Each algorithm uniquely corresponds to a directed graph, and each parallel computer
system, also uniquely corresponds to a directed graph. Therefore, solving
many problems regarding algorithms and parallel computer systems, is reduced to
solving problems on the graph level. In this thesis, we deal with the problem how
to determine a mapping of graph of algorithm, which corresponds to a given problem,
so that the result of mapping is a graph of corresponding parallel computer
system, on which the problem is going to be effectively implemented. Practically,
on the basis of the algorithm of problem, we design a parallel computer system of
special purpose, suitable for solving the problem.
The main topic of interest in this thesis are 3D coordinate graphs, as well as
possibilities of their mapping to the plane (2D) and line (1D). The geometrical
relationship between three-dimensional objects and their two-dimensional images,
ie. projections in some planes, is very important and... has many applications in
various fields of science and technology. During determination of those mappings,
principle of neighborhoods between nodes in a given graph and in its image has to
be preserved, and also minimal number of nodes in a graph, which corresponds to
a parallel computer system.
Without loss of generality, we deal with graphs that can be presented in a
three-dimensional coordinate system. This thesis gives an answer to the question
how the three-dimensional mesh, ie. coordinate graph, maps to a plane and a line.
First, we determine all possible projection directions for design, with respect to
certain limitations. For each possible projection direction we define the mapping,
which ensures that the resulting image have the optimal number of nodes, taking
into account that principle of neighborhood between nodes is preserved.
Beside compilation of previous results, the main results and contributions of this thesis has been provided via formal mathematical forms through 1 lemma, 7
theorems and 3 corollaries. The most important theorems, that are base of whole
process of projection of coordinate graphs are given with complete proofs. These
theorems are still not published and they represents an original contribution of
this thesis.
In other parts of the dissertation, using introduced mapping, we design all
possible 1D systolic arrays for solving one of the basic and most used problems
in linear algebra, matrix product. Then, we implement the same methods for
synthesis of 1D systolic arrays for solving some standard problems in graph theory:
finding the transitive closure of directed graph, finding all shortest paths between
nodes in graph, finding all possible minimum cost spanning trees in a given graph.
Several years of research in this area, showed us the path for solving the complete
set of tasks. It is shown, first on the theoretical level, and then through all
those applications, that method of projection of graphs, introduced in this thesis,
is the best, and also enables an automatic synthesis of arrays, through explicit
formulas, and also gives optimality of the obtained arrays, in terms of space, or
time or space-time.
