Abstract
We show that each of the following problems can be solved
fast and with optimal speedup with high probability on a
randomized CRCW PRAM using
$O(n)$ space:
Space allocation: Given $n$ nonnegative integers
$x_1,\ldots,x_n$, allocate $n$ blocks of consecutive
memory cells of sizes $x_1,\ldots,x_n$ from a base
segment of $O(\sum_{i=1}^n x_i)$ consecutive
memory cells;
Estimation: Given $n$ integers %$x_1,\ldots,x_n$
in the range $1\Ttwodots n$, compute ``good'' estimates
of the number of occurrences of each value
in the range $1\Ttwodots n$;
Integer chain-sorting: Given $n$ integers $x_1,\ldots,x_n$
in the range $1\Ttwodots n$, construct a linked list
containing the integers $1,\ldots,n$ such that for all
$i,j\in\{1,\ldots,n\}$, if $i$ precedes $j$ in the
list, then $x_i\le x_j$.
\noindent
The running times achieved are $O(\Tlogstar n)$ for
problem (1) and $O((\Tlogstar n)^2)$ for
problems (2) and~(3).
Moreover, given slightly superlinear processor
and space bounds, these problems or variations
of them can be solved in
constant expected time.
