# Computing boolean functions on anonymous networks

## Abstract

We study the bit-complexity of computing boolean functions on anonymous networks. Let *N* be the number of nodes, δ the diameter and *d* the maximal node degree of the network. For arbitrary, unlabeled networks we give a general algorithm of polynomial bit complexity *O*(*N*^{4} · δ · *d*^{2} · log*N*) for computing any boolean function which is computable on the network. This improves upon the previous best known algorithm which was of exponential bit complexity \(O(d^{N^2 } )\). For symmetric functions on arbitrary networks we give an algorithm with bit complexity *O*(*N*^{2} · δ · *d*^{2} · log^{2}*N*). This same algorithm is shown to have even lower bit complexity for a number of specific networks. We also consider the class of distance regular unlabeled networks and show that on such networks symmetric functions can be computed efficiently in *O*(*N* · δ · *d* · log*N*) bits.

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