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#### Sorting in linear time?

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##### Fulltext (public)

MPI-I-95-1-024.pdf

(Any fulltext), 23MB

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##### Citation

Andersson, A., Nilsson, S., Hagerup, T., & Raman, R.(1995). *Sorting
in linear time?* (MPI-I-1995-1-024). Saarbrücken: Max-Planck-Institut für Informatik.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-A1DE-D

##### Abstract

We show that a unit-cost RAM with a word
length of $w$ bits can sort $n$ integers
in the range $0\Ttwodots 2^w-1$ in
$O(n\log\log n)$ time, for arbitrary $w\ge\log n$,
a significant improvement over
the bound of $O(n\sqrt{\log n})$ achieved
by the fusion trees of Fredman and Willard.
Provided that $w\ge(\log n)^{2+\epsilon}$
for some fixed $\epsilon>0$, the sorting can even
be accomplished in linear expected time
with a randomized algorithm.
Both of our algorithms parallelize without
loss on a unit-cost PRAM with a word
length of $w$ bits.
The first one yields an algorithm that uses
$O(\log n)$ time and\break
$O(n\log\log n)$ operations on a
deterministic CRCW PRAM.
The second one yields an algorithm that uses
$O(\log n)$ expected time and $O(n)$ expected
operations on a randomized EREW PRAM,
provided that $w\ge(\log n)^{2+\epsilon}$
for some fixed $\epsilon>0$.
Our deterministic and randomized sequential
and parallel algorithms generalize to the
lexicographic sorting problem of sorting
multiple-precision integers represented
in several words.