A lower bound for linear approximate compaction

The λ-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+λ)k array locations, where k is the number of 1´s in the input. λ is an accuracy parameter. This problem is of fundamental importance in parallel computation because of its applications to processor allocation and approximate counting. When λ 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)³) time for λ=¹/_loglogn, using ⁿ/_(loglogn)3 processors. This is close to the best possible. Specifically, the authors, prove that LAC requires Ω(log log n) time using 𝒪(n) processors. They also give a tradeoff between λ and the processing time. For ∈<1, and λ=n^∈, the time required is Ω(log¹/_∈)