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, anonymous networks we give a general algorithm of polynomial bit complexity O ( N 3 · 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 3 · δ · d 2 · log 2 N ). This same algorithm is shown to have even lower bit complexity for a number of specific networks, for example tori, hypercubes, and random regular graphs. 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.