**3.1 Construction of Hamiltonian cycles in toroidal graphs with edge defects**

The capability of recurrent neural networks to converge to stable states can be used for mapping program graphs to CS graphs with violations of regularity caused by deletion of edges and/or nodes. Such violations of regularity are called defects. In this work, we study the construction of Hamiltonian cycles in toroidal graphs with edge defects. Experiments in 2D-tori with a deleted edge and with *n* = 9 to *n* = 256 nodes for *p* = *n* were conducted. The experiments show that the construction of Hamiltonian cycles in these graphs by the above-described algorithm is possible, but the value of the step *t* at which the cycle can be constructed depends on the choice of the deleted edge. The method of automatic selection of the step *t* is described at the beginning of Section 3. Table 5 illustrates the dependence of the step *t* on the choice of the deleted edge in constructing an optimal Hamiltonian cycle by the SOR method in a 2D-torus with *n* = 16 nodes for 0.125 .

Examples of Hamiltonian cycles constructed by the SOR method in a 2D-torus with *n* = 16 nodes are given in Figs. 12a and 12b. Figure 12a shows the cycle constructed in the torus without edge defects for 0.5 and 0.25 *t* . Figure 12b shows the cycle constructed in the torus with a deleted edge (0, 12) for 0.125 and 0.008 *t* .

Optimization of Mapping Graphs of Parallel

algorithms proposed in (Tarkov, 2005).

using the following method:

Fig. 13. Unification of cycles

Programs onto Graphs of Distributed Computer Systems by Recurrent Neural Network 219

The time of execution of the above-described algorithm can be substantially reduced by

For example, the initial graph of the system can be split into connected subgraphs by the

2. Construct a Hamiltonian cycle in each subgraph by the algorithm described above.

3. Unite the Hamiltonian cycles of the subgraphs into one Hamiltonian cycle.

**3.2 Construction of Hamiltonian cycles by the splitting method** 

1. Split the initial graph of the system into *k* connected subgraphs.


Table 5. Dependence *t* of the step on the choice of the deleted edge

Results discussed in this section should be considered as preliminary and opening the research field studying the relation between the quality of nesting of graphs of parallel algorithms to graphs of computer systems whose regularity is violated by node and edge defects and the parameters of neural network algorithms implementing this nesting.

Fig. 12. Examples of Hamiltonian cycles in 2D-torus
