A lower bound for linear approximate compaction

The {\em $\lambda$-approximate compaction} problem is: given an input array of $n$ values, each either 0 or 1, place each value in an output array so that all the 1's 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 {\em Linear Approximate Compaction} (LAC). On the CRCW PRAM model, %there is an algorithm that solves approximate compaction in \order{(\log\log n)^3} time for $\lambda = \frac{1}{\log\log n}$, using $\frac{n}{(\log\log n)^3}$ processors. Our main result shows that this is close to the best possible. Specifically, we prove that LAC requires %$\Omega(\log\log n)$ time using \order{n} processors. We also give a tradeoff between $\lambda$ and the processing time. For $\epsilon < 1$, and $\lambda = n^{\epsilon}$, the time required is $\Omega(\log \frac{1}{\epsilon})$

22. Workshop Komplexitätstheorie und effiziente Algorithmen

his publication contains abstracts of the 22nd workshop on complexity theory and efficient algorithms. The workshop was held on February 8, 1994, at the Max-Planck-Institut für Informatik, Saarbrücken, Germany

Approximate and Exact Deterministic Parallel Selection

The selection problem of size n is, given a set of n elements drawn from an ordered universe and an integer k with 1 k n, to identify the kth smallest element in the set. We study approximate and exact selection on deterministic concurrent-read concurrent-write parallel RAMs, where approximate selection with relative accuracy ? 0 asks for any element whose true rank differs from k by at most n. Our main results are: (1) Exact selection problems of size n can be solved in O(logn=log log n) time with O(n log log n=logn) processors. This running time is the best possible (using only a polynomial number of processors) , and the number of processors is optimal for the given running time (optimal speedup); the best previous algorithm achieves optimal speedup with a running time of O(logn log n=log log n). (2) For all t (log log n) 4 log n, approximate selection problems of size n can be solved in O(t) time with optimal speedup with relative accuracy 2 \Gammat loglog log n=(log logn) ..