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キーワード:
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要旨:
The selection problem of size $n$ is,
given a set of $n$ elements drawn from an ordered
universe and an integer $r$ with $1\le r\le n$, to
identify the $r$th 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 $\lambda>0$ asks for any element
whose true rank differs from $r$ by at most $\lambda n$.
Our main results are:
(1) For all $t\ge(\log\log n)^4$, approximate
selection problems of size $n$ can be solved in
$O(t)$ time with optimal speedup with relative accuracy
$2^{-{t/{(\log\log n)^4}}}$;
no deterministic PRAM algorithm for approximate
selection with a running time below
$\Theta({{\log n}/{\log\log n}})$
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 $O({{\log n\log^*\! n}/{\log\log n}})$.
