley graphs of groups with nilpotent subgroups and upper bounds on the size of node bisectors of Cayley graphs of groups with solvable subgroups.

On the diameter and bisector size of Cayley graphs

It is shown that the expected benefits of multigauging can be attained without any hardware modification and that additional advantages may be gained from enabling emulations. The authors start with a (physical) bit-serial architecture and build (software) support for multigauge computation on top of its native instruction set. Assumptions about this instruction set are modest and confined solely to its functionality, not its implementation. Multigauge behavior is achieved as a high-level abstraction, which is independent of the physical design. The danger that (hardware-enabled) multigauge behavior might preclude certain types of hardware optimization is avoided. >

Marc Baumslag

Papers

Group action graphs and parallel architectures

A unified approach to off-line permutation routing on parallel networks

A unified framework for off-line permutation routing in parallel networks

On the diameter and bisector size of Cayley graphs

Achieving multigauge behavior in bit-serial SIMD architectures via emulation