On the Runtime and Robustness of Randomized Broadcasting

Abstract

One of the most frequently studied problems in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol. At some time t an information r is placed at one of the nodes of a graph. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it.

In this work, we develop tight bounds on the runtime of the algorithm described above, and analyze its robustness. First, it is shown that on Δ-regular graphs this algorithm requires at least $\log_{2-\frac{1}{\Delta}} N + \log_{ (\frac{\Delta}{\Delta-1})^{\Delta}} N -- o(\log N)$ rounds to inform all N nodes. For general graphs, we prove a slightly weaker lower bound and improve the upper bound of Feige et. al. [8] to (1+o(1)) N ln N which implies that K _{1,N − − 1} is the worst-case graph. Furthermore, we determine the worst-case-ratio between the runtime of a fastest deterministic algorithm and the randomized one.

This paper also contains an investigation of the robustness of this broadcasting algorithm against random node failures. We show that if the informed nodes are allowed to fail in some step with probability 1–p, then the broadcasting time increases by a factor of at most 6/p. Finally, the previous result is applied to state some asymptotically optimal upper bounds for the runtime of randomized broadcasting in Cartesian products of graphs and to determine the performance of agent based broadcasting [6] in graphs with good expansion properties.

This work is partially supported by German Science Foundation (DFG) Research Training Group GK-693 of the Paderborn Institute for Scientific Computation (PaSCo) and by Integrated Project IST-15964 ”Algorithmic Principles for Building Efficient Overlay Computers” (AEOLUS) of the European Union.