Approximate and exact deterministic parallel selection

Abstract

The selection problem of size n is, given a set of n elements drawn from an ordered universe and an integer r with 1<r ≤n, to identify the rth 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 λ>0 asks for any element whose true rank differs from r by at most An. Our main results are: (1) For all t≥(log log n)⁴, approximate selection problems of size n can be solved in O(t) time with optimal speedup with relative accuracy $2^{{{ - t} \mathord{\left/{\vphantom {{ - t} {\left( {\log \log n} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {\log \log n} \right)}}^4 }$; no deterministic PRAM algorithm for approximate selection with a running time below Ο(log n/log log n) was previously known. (2) Exact selection problems of size n can be solved in O(log n/log log n) time with O(n log log n/log n) 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(log n log^* n/log log n).

Supported by the ESPRIT Basic Research Actions Program of the EC under contract No. 7141 (project ALCOM II). Authors' email addresses: shiva@mpi-sb.mpg.de, torben@mpisb.mpg.de, raman@umiacs.umd.edu.