Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Switching codes and designs

AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters

Fault-Tolerant Load Management for Real-Time Distributed Computer Systems

This paper presents a fault-tolerant scheme applicable to any decentralized load balancing algorithms used in soft real-time distributed systems. Using the theory of distance-transitive graphs for representing topologies of these systems, the proposed strategy partitions these systems into independent symmetric regions (spheres) centered at some control points. These central points, called fault-control points, provide a two-level task redundancy and efficiently re-distribute the load of failed nodes within their spheres. Using the algebraic characteristics of these topologies, it is shown that the identification of spheres and fault-control points is, in general, is an NP-complete problem. An efficient solution for this problem is presented by making an exclusive use of a combinatorial structure known as the Hadamard matrix. Assuming a realistic failure-repair system environment, the performance of the proposed strategy has been evaluated and compared with no fault environment, through an extensive and detailed simulation. For our fault-tolerant strategy, we propose two measures of goodness, namely, the percentage of re-scheduled tasks which meet their deadlines and the overhead incurred for fault management. It is shown that using the proposed strategy, up to 80% of the tasks can still meet their deadlines. The proposed strategy is general enough to be applicable to many networks, belonging to a number of families of distance transitive graphs. Through simulation, we have analyzed the sensitivity of this strategy to various system parameters and have shown that the performance degradation due to failures does not depend on these parameter. Also, the probability of a task being lost altogether due to multiple failures has been shown to be extremely low

Semi-Distributed Load Balancing for Massively Parallel Multicomputer Systems

This paper presents a semi-distributed approach, for load balancing in large parallel and distributed systems, which is different from the conventional centralized and fully distributed approaches. The proposed strategy uses a two-level hierarchical control by partitioning the interconnection structure of a distributed or multiprocessor system into independent symmetric regions (spheres) centered at some control points. The central points, called schedulers, optimally schedule tasks within their spheres and maintain state information with low overhead. We consider interconnection structures belonging to a number of families of distance transitive graphs for evaluation, and using their algebraic characteristics, show that identification of spheres and their scheduling points is, in general, an NP-complete problem. An efficient solution for this problem is presented by making an exclusive use of a combinatorial structure known as the Hadamard Matrix. Performance of the proposed strategy has been evaluated and compared with an efficient fully distributed strategy, through an extensive simulation study. In addition to yielding high performance in terms of response time and better resource utilization, the proposed strategy incurs less overhead in terms of control messages. It is also shown to be less sensitive to the communication delay of the underlying network

Arithmetic completely regular codes

In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.Comment: 26 pages, 1 figur