#### Name

Pavlov Dmitriy Alekseevich

#### Scholastic degree

•

#### Academic rank

associated professor

#### Honorary rank

—

#### Organization, job position

Kuban State Agrarian University

#### Web site url

—

## Articles count: 6

The fractal and prefractal graph are described in the
article. The basic definitions and notation are
proposed, the procedure for constructing prefractal
graph, the operation of replacement vertex by seed is
given

In the article we investigate the multicriteria task
arising at the organization of distributed calculations
in a corporate network. As a mathematical tool to
solve the problem we use prefractal graphs, which
naturally reflect the structure of relationships in
global and corporate networks. The corporate network
with the distributed computing system at the solution
of a particular task has to be reliable, quickly and
qualitatively to make decisions. And every computer
in the network should be a part in the solution of the
problem, since it is fixed for a certain function. The
problem is reduced to cover the prefractal graphs with
disjoint simple paths along the edges and vertices.
On the set of all admissible coverings we constructed
a vector-target function with specific criteria. All
these criteria have a specific meaningful
interpretation, allowing organizing the calculation of
maximum reliability, with minimum time information
processing and loading balancing between the
network elements. In the article we constructed
polynomial algorithms for finding optimal solutions
according to specific criteria. For the criteria which
are not optimizing the allocated coverings, estimates
of the lower and upper bounds are given. For all the
algorithms we constructed and substantiated
estimation of computational complexity, confirming
the advantage of using algorithms on prefractal
graphs to classical algorithms on graphs

Most of the tasks of planning and organizing transport
routes are pointed to solving optimization problems
on graphs in multi-criteria statements, for which the
only optimal solution is missing. In conditions of
multicriteria, it becomes necessary to search for a set
of alternatives instead of an optimum. The quality of
the admissible solutions is estimated by the vector
objective function. The article proposes to investigate
the problem using a special class of graphs -
prefractal graphs, which allow describing in a natural
way the structure of the hierarchy of territorial links,
as well as enable to take into account structural
dynamics in terms of system growth. A multicriteria
mathematical formulation of the problem of covering
a prefractal graph by simple intersecting chains is
constructed, to which the investigated problem of
organizing routes in large-scale transport networks
reduces. The main social and economic requirements
for the transport system are formulated and included
in the model in the form of criteria

This article explores the multicriteria problems arise
in the organization of routes in large-scale transport
management system. As a mathematical tool for constructing
a model, we were using the prefractal
graphs. Prefractal graphs naturally reflect structure of
the device of communications of transport system,
reflecting its important features – locality and
differentiation. Locality is provided with creation of
internal routes (city, raionwide, etc.). Differentiation
is understood as division of routes on intra regional,
interregional and international. The objective is
reduced to a covering of prefractal graphs by the
simple paths which are crossed on edges and nodes.
On the set of feasible solutions, vector criterion
function with certain criteria is based. In concepts of
transport system, the given criteria have concrete
substantial interpretation, the transport routes allowing
to design considering features of system.
In this article, we construct polynomial algorithms for
finding optimal according to certain criteria decision.
By the criteria which aren't optimizing the allocated
routes their estimates of the lower and upper bounds
are given. On all given algorithms the estimates of
computing complexity confirming advantage of use
of methods of prefractal and fractal graphs before
classical methods of the theory of graphs are
constructed and proved

In the article we present a spatial structure of largescale
transport systems. The model of a transport
network can be presented in the form of a graph, with
a set of the nodes corresponding to elements of a
network and a set of edges – to sections of roads the
connecting these nodes. As the model of a card of
roads, it is offered to use prefractal graphs which
naturally reflect structure of communications when
reviewing a transport network in different scales (the
states, regions, areas). Prefractal graphs allow
describing structural dynamics of the studied system
in the discrete time. One of the most widespread
scenarios of structural dynamics is the growth of
structure. The statement of tasks of the organization
of transport routes contains requirements criteria to
finding of optimal solutions. Often these requirements
and criteria are contradicting each other. It leads to
appearance of a multicriteria problem definition.
The multicriteria problem definition on a class of
prefractal graphs is considered. The optimum
algorithm of separation of the greatest maximum
paths by the given criterion is constructed and
estimates by remaining criteria are given. In operation
computing complexity of the constructed algorithm of
separation of the greatest maximum paths on a
prefractal graph is calculated and advantage of
operation of algorithm on last before algorithm of
separation of the greatest maximum paths on normal
graphs is justified. The constructed algorithm on
prefractal graphs has polynomial complexity

This work is devoted to a new method for designing
large-scale structures of transport networks. The
model of a large-scale transport network is built on
prefractal graphs. The model of a large-scale transport
network is based on the principle of hierarchical
organization of territories. A prefractal graph is a
finite analogue of a fractal graph combining the
properties of a fractal and a graph. Some problems of
discrete optimization on prefractal graphs become
polynomially solvable under certain conditions.
Reducing the complexity of extreme problems on
prefractal graphs is due to the fact that on these
graphs for some problems, along with the selfsimilarity
property, the property of heredity appears.
Using this property, it is possible to construct parallel
algorithms for problems on prefractal graphs, the
complexity of which is orders of magnitude lower
than for known successive algorithms