Arden and Lee proposed a class of chordal ring networks of degree three as communication networks for multicomputer systems, derived a formula for the diameter, and produced an algorithm for finding the shortest paths for them. In this paper, it is shown that the formula for the diameter and the routing algorithm presented by them are inaccurate. We have obtained a large dataset containing parameters for describing optimal diameter chord rings for all the numbers of nodes up to 60 000 and found the exact lower bound for the diameter of chordal ring networks. A new method is proposed and the algorithms for automatic search for analytical descriptions of families of optimal chordal rings are realized based on an analysis of the database. Using the latter, analytical descriptions of over 500 new families of optimal chordal ring networks for many values of the number of nodes are found (only six families have been known until now in the literature).

Optimal circulant networks are of practical interest as models of reliable low-latency communication networks for multiprocessor cluster systems and on-chip networks. The authors are the first to construct a large dataset of optimal diameter doubleloop circulant networks with up to 50 thousand nodes, containing a complete set of optimal graph generators. The analysis of the dataset has been carried out in order to study the problem of finding analytically defined families of optimal graphs. Two new algorithms for automatically finding analytical descriptions of optimal graphs families described by polynomials in diameter have been developed. Using the implemented algorithms, a large number of new analytically described families of optimal networks have been found and tested using validation over the entire range of changes in the diameters of the dataset graphs. The found families of optimal networks can be used when scaling information transmission algorithms in double-loop circulant structures.

Optimal ring circulant networks of degree four are considered as models of reliable communication networks with minimal delays for networks on a chip and multipro-cessor cluster systems. Based on the analysis of a data set of optimal descriptions of double-loop networks, a search has been carried out for analytically determined. infinite families of optimal graphs. By integrating data visualization and analytical descriptions of optimal graphs, new infinite families of optimal networks with a linear generator of the forms 4d+a, where d is the diameter of the graph, have been constructed and theoretically justified. The proposed approach to obtaining families of optimal networks is new and is of interest for further studies of the properties of optimal double-loop networks.

Optimal circulant networks are of practical interest as graph models of reliable communication networks for supercomputer systems and networks on a chip. A large dataset of parameters of optimal double-loop circulant networks with a number of nodes of up to 50,000 and 451,000 points has been constructed. The dataset contains, for each number of vertices, all generators corresponding to optimal or suboptimal (in the absence of optimal) descriptions of the graph. This made it possible to find analytical dependencies of the parameters that define families of optimal graphs. Based on the analysis of the constructed dataset, the solution to the problem of determining optimal two-dimensional circulant networks is studied. A process is automated for determining analytical, parametric descriptions of the families of optimal double-loop networks, described polynomials of diameter. Having applied the methods of integrating differential evolution and reduced local search to the analysis of the generated dataset, we rediscovered a number of previously known families and found more than 200 new, different from all those known in literature, families of optimal double-loop networks with generators of linear and quadratic types of polynomials. Our new ap­proach can be applied to studing datasets of other promising classes of graphs and networks.

The solution of the problem of organizing optimal communications in circulant networks of degree four is considered. For a family of optimal circulant networks with the minimum diameter and average distance for any number of nodes in a graph, we propose an optimal pair routing algorithm of constant complexity based on using the relative addressing of nodes in a network. The new algorithm is an analytical extension to any number of nodes in the network of the routing method proposed for dense Gaussian networks, and it does not use division operations, which are very expensive to implement in fixed-point format. This extension is based on the proposed scheme of transformations on the plane of geometrical patterns of optimal circulant networks. The developed routing algorithm is the basis for generating the series of routing algorithms for different subfamilies of the optimal two-dimensional circulants. The general routing algorithm and its modification for a separate subclass of circulants are implemented in the HDL NoC model with circulant topology. All the algorithm parameters important for networks-on-chip, including the consumption of memory, logical resources, and execution time are comprehensively investigated. The results of a comparative analysis of the new algorithms with other routing algorithms, previously implemented in the networks-on-chip, are presented.

Parallel versions of a genetic algorithm based on the hybrid MPI—OpenMP model are implemented to optimize circulant networks, which are of practical interest in the design of supercomputer systems and systems on a chip. An analysis of the efficiency of parallel programs with different numbers of MPI processes and OpenMP threads on a cluster of Kunpeng processors has been carried out. The speed-up of several hybrid parallel computing schemes was experimentally evaluated and analyzed. Two bottlenecks in terms of efficiency in parallel execution of the algorithm are identified and methods for their solution are proposed. By means of the parallel genetic algorithm the descriptions of circulant networks with better average distance and bisection width for the known large circulant networks were obtained.

