Building communication networks: on the application of the Kruskal’s algorithm in the problems of large dimensions

Abstract

The paper deals with the development of the topology of ultra-large communication networks, i.e. networks containing several thousand vertices. In this case, the coordinates of the vertices of the undirected graph are somehow predetermined and a set of edges must be constructed. The main point of the options we are considering for developing the network topology is the minimum of the sum of weights of the edges; however, we note in advance that this criterion of minimality is often not the only objective function in the practical problems we are considering. In our previous papers, two realistically considered tasks were formulated. However, everything is not so simple, and we cannot use the direct version of Kruskal’s algorithm. The complexity of this algorithm depends on the representation of the data, i.e. the data structures used. In our situation (when the number of considered vertices is approximately 5000 to 10000), the operation of a simple version of the algorithm takes about a half an hour, which, of course, is acceptable for a one-time solution to the problem under consideration, but it is unacceptable in the case when such solutions are constructed repeatedly (in particular, iteratively). Some temporary improvements to the practical operation of the algorithm provide different options for using complex data structures. The subject of this article can be formulated as follows. We are moving from exact algorithms (in particular, Kruskal’s algorithm) to some heuristics. Moreover, for the starting problem that we are considering, we cannot work without heuristic algorithm at all. However, we describe two specific variants of a simple implementation of Kruskal’s algorithm for problems of large dimensions; in our titles, those are “the 1st and the 2nd algorithms of the usual implementation”. We have formulated two heuristics (“the 3rd and 4th algorithms”). In our opinion, one of these algorithms turned out to be quite acceptable; we present some practical results of computational experiments. And it is very important that these two heuristics will be useful not only for such a “0-th problem”, but also for much more complex problems.