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Steven Deller <[log in to unmask]>
Reply To:
Steven Deller <[log in to unmask]>
Mon, 18 Dec 2000 11:45:38 -0500
text/plain (273 lines)
Both VADS and APEX provide *generic* versions of Quicksort, Heapsort and
Insertion sort in a package named "ordering".  I wrote these about 12 years
ago and have used them successfully in numerous situations.  The spec for
the package discusses tradeoffs, so I have copied it below.

You really need to do some analysis before picking the algorithm.

Are your objects large?
If they are very large then SWAPs will dominate COMPAREs and you need to
pick an algorithm that minimizes SWAPs. It may even pay to build an
"indirect" array, sort that, and then do only the MOVEs required.

What is the *guaranteed* maximum size?
If there is no limit, that excludes certain algorithms.  If there is a fixed
limit, then some things can be done to make the algorithm faster, including
converting SWAPs to single MOVEs.

What is the stack or local space size restrictions?
If you have limited stack then before using Quicksort, you need to make sure
you have enough stack for when the algorithm encounters *worst* case inputs.
Quicksort is recursive and can go to a depth of O(N**2) if you are not

What is the *expected* sorting size?
If it is quite small, then you need to minimize OVERHEAD for your
algorithms, possibly at the expense of more COMPAREs or even more SWAPs.

Must you have reproducible times, i.e. times  independent of the initial
If so, then Quicksort is *NOT* appropriate.  Quicksort has an *average*
O(NlnN) performance, but there are input patterns that can produce O(N**2)
and all values in between.  The original Quicksort algorithm was worst case
for pre-SORTED data.

Will the list have *equal* values?
If the compare may have equal values, and you must have a "stable" sort,
then Insertion sort or some other O(N**2) algorithm is required.  A stable
sort keeps the order of equal items.  It is used when the list has already
been sorted by some other process or algorithm and it is important that the
original order not be disturbed by the new sort.

If you are sure the lists will never be more than 20 long, you need to
seriously look at the OVERHEAD for each algorithm.  I'd suggest NOT using
Quicksort, since it only averages a speed of O(NlnN), and with only a few
entries, it is quite possible to have an input pattern that produces

I've had good luck with Heapsort, which is guaranteed to be O(NlnN), but
that does not minimize MOVEs so you might want to try an "indirect" Heapsort
if your objects are of any significant size.

An "indirect" sort creates an array with the indices to be sorted.  You then
sort the indices using them as indices into the original array.  Finally you
walk the index array to determine where objects are to be located (you walk
it in a way that only requires one MOVE of the pointed to objects and only
for those that are "mislocated" -- that in itself is an interesting little

[log in to unmask]

