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キーワード:
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要旨:
The following problems are shown to be solvable
in $O(\log^{\ast }\! n)$ time
with optimal speedup with high probability on a
randomized CRCW PRAM using
$O(n)$ space:
\begin{itemize}
\item
Space allocation: Given $n$ nonnegative integers
$x_1,\ldots,x_n$, allocate $n$ nonoverlapping
blocks of consecutive
memory cells of sizes $x_1,\ldots,x_n$ from a base
segment of $O(\sum_{j=1}^n x_j)$ consecutive
memory cells;
\item
Estimation: Given $n$ integers %$x_1,\ldots,x_n$
in the range $ 1.. n $, compute ``good'' estimates
of the number of occurrences of each value
in the range $1.. n$;
\item
Semisorting: Given $n$ integers $x_1,\ldots,x_n$
in the range $1.. n$,
store the integers $1,\ldots,n$ in an array of $O(n)$
cells such that for all $i\in\{1,\ldots,n\}$,
all elements of $\{j:1\le j\le n$ and $x_j=i\}$
occur together, separated only by empty cells;
\item
Integer chain-sorting: Given $n$ integers $x_1,\ldots,x_n$
in the range $1.. n$, 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 $x_i\le x_j$.
\end{itemize}
\noindent
Moreover, given slightly superlinear processor
and space bounds, these problems or variations
of them can be solved in
constant time with high probability.
As a corollary of the integer chain-sorting result,
it follows that $n$ integers in the range $1.. n$
can be sorted in $O({{\log n}/{\log\log n}})$ time
with optimal speedup with high probability.
