On the Message Complexity of Global Computations

Abstract

It is well known that, for most non-trivial problems (such as Election, Spanning-Tree Construction, Traversal, Broadcast, etc.), any generic solution requires at least Ω(m) messages in the worst case, where m is the number of links among the n entities. However, all the existing proofs of this fact assume that the network size (i.e., the parameters n and m) are not known to the protocol.

A natural question arises whether this rather strong assumption, which is crucial for the proofs, is truly necessary for establishing a lower bound to these problems.

In this paper we answer this question and prove that the Ω(m) bound is inherent for all these problems, as well as many more. In fact, we consider the class of global problems, that is those whose solution requires the involvement of every entity in the communication (sending or receiving messages). The relationship between n and m plays an important role in establishing the lower bound. We show that for most networks (where $m \le \frac{1}{2}(n-2)(n-3)+1$) a generic solution for any problem in this class requires at least m messages even if n, m, and the degree of each node are known. This result holds for almost all values of m (e.g., when $\frac{1}{2}(n-2)(n-3)+ 1 < m \le \frac{1}{2}(n-1)(n-2)+1$ the number of required messages is m − 1), even if there is a single initiator and the entities have distinct identifiers, and both these facts are known. Moreover, the results hold even if the protocol can maintain a global view of the network.

As the networks become more dense, namely the network approaches a complete graph, the number of required messages is gradually reduced. For extreme values of m ( i.e., $m = \frac{1}{2}n(n-1) - c > \frac{1}{2}(n-1)(n-2)+1$), where c ≥ 0 is constant, the lower bound gradually approaches Ω(n); this is understandable since we establish it for the single initiator scenario. However, we prove that in networks of such a size, single initiators problems such as Broadcast and Traversal can be solved with precisely that order of magnitude. This means that for those problems the knowledge of n and m generates a significant and sudden complexity drop from Θ(n ²) to Θ(n).