A lower bound for linear approximate compaction

The lambda -approximate compaction problem is: given an input array of n values, each either 0 or 1, place each value in the output array so that all the 1s are in the first (1+ lambda )k array locations, where k is the number of 1's in the input. lambda is an accuracy parameter. This problem is of fundamental importance in parallel computation because of its applications to processor allocation and approximate counting. When lambda is a constant, the problem is called linear approximate compaction (LAC). On the CRCW PRAM model, there is an algorithm that solves approximate compaction in O((log log n)/sup 3/) time for lambda =/sup 1///sub loglogn/, using /sup n///sub (loglogn)3/ processors. This is close to the best possible. Specifically, the authors, prove that LAC requires Omega (log log n) time using O(n) processors. They also give a tradeoff between lambda and the processing time. For in <1, and lambda =n/sup in /, the time required is Omega (log/sup 1///sub in /).<>