# MPI-I-93-123

## Fast parallel space allocation, estimation and integer sorting (revised)

### Bast, Holger and Hagerup, Torben

**MPI-I-93-123**. June** **1993, 85 pages. | Status:** **available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:

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.

Acknowledgement:** **

References to related material:

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**URL to this document: **http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1993-123

**BibTeX**
`@TECHREPORT{``BastHagerup93``,`

` AUTHOR = {Bast, Holger and Hagerup, Torben},`

` TITLE = {Fast parallel space allocation, estimation and integer sorting (revised)},`

` TYPE = {Research Report},`

` INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},`

` ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},`

` NUMBER = {MPI-I-93-123},`

` MONTH = {June},`

` YEAR = {1993},`

` ISSN = {0946-011X},`

`}`