Parallel versions of the reduced exhaustive search algorithm based on the Python tools are implemented to optimize chordal ring networks, which are of practical interest in the design of systems on a chip and supercomputer systems. An analysis of the effectiveness of parallel programs with different numbers of MPI processes on Kunpeng processors was carried out. The speed-up of several parallel computing schemes was experimentally evaluated and analyzed. The large dataset of all optimal chordal networks with numbers of up 6 · 10^4 nodes was generated for the first time. A preliminary analysis of experimentally obtained dataset has been carried out and the existence of new families of optimal chordal ring networks with analytical descriptions of parameters has been discovered.

We consider a solution of the problem of constructing a series of families of circulant networks of degree six specified analytically with the help of two parameters one of which is the network diameter. Based on the analysis and generalization of the properties of a new description of an extremal family of circulants, a general series of families of circulant graphs of degree six of arbitrary diameters is constructed that includes extremal circulant graphs of degree six and new infinite families of circulants with an even number of vertices. In the found series of families, descriptions of a series of circulant graphs of any given diameter are analytically determined. Optimality ranges of series graphs are algorithmically identified, where ‘optimal’ is understood as a circulant graph of degree six with the minimum possible diameter for a given number of vertices. The resulting series of families of circulant networks is promising as a scalable topology model for networks on a chip.

Parallel versions of the genetic algorithm based on the MPI model are implemented to optimize circulant networks, which are of practical interest in the design of systems on a chip and supercomputer systems. The problem of finding the optimal circulant networks with minimal average distance for the given number of nodes and given degree is investigated. An analysis of the effectiveness of parallel programs for synthesis of optimal circulant networks with different numbers of MPI processes on a cluster of Kunpeng processors was carried out. The speedup of parallel computing schemes on several cores and nodes were experimentally evaluated and analyzed. The results of the synthesis of optimal circulant networks of various degrees of nodes with a minimum average distance are presented. The proposed parallel genetic algorithm allows to obtain optimal circulants with a minimum average distance and better bisection width for the known circulant networks with large number of nodes and degrees.

As a promising topology of networks on a chip, we consider a family of Dense Gaussian Networks, which are optimal circulant degree four graphs of the form C(D2 + (D + 1)2; D, D + 1). For this family, an algorithm for finding the shortest paths between graph vertices is proposed, which uses relative addressing of vertices and, unlike a number of the known algorithms, allows to calculate the shortest paths without using the coordinates of neighboring lattice zeros in a dense tessellation of graphs on the ℤ2 plane. This reduces the memory and execution time costs compared to other algorithms when the new algorithm is implemented on a network-on-chip with a Dense Gaussian Network topology.

The article considers a family of dense Gaussian networks. Theese graphs are optimal circulant degree four graphs of the form $\boldsymbol{C}(\boldsymbol{D}^{2}+(\boldsymbol{D}+1)^{2},\ \boldsymbol{D},\ \boldsymbol{D}+1)$ . For this family, as a topology of networks-on-chip, an $\boldsymbol{O}(1)$ algorithm for finding the shortest paths between vertices of a graph is developed. The proposed routing algorithm uses relative addressing of vertices. Its difference over the known algorithms is calculation of the shortest paths without using the coordinates of neighboring lattice zeros in a dense tessellation of graphs on the $\mathbf{Z}^{2}$ plane. The proposed algorithm was modeled and synthesized with the RTL-implementation of a NoC with dense Gaussian network topology. The dependence of memory estimates and the number of logical elements on the number of nodes in the network, required for the operation of the algorithm, is obtained. These indicators were compared with six other algorithms previously developed for networks-on-chip. Simulation results demonstrate that the new algorithm is more preferable for use in networks-on-chip than a number of other routing algorithms.

This paper studies the effect of changing the diameter of the network in circulant networks of dimension two with unreliable elements (nodes, links). The well-known (Δ,D,D′, s)-problem is to find (Δ, D)-graphs with maximum degree Δ and diameter D such that the subgraphs obtained from the original graph by deleting any set of up to s vertices (edges) have diameter at most D′. For a family of optimal circulants of degree four we found the ranges of the orders of the graphs that preserve the diameter of the graph for one (two) vertex or edge failures. It is proved that in the investigated circulant networks in case of failure of one or two edges (vertices), the diameter can increase by no more than one, and in case of failure of three elements by no more than two (edge failures) or three (vertex failures). It is shown that two-dimensional optimal circulants, in comparison with two-dimensional tori, have a better diameter in case of element failures.

