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FlashSort Algorithm
www.neubert.net - Dr. Neubert's Website
The Entropy Reduction Laboratory
The FlashSort Algorithm
The Problem
of in-situ sorting with
minimal auxiliary space
in minimal time.
Introduction
In "Mathematical Analysis of Algorithms",
(Information Processing '71, North Holland Publ.'72)
Donald Knuth remarked "... that research on computional
complexity is an interesting way to sharpen our tools for
more routine problems we face from day to day."
With respect to the sorting problem, Knuth points out,
that time effective in-situ permutation
is inherently connected with the problem of finding the cycle
leaders, and in-situ permutations could easily
be performed in O(n) time if we would be allowed
to manipulate n extra "tag" bits specifying
how much of the permutation has been carried out
at any time. Without such tag bits, he concludes
"it seems reasonable to conjecture that every algorithm
will require for in-situ permutation at least n log n
steps on the average".
Now this conjecture is shown not to be valid. A new
efficient way to find cycle leaders is presented and
in-situ permutations can be performed in optimal time.
The algorithm FlashSort sorts in O(n)
time without the manipulation of n extra "tag" bits.
Here an auxiliary vector of only length m is required,
where m is a small fraction of the number of elements n.
Classification
Accumulation
run the loops:
- find cycle leader
- in situ permutation
short range sorting
The FlashSort Algorithm
FlashSort sorts n elements in O(n) time. Flash-Sort
uses a vector L(k) of length m in a first step for
the classification of the elements of array A. Then,
in a second step, the resulting counts are accumulated
and the L(k) point to the class boundaries. Then the
elements are sorted by in situ permutation. During the
permutation, the L(k) are decremented by a unit step at each
new placement of an element of class k in the array A.
A crucial aspect of FlashSort is that for identifying new cycle
leaders. A cycle ends, if the the vector L(k) points to the
position of an element below the classboundary of class k.
The new cycle leader is the element situated in the lowest position
complying to the complimentary condition, i.e. for which L(k)
points to a position with L(k(A(i))) >= i.
Evidently, in addition to the array A of length n which
holds the n elements to be sorted, the only auxiliary vector
is the L(k)-vector. The size of this vector is equal to the
number m of classes which is small compared to n,
e.g. m typically may be set to m=0.1 n in case of floating point
numbers.
Finally,a small number of partially distinguishable elements are
sorted locally within their classes either by recursion or by a simple
conventional sort algorithm.
In these papers you find
a more detailed description of the algorithm.
Karl-Dietrich Neubert
The FlashSort Algorithm
This is an MS 6.0 Word document as
published in the Proceedings of the
euroFORTH'97
-Conference, Oxford, England,Sept.26-28 1997
Karl-Dietrich Neubert,
FlashSort1 Algorithm in
Dr. Dobb's Journal
Feb.1998,p.123
CLICK HERE
TO SUBSCRIBE TO DDJ
A collection of FlashSort demos
For animations of the FlashSort principle and runtime, select
Collection of FlashSort demos
A collection of FlashSort codes
For codes in various languages and various modifications
of the FlashSort algorithm see
Collection of FlashSort codes.
Back
To go back to the Welcome page click
here.
This page and each part of it Copyright © 1998
Karl-Dietrich Neubert.
All Rights Reserved
Design by
Vladimir Marek.
Last update of the page: March 18, 2003
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Sorting | by | in | place | permutation | with | time | complexity | O(n) | using | an | auxiliary | vector. | | Papers, | animations | and | implementations. |
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http://www.neubert.net/FSOIntro.html
FlashSort 2008 August
dvd rental
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Sorting by in place permutation with time complexity O(n) using an auxiliary vector. Papers, animations and implementations.
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