Configuring a Distributed Network on the Shortest Paths

Abstract

We consider a distributed system based on a strongly connected directed numbered rooted graph without multiple edges and loops in which the arcs outgoing from the vertex are numbered from 1 to the out-degree of this vertex. The task is to ensure the message passing from each sender to each receiver by the shortest paths. For this it is proposed to construct at each vertex a of the graph a mapping that assigns to each vertex number b the arc number with which the shortest path starts from a to b. This mapping is useful for solving many problems, in particular, the task of self-configuration a program-configurable network. The network switch stores in program-configurable tables the mapping of the address of the target host to the number of the output port. The time for passing packets is minimal, if the shortest paths are used in the mapping. Another problem is the simulation of distributed algorithms developed for distributed system based on an undirected connection graph on a class of systems based on a directed connection graph. Here we have to solve the backtracking problem: instead of passing a message from the end of the arc a→b to its beginning, the message is passing along the directed path b→ ... →a. It is assumed that the mapping construction algorithm should start and end at the root of the graph. Two algorithms for constructing a mapping are proposed: a "fast" algorithm with estimates T = O(n) and N = O(n) and an "economical" algorithm with estimates T = O(n 2) and N = O(1), where T is the algorithm running time, N is the number of messages simultaneously transmitted along the arc, n is the number of vertices of the graph. It is proved that these estimates are lower bounds on the class of algorithms that correctly operate on the class of all graphs under consideration.