DEVELOPMENT OF MAXIMUM CLIQUE DEFINITION METHOD IN A NON-ORIENTED GRAPH

Implementation of information and measuring systems with analog-digital converters of high capacity and quick operation considerably enlarge the data flows [1]. The facilities of diagnosing the railway automatic equipment and mobile complete units [2] require communication resources in technological communication networks. The change to digital networks based on optical dielectric waveguides makes the technicians optimize the resources providing the network traffic of branch subnetworks [3]. The loading irregularity of railway computing recourses is reasonable to be used for distributed computations. For this purpose, the computing system is simulated as a non-oriented graph. Programmatic definition of maximum cliques will let us optimize the computing resources and withhold us from premature hardware infrastructure investment.


Introduction
Implementation of information and measuring systems with analog-digital converters of high capacity and quick operation considerably enlarge the data flows [1].The facilities of diagnosing the railway automatic equipment and mobile complete units [2] require communication resources in technological communication networks.
The change to digital networks based on optical dielectric waveguides makes the technicians optimize the resources providing the network traffic of branch subnetworks [3].The loading irregularity of railway computing recourses is reasonable to be used for distributed computations.For this purpose, the computing system is simulated as a non-oriented graph.Programmatic definition of maximum cliques will let us optimize the computing resources and withhold us from premature hardware infrastructure investment.

Literature review and problem statement
The development of the distributed computing system monitoring method based on the definition of the shortest paths and the shortest Hamiltonian cycles in a graph is analyzed in the paper [4].The model of the systems under discussion is equivalent to formal definitions of sparse graphs.It may be replaced by the Maximum Clique Problem (MCP), which is known as one of the hardest mathematical problems of the graph theory.As yet, we have no algorithms for solving the problem within polynomial time by parallel computing.
Meanwhile, the problem under discussion has lots of applications.In bioinformatics, MCP is used with a computer-based analysis of genomic databases, for example, when searching the potential regulatory structures of ribonucleic acids.In social networks, MCP is used while clustering the data, i. e. when dividing different communities into groups (clusters) that have general properties.Clustering allows each cluster to be processed using an individual helper server.In chemistry, MCP is the basis for searching "the maximum generic substructure" in a graph describing the structure of a chemical compound.Besides, MCP is a mathematical model of a number of problems arising from electronics design automation.The input data amount is vast (input graphs may contain up to million vertices).Thus, the present-day research direction of MCP is the development of fresh approaches to finding the exact and approximate solutions with regard to graph peculiarities resulted in applications.All the algorithms for MCP solution are classified as exact and approximate.Among the exact ones, there are cliquebased algorithms, for instance, [5]; as well as backtracking algorithms, for instance, [6].The exact algorithms operate in a time exponential from an input data amount.Among the approximate ones, there are combinatorial optimization and genetic algorithms [7].These algorithms have an indefinite
In the paper [8], the authors represent an advanced parallel binary algorithm for MCP.However, such an algorithm, even when it is advanced, has not been intended for high-dimensionality graphs.In the papers [9,10], the authors have investigated a biphase algorithm for a clique problem solution, and clique issues for sparse graphs of high dimensionality, as well as decomposition of minimal clique separator.However, the presented methods leave out the admissibility of parallel computing.
The paper [11] presents the correction procedure of only a single computer error and in the papers [12,13] the group of authors has suggested MCP solution for a degenerate graph and the approaches to finding the maximum quasicliques.So, it seems promising to develop the method of determining the maximum cliques in non-oriented graphs that makes possible the execution of different applications in information systems in a real-time scale.

The aim and objectives of the study
The aim of the paper is developing the method for defining the maximum cliques in non-oriented graphs with small time complexity.This will expand the admissibility of data clustering in computer systems and raise the effectiveness of electronics development.
To achieve the aim, it is necessary to do the following: -to introduce procedure B making possible the definition of estimates of the largest graph clique size value; -to introduce procedure A allowing the cliques to be formed on the base of each graph vertex with the help of procedure B and the maximum graph clique to be selected.

