Dynamic faults have small effect on broadcasting in hypercubes

Abstract

We consider a broadcasting problem in the n-dimensional hypercube in the shouting communication mode, i.e. any node of a network can inform all its neighbours in one time step. In addition, during any time step a number of links of the network can be faulty. Moreover, the faults are dynamic. Given a number m⩽ n−1, the problem is to determine the minimum broadcasting time if at most m faults are allowed in any step. The case m= n−1 was studied in Chlebus et al. (Networks 27 (1996) 309), De Marco and Vaccaro (Inform. Process. Lett. 66 (1998) 321), Fraigniaud and Lazard (Inform. Process. Lett. 39 (1991) 115) and completely solved in Dobrev and Vrťo (Inform. Process. Lett. 71 (1999) 81). A related problem, what is the maximal m s.t. the minimum broadcasting time remains n was proposed in De Marco and Vaccaro (Inform. Process. Lett. 66 (1998) 321). We prove that for m⩽ n−3 the minimum broadcasting time is n. If m= n−2 the broadcasting time is always at most n+1, for n>3, and the upper bound is the best possible. Our method is related to the isoperimetric problem in graphs and can be applied to other networks.