Authors B. K. Lebedev, O. B. Lebedev, E. О. Lebedeva
Month, Year 07, 2017 @en
Index UDC 004.896
Abstract We propose new technologies, principles and mechanisms for solving the problem of con-structing a Minimal Steiner Tree using mathematical methods, which lay down the principles of natural decision-making mechanisms. The problem of constructing a Steiner tree is represented in the form of an adaptive system, based on the integration of the principles of self-organization and the ant approach to finding a solution. The well-known Steiner problem consists in the following. Given is a set of P points in the plane. An orthogonal grid is formed by conducting a horizontal and vertical line through a set of points P located on a plane. An orthogonal grid G=(V, E) is formed, where V is the set of points (nodes) of intersections of grid lines. The initial problem is equivalent to the problem of finding a tree G*=(V*, E*) in the graph G=(V, E) having the minimal total weight F of edges and including a given set of vertices P of G; PV*V, E * E. The construction of the Minimal Steiner Tree is reduced to the problem of constructing and selecting (n-1) s-routes connecting the n major vertices. The problem is solved in two stages. At the first stage, a set of alternative variants of s-routes is formed, of course, of greater dimension than (n-1) covering the Minimal Steiner Tree. In the second step, the (n-1) s-routes covering the Minimal Steiner Tree are selected from the generated set. The paper discusses combinatorial and parallel sequential approaches to the construction of the Minimal Steiner Tree: each of the s-routes is constructed sequentially, but all (n-1) s-routes covering the Minimal Steiner Tree are built in parallel. Both approaches are based on the method of the ant colony. In the general case, the search for a solution to the problem of constructing a Minimal Steiner Tree is performed by a population of clusters of agents A={Aσ|σ = 1, 2, ..., nσ}. At each iteration, the agents of each cluster Aσ = {aσk | k = 1, 2, ..., n-1} construct their concrete Steiner tree. In other words, the number of decisions generated by agents at each iteration is equal to the number of agent clusters. The method of double indirect communication is used for the first time. The method is based on the composite structure of pheromone and its differentiated deposition method. Each agent in cluster Aσ marks the path (route edges) with two kinds of pheromone: pheromone-1 and pheromone-2. The testing was performed on benchmarks. Compared with the existing algorithms, the improvement of results is achieved by 6–7 %. The probability of obtaining the global optimum was 0.94. The time complexity of the algorithm for fixed values M and T of the iterations number and population size lies in the range O(n).

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Keywords Steiner tree; new technologies; principles; natural mechanisms; decision making; adaptive system; clusters of agents; ant colony; algorithm; optimization.
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