# 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*).

## Keywords

distributed algorithms global computations generic protocols asynchronous systems message complexity## Preview

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