On the Network Complexity of Selection

The selection problem is to determine the kth largest out of a given set of n keys, and its sequential complexity is well known to be linear. Thus, given a p processor parallel machine, it is natural to ask whether or not an O(n=p) selection algorithm can be devised for that machine. The main result of this paper is an ((n=p) log log p + log p) lower bound for selection on any network that satisses a particular low expansion property. The class of networks satisfying this property includes all of the common network families such as the tree, multi-dimensional mesh, hypercube, butterry and shuue exchange. When n=p is suuciently large (for example, greater than log 2 p on the butterry, hypercube and shuue exchange), this result is matched by the upper bound presented in 4].