1. Formal characterization and solution of the problem
Let us consider the MCP solution because of getting the estimates of the largest value of graph clique sizes based on a sufficiently clear assertion 1.
Assertion 1.If in graph G (V, E) there is a clique of k size, then the number of vertices i with 1 i d k ≥ − power should be at least k, i. e. we shall say that there is an estimate of the largest value of maximum clique size max k ∆ = in G (V, E) graph.Let us assume that there is G(V, E) graph with n vertices.Let us build up n subgraphs as follows: select the vertex i=1, point out all the vertices related to it, and connect all the vertices with ribs according to the links of the source graph G (V, E).As a result, we've got subgraph G 1 without a vertex, subgraphs G 2 , G 3 ,…, G n are build up in the same way.Thus, generation of some arbitrary subgraph G i based on some vertex i is resulted in selecting the subset of vertices related to vertex i.They are connected with ribs according to those vertex links that were held in G graph.It is clear, that if there is the maximum clique in source graph G (V, E), then the above clique is in one of subsets G i .
Let us consider the possible generation of procedure В for defining the estimates of the largest clique size value in G i subgraphs.Let us suppose that G i graph is preset with v d vertices selected in a descending order of powers.At that, the vertices with equal powers d r are pooled in subsets { } r a with r r p a = cardinality (where r a -the numbers of corresponding vertices in G i graph), and G i is the number of vertices in G i graph with equal powers d r .The total number of such subsets is denoted by γ.As a result of sorting, we obtain a sequence of vertices satisfying the inequation: The sequence (1) will be stored as list .S g Because of assertion 1, the following procedure of defining the largest value of estimate max ∆ as per list S j can be suggested.Procedure B of definition max ∆ : Step 1. Sorting the vertices and forming the subsets { }. = + , and the procedure stops, if not, let us proceed to the following step.
Step 4. : 1 r r = + and let us proceed to step 3.It is well to bear in mind, that for the inequation analyzed in step 3 when r=1, the sum is  After sorting the vertices (Table 1), we obtain two subsets + ≥ + The inequation isn't done, therefore 1 : and we pass into 2 .
Let us check inequation P r=1 +P r=2 ≥d r=2 +1, then we obtain 3+4≥3+1, that inequation is done, hence max ∆ =4.Thus, the maximum graph clique cannot exceed the size equal to 4. If the size of the graph clique is bigger, for instance size 5, then the number of vertices with power 4 should be more or equal to 5, and for clique size 4 the number of vertices with powers 3 should be more or equal to 4. The sorting results of graph cliques are presented in Table 1.
If the estimates of the largest values max then the subset of those vertices will be as Then the following assertion is true.

Table 1
The results of sorting the graph vertices the total number of such subsets is denoted by variable β.As the result of sorting, we obtain the sequence of vertices satisfactory to the inequation The sequence (2) will be stored as list .S b With the help of procedure B, we define the subset of vertices r * with the following inequation The correctness of assertion 3 is clear as far as on the base of the vertices in list S b after r * it is impossible to build up the cliques of a bigger size than the cliques on the base of vertices in the list prior to r * .

2. The problem-solving method
Using procedure В, let us build A procedure of forming the maximum clique in some graph G.The basic data for procedure А is graph G itself and list S of the graph vertices.
Procedure A: Step 1. Check in graph G the availability of Y={i} vertices related to all the vertices of G graph.Delete the set of Y vertices and incident to them ribs out of the graph and bring them into U set.After that, check all the vertices for deleting out of graph G.If they are deleted, the procedure stops as the source graph presents the clique of maximum size, if not, proceed to the next step.
Step 2. Check all the remainder of graph vertices with like powers.If so, check the remainder of vertices for vertex j, related to all the vertices in U.If so, the procedure is stopped as the set of vertices { } j U  form the maximum clique in graph, if not, proceed to the next step.
Step 3. Select i vertex out of list S and on the base of vertices adjacent to i form subgraph G i (V i , E i ).
Check subgraph G i (V i , E i ) for vertices D={i} related to all the vertices of G i (V i , E i ) subgraph, and delete the set of vertices D and incident to them ribs out of the graph and bring them into the set U i .After that, check the vertices for deleting out of subgraph G.If so, then max i ∆ = , i V if not, proceed to the next step.

and bring max i
∆ into set H, here assign the estimates equal to the maximum estimate in set H to vertices j entered into U, if they are available in that set, and bring them into H.
Step 5. Check list S for elements.If there are no elements, proceed to step 3, if so, proceed to the next step.
Step and denote the total number of those subsets by variable β.As a result of sorting out, we obtain the sequence of vertices satisfactory to inequation: It is saved as list S b and then let us proceed to the next step.
Step 8.With the help of procedure В, define the set of vertices r * which is the beginning of inequation

3. Example of problem solution by the developed method
Let us consider an example of procedure A performance for the graph shown in Fig. 2. Next, start generating graphs G i and find max .
i ∆ for every graph G i on the basis of procedure В.For this purpose, select the vertex i=1 which is related to vertices 2, 3, 4, 5, 10 out of the list, and on the ground of the vertices data subgraph G 1 is obtained (Fig. 3).As is seen from Table 2, when procedure B starts the inequation Here a new list S β ={8, 11, 10, 9, 7, 6, 5} appears and graph G is converted into graph G / (Fig. 4) with vertices list S=S β .And then procedure A cycle repeats for new graph G / with vertices list S=S β .

Assessment of problem solution algorithm complexity
Let us assess the complexity of performing procedure А.To form the sequences (1) and ( 2) for each vertex i based on procedure В, it is necessary to do no more than 2 log n n comparison operations and n addition operations.Hence, the maximum number of comparison operations for analysis of all the vertices will not exceed 2 2 2( log ).n n n + As far as the cycle related to the omission of feature ν cannot be repeated more than n times, then the general complexity of procedure А implementation will not exceed (2 ( log In this case, in compliance with assertions 2 and 3, we obtain either an optimal solution of the MCP or sufficiently close to it.