The problem of the organization of optimal communications in circulant networks is considered. For a family of optimal two-dimensional circulant networks with the minimum diameter and average distance, the relative addressing of nodes of a network is introduced as shortest paths vectors from a fixed node. On its basis, a constant complexity pair routing algorithm using a set of shortest paths is proposed. This algorithm reduces the execution time in comparison with known routing algorithms due to the elimination of division operations and reduces the required memory to few words. The new algorithm is an analytical generalization to any number of nodes in a network of the method proposed for the family of Gaussian networks. This generalization is based on the proposed scheme of transformations on the plane of geometrical patterns of graphs of the family. The routing algorithm can be applied in networks-on-chip with a two-dimensional circulant topology.

This paper is considering the solution of the deadlock problem when applying different routing algorithms to networks-on-chip (NoCs) with circulant topology. The paper describes two different solutions for this problem.There is an effective method based on special aspects of circulant topologies which provides significant increase of peak throughput compared with mesh topology widely applied in NoCs.The comparison of peak throughput based on high-level simulation results showed more than 60 % increase for circulant topologies.In addition, there is a review of universal deadlock avoidance method based on bypassing blocked network segments through acyclic sub-network.The presented methods were evaluated using a high-level universal simulator for NoCs developed on the base of Noxim simulator.

An approach for constructing and optimizing graphs of series of analytically described circulant graphs of degree six with general topological properties is proposed. The paper presents three series of families of undirected circulants having the form C (N (d, p);1, s(2) (d, p), s(3) (d, p)), with an arbitrary diameter d > 1 and a variable parameter p(d), 1 <= p(d) <= d. The orders N of each graph in the families are determined by a cubic polynomial function of the diameter, and generators s(2) and s(3) are defined by polynomials of the diameter of various orders. We have proved that the found series of families include degree six extremal circulant graphs with the largest known orders for all diameters. By specifying the functions p(d), new infinite families of circulant graphs including solutions close to extremal graphs are obtained.

A new pair routing algorithm for transmitting messages in multiprocessor systems and networks-on-chip based on circulant networks of arbitrary dimension is proposed. It allows using all reserve shortest paths in the presence of destructive factors (deadlocks, livelocks, starvation, failures) at the nodes and channels of the communication network. A distinctive feature of the proposed algorithm is the absence of using the routing tables with fixed shortest paths when message is transmitted. It becomes possible to determine the set of the shortest paths for routing due to the relative addresses of destination nodes based on a parametric description of the network. Estimates of the number of reserve shortest paths are obtained, and an effective algorithm for using these paths to prevent dynamic topology changes and network congestions is proposed. To reduce the required memory in networks-on-chip with a circulant topology, we proposed a version of the routing algorithm for two-dimensional optimal circulants. We experimentally found the minimum number of reference nodes (nodes containing mapping tables) for them and estimates of memory for mapping tables, as well as the average path length for the routing algorithm using the reference nodes.

A new algorithm of metaheuristic programming with gene expression based on various natureinspired algorithms as a search engine for solving the problem of automatic synthesis of nonlinear models in an analytical form is proposed. We show that the proposed algorithm is superior to the standard gene expression programming algorithm.

For analytically defined families of three-dimensional circulant networks with a parametric description, an analytical algorithm for finding shortest paths which has a common scheme for all networks of the family based on a given generating function was developed. A comparative analysis of three routing algorithms (analytical Two-terminal routing algorithm, Coefficients search on graph generators, and Dijkstra's algorithm) for a variety of circulant networks from an analytically defined family was carried out. Estimates of the effectiveness of the considered routing algorithms for use in networks-on-chip were obtained.

A set of families of undirected triple loop networks of the form C(N (d, p), 1, s(2)(d, p), s(3)(d, p)) with the given diameter d > 1 and a parameter p = 1, 2,..., d-1 is obtained. For each such family, the order N of every graph in the family and its generators s(2) and s(3) are defined by a cubical polynomial function of the diameter. The found set includes circulant graphs of degree 6 with the largest known orders for any diameters d equivalent to 0 (mod 3) and d equivalent to 2 (mod 3). Examples of constructing new families of triple loop networks based on the definition of functions p = p(d) are presented.

For a family of optimal two-dimensional circulant networks with an analytical description, two new improved versions of the shortest path search algorithm with a constant complexity estimate are obtained. A simple, based on the geometric model of circulant graphs, proof of the formulas used for the shortest path search algorithm is given. Pair exchange algorithms are presented, and their estimates are given for networks-on-chip (NoCs) with a topology in the form of the considered graphs. New versions of the algorithm improve the previously proposed shortest path search algorithm for optimal generalized Petersen graphs with an analytical description. The new proposed algorithm is a promising solution for the use in NoCs which was confirmed by an experimental study while synthesizing NoC communication subsystems and comparing the consumed hardware resources with those when other previously developed routing algorithms.