-- UNIT: package spec ORDERING
-- FILES: ordering_s.a in verdixlib
--        related file is ordering_b.a in verdixlib
-- PURPOSE: provide ordering functions, including sorting and permuting
--              and PERMUTE.
--              Usage: these packages are instantiated with the Element type
--                     of elements to be sorted, the Index discrete type
--                     indexes them, and the Boolean function on the
--                     that orders them.
-- package ORDERING
-- Provides several sorting algorithms and a permuting algorithm, which are
-- generic with respect to
--   1. Element type in a List to be sorted, an arbitrary type.
--   2. Index type for indexing into the List, a discrete type.
--      A type List is defined from the Element type and the Index type.
--   3. A relation (Boolean result function) between Elements of the List.
--      For an ascending sort, the relation should be True whenever the
--      first Element is strictly less than the second Element.  For a
--      descending sort, the relation should be True whenever the first
--      Element is strictly greater than the second Element.
-- Only "in-place" algorithms are provided, i.e. algorithms whose space
-- requirement is N or ( N + log_2(N)*e ), where e is some small value.  The
-- log_2(N)*e additional space is taken from the stack.
-- The following sorting algorithms are provided, and an indication is given
-- about -- when the algorithm should be chosen:
-- 1. Quicksort (median-of-three): least average sort time of all, but
--    there are data organizations for which sorting time is O(N**2).  the
--    median-of-three approach is used for the split partition value, so
--    sorted data is NOT a worst case data organization (unlike the basic
--    quicksort algorithm).
-- 2. Heapsort: guaranteed O(NlnN) time.  Note that the "heap" in this
--    algorithm applies to the type of data structure applied to the
--    list to be sorted; there is NO allocation of space from the heap.
-- 3. Insertion sort: one of the simplest sorts.  It is O(N**2) in time
--    complexity, but has the one advantage that it is stable, i.e. elements
--    with equal values retain their order in the list.
-- The following sorting algorithms are NOT provided.  An indication of
-- why the algorithms not provided is given.
-- 1. Shellsort (diminishing increments): This algorithm is consistently
--    bettered by Heapsort, and usually bettered by Quicksort.  Either of
--    those should be used instead.
-- 2. Listmerge sort: This algorithm requires a special list that has a
--    link field in each element that can hold an index value.  Its major
--    advantage is that it is a stable sort in O(NlnN) in time, i.e. it
--    makes a reasonable time-space tradeoff to achieve a stable sort.
--    However, its space requirements violate the "general element",
--    "in-place" sort requirements defined for this package.  A reasonable
--    alternative to providing a stable sort in (NlnN) time is to add
--    a single field to each initial element, initialized to the value
--    of the element's position in the array, and make the "<" function
--    include a compare of this field if the original compared fields
--    are equal.  Thus equal elements are forced non-equal, and the order
--    of equal elements in the array is maintained.
-- 3. Bubblesort.  This algorithm takes O(N**2) time, and has a higher
--    constant factor than insertion sort.
-- 4. Selectionsort.  This algorithm takes O(N**2), and is not stable
--    when done in-place with exchanges, unless considerable complications
--    are added to the algorithm.
-- It is helpful to organize the sorting algorithms to aid understanding.
-- The following taxonomy is based on the article:
--   An Inverted Taxonomy of Sorting Algorithms, Susan M. Merritt,
--   Communications of the ACM, Vol. 28, No. 1, January 1985, pp. 96-99.
-- All sorting algorithms are based on the abstraction "split into parts,
-- sort the parts, join the sorted parts".  Specific sorting algorithms are
-- conveniently classified as hard-split/easy-join or easy-split/hard-join,
-- based on whether the "work" is done at the start or end of the algorithm.
-- The abstract algorithm is clearly recursive.  Usual programming style
-- converts the recursions to iterations for speed.
-- If we independently specify the splitting criteria then we get the
-- following table:
--   Splitting criteria        hard-split/easy-join
--   ------------------        --------------------      -------------------
--   Equal-sized parts         Quicksort                 Merge Sort
--   One part is one element   Selection Sort            Insertion Sort
--   One element, in place     Bubble Sort               Sinking Sort
-- The Heapsort algorithm is viewed as a Selection Sort that minimizes
-- the comparisons to find the next element by using a data structure
-- (a heap).  The Shell Sort algorithm is viewed as a version of the
-- Insertion Sort that operates independently on various subsets of the
-- input data. An additional class of algorithms* may be viewed as
-- splitting criteria that are varying combinations of Quicksort with
-- Bubble Sort.  (Perhaps the combination of Merge Sort and Sinking Sort
-- would give rise to a similar set of algorithms.)
-- * A Class of Sorting Algorithms Based on Quicksort, Roger L. Wainwright,
--   Communications of the ACM, Vol. 28, No. 4, April 1985, pp. 396-403.
-- package PERMUTE
-- Provides a generic Permute package.  The generic parameters are an
-- of any type, an index of discrete type, and an ordering function.  The
-- package defines a list data type and procedure.   The procedure takes a
-- single argument of list type, and rearranges it to the next lexical
-- permutation.  The exception "no_more_permutations" is raised when the
-- input list is lexically the last permutation.
-- The algorithm is derived from that presented in Dijkstra's "A Discipline
-- Programming".
.......................................................................... -

package ORDERING is

    -- hard-split / easy-join sorting algorithms

        type Element is private;
        type Index is (<>);
        with function "<" ( left, right: Element ) return Boolean is <>;
    package QuickSort is
        type List is array (Index range <>) of Element;
        procedure QuickSort ( L: in out List );
    end QuickSort;
    pragma Share_Body ( QuickSort, False ) ;

        type Element is private;
        type Index is (<>);
        with function "<" ( left, right: Element ) return Boolean is <>;
    package HeapSort is
        type List is array (Index range <>) of Element;
        procedure HeapSort ( L: in out List );
    end HeapSort;
    pragma Share_Body ( HeapSort, False ) ;

    -- easy-split / hard-join sorting algorithms

        type Element is private;
        type Index is (<>);
        with function "<" ( left, right: Element ) return Boolean is <>;
    package InsertionSort is
        type List is array (Index range <>) of Element;
        procedure InsertionSort ( L: in out List );
    end InsertionSort;
    pragma Share_Body ( InsertionSort, False ) ;

    -- permute list to next lexical permutation algorithm

        type Element is private;
        type Index is (<>);
        with function "<=" (left, right: Element) return Boolean is <>;
    package Permute is
        type List is array (Index range <>) of Element;
        procedure Permute ( L: in out List );
        No_more_permutations : exception ;
    end Permute ;
    pragma Share_Body ( Permute , False ) ;


.......................................................................... -
-- This software is copyright 1985 by the VERDIX Corporation.
-- All rights reserved.  No part of the material protected by
-- this copyright notice may be reproduced or utilized in any
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-- for a particular purpose exist.
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-- which this software may be used, no warranty of fitness for
-- a particular purpose is offered.  The user is advised to
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-- must assume the entire risk and liability of using this
-- software.
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-- or inconsequential damages or lost profits.