Discussion of investigation results of the developed problem solution method
Hence, procedure А that allows solving the problem of maximum independent set and minimum vertex cover is proposed.As far as graphs G i may be formed independently, then the above procedure А may be vectorized.If system n is used for forming G i processor, then the time complexity of procedure А falls down to It is also interesting to discuss the connection of the problem of defining the maximum independent sets, vertex covers and cliques with the problem of graph isomorphism and, in particular, the isomorphic embedding problem, i. e. whether graph G(V, Е) possesses isomorphic subgraph Н(V \ , Е \ ).
It is more suitable to discuss not the Vizing product itself, but an additional to it graph.Later on, such a product will be called a modular product.Modular product GàG | of graphs G(V, E) and G(V | , E | ) is called a graph specified by the following conditions: It should be noted that because of the known connection of the clique problem with the problems of maximum independent set and minimum vertex cover [7], procedure A may be also used for solving those problems [4,14].

Conclusions
Procedure А with small time complexity 3 2 (2 log ), O n n that allows from one viewpoint solving such problems as defining maximum cliques in non-oriented graphs, defining minimum independent sets and minimum vertex covers in graphs, as well as isomorphism of graphs and isomorphic embedding is generated.
Procedure A may be effectively vectorized that allows the time complexity of its operation to be decreased to 2 2 (2 log ), O n n and the mentioned problems to be solved in a real-time scale.

Introduction
Application of intelligent information technology for data analysis makes it possible to increase a functional effi-ciency of systems of control by weakly formalized processes.Processes of control of power units of thermal power plants relate to such processes.A promising way to increase a functional efficiency of weakly formalized controlled processes

INFORMATION-EXTREME MACHINE LEARNING OF THE CONTROL SYSTEM OVER THE POWER UNIT OF A THERMAL POWER MAIN LINE A . D o v b y s h
Doctor of Technical Sciences, Professor, Head of Department* E-mail: a.dovbysh@cs.sumdu.edu.ua

D . V e l y k o d n y i
PhD* Е-mail: d.velykodnyi@cs.sumdu.edu.ua

I . S h e l e h o v
PhD, Associate Professor* Е-mail: igor-i@ukr.net

∑
If the inequation is done, then max : 1r d ∆
sizes are defined and they may be formed on the base of vertices { },(1, )   i i n = of some graph G, and in max { } i H = ∆ set there are k vertices with the estimates

∑
Here all the vertices in list S b after r * are deleted out of graph G with the incidence ribs and those vertices are deleted out of the list.Assertion 3.If subset G / is built on the base of list , S b i. e. it is the result of vertices deletion where r>r * , then it possesses the clique of maximum size.

∑
Here delete all the vertices in list , S b after r * out of the graph G with incident to them ribs and delete these vertices out of list .S b Then, S:=S b and proceed to Step 1.

Fig. 2 .
Fig. 2. Example of graph G for procedure A

Fig. 3 .
Fig. 3. Subgraphs and vectors: а -subgraph G 1 ; b -subgraph G 1 | ; c -vectors of vertices and number of ribs As it follows from Fig. 3, а, vertex 3 in a subgraph relates to all the subgraph vertices, so it is deleted along with the incident ribs and brought into set U1={3}.As the result, subgraph G 1 | is obtained, Fig. 3, b.In Fig. 3, c, it is shown that as per procedure В | max 1 3 i= ∆ = and so on their basis form sequence , S b Table2.
r * =2 and, hence, vertices {3, 4, 2, 1} matched to r>2 values are deleted out of the source graph G and the list.

Fig. 5 .
Fig. 5. Graph G / after deleting the vertices It is easy to see that | max 5 2; i= ∆ = the problem of maximum clique definition in different applications be solved in a real-time scale.

Table 2 Sequence
S β dependence on Δ max i Cartesian product of sets V and V | ; 2. Vertices (u, u | ) and (v, v | ) of graph GàG | are adjacent, if and only if at a time u¹v and u | ¹v or uvÎV, u | v | Î V | , or uvÏV, u | v | ÏV | .Later, for convenience, we'll denote  In general, it is clear that density j (GàG | )£n, as none of cliques of graph GàG | can contain two vertices out of one line or one column and n£n | .Equality j(GàG | )=n is valid only if graph G | includes an isomorphic subgraph G. Let' assume that the vertices of both the graphs are natural numbers V={1, 2, .., n}, V | ={1, 2, …, n, …, n | }.Let us suppose that at first G | includes subgraph G || (V || , E || ) isomorphic G and that isomorphism is generated by vertices correspondence.li are any two different vertices (k¹l), so as correspondence « between GàG | vertices is isomorphism, then either k and l are adjacent in G, and i k and i l are adjacent in G || , or k and l are nonadjacent in G, and i k and i l are nonadjacent in G || .In both cases, the vertices  isomorphism of graph G | subgraph G || generated by the set of vertices V || ={i 1 , i 2 ,…, i n }, so as follows from the definition GàG |  Î kl V if and only if  Î | .So, all the n-cliques of graph GàG | one-to-one uniquely correspond to all possible G isomorphic embeddings as a subgraph in G. l