devn00b
/
EQ2EMu


			
							123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532
							<!DOCTYPE html public "-//w3c//dtd html 4.0 transitional//en">
<HTML>
<HEAD>
   <LINK REL="stylesheet" TYPE="text/css" HREF="../../../boost.css">
   <META http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
   <META name="GENERATOR" content="Mozilla/4.77 [en] (X11; U; Linux 2.2.19 i686) [Netscape]">
   <META name="Author" content="Herve Bronnimann">
   <META name="Description" content="Small library to propose minmax_element algorithm.">
   <title>Boost Minmax library</title>
</HEAD>
<body text="#000000" bgcolor="#FFFFFF" link="#0000EE" vlink="#551A8B" alink="#FF0000">

<h2><img src="../../../boost.png" WIDTH="276" HEIGHT="86">Header &lt;<A
HREF="../../../boost/algorithm/minmax.hpp">boost/algorithm/minmax.hpp</A>&gt; </H2>

<quote>
<b>
<a href="#minmax_element">Motivation</a><br>
<a href="#synopsis">Synopsis</a><br>
<a href="#description">Function templates description</a><br>
<a href="#definition">Definition</a><br>
<a href="#reqs">Requirements on types</a><br>
<a href="#precond">Preconditions</a><br>
<a href="#postcond">Postconditions</a><br>
<a href="#complexity">Complexity</a><br>
<a href="#example">Example</a><br>
<a href="#notes">Notes</a><br>
<a href="#rationale">Rationale</a><br>
<a href="#perf">Note about performance</a><br>
<a href="#acks">Acknowledgements</a>
</b>
</quote>


<a name="minmax_element">
<h3>
Motivation</h3>

<p>The minmax library is composed of two headers, <a
href="../../../boost/algorithm/minmax.hpp">&lt;boost/algorithm/minmax.hpp></a>
and <a
href="../../../boost/algorithm/minmax_element.hpp">&lt;boost/algorithm/minmax_element.hpp></a>.
(See <a href="#two_headers">rationale</a>.)
The problem solved by this library is that simultaneous min and max
computation requires
only one comparison, but using <tt>std::min</tt> and <tt>std::max</tt>
forces two comparisons. Worse, to compute the minimum and
maximum elements of a range of <tt>n</tt> elements requires only
<tt>3n/2+1</tt> comparisons, instead of the <tt>2n</tt> (in two passes)
forced by <tt>std::min_element</tt> and <tt>std::max_element</tt>.
I always thought it is a waste to have to call two functions to compute the
extent of a range, performing two passes over the input, when one should
be enough. The present library solves both problems.</p>

<p>The first file implements the function templates
<tt>minmax</tt>
as straightforward extensions of the C++
standard. As it returns a pair of <tt>const&amp;</tt>, we must use the <a
href="../../tuple/index.html">Boost.tuple</a> library to construct such
pairs. (Please note: the intent is not to fix the known defaults of
<tt>std::min</tt>
and <tt>std::max</tt>, but to add one more algorithms that combines both; see the
<a href="#no-fix">rationale</a>.)</p>

<p>The second file implements the function templates
<tt>minmax_element</tt>. In a
second part, it also proposes variants that can usually not be computed by
the minmax algorithm, and which are more flexible in case some elements are equal.
Those variants could have been also provided with policy-based design,
but I ruled against that (see <a href="#no-policy">rationale</a>).
</p>

<p>If you are interested about
<a href="doc/minmax_benchs.html">performance</a>,
you will see that <tt>minmax_element</tt> is just slightly less efficient
than a single <tt>min_element</tt> or <tt>max_element</tt>, and thus
twice as efficient as two separate calls to <tt>min_element</tt> and
<tt>max_element</tt>. From a
theoretical standpoint,
all the <tt>minmax_element</tt> functions perform at most
<tt>3n/2+1</tt>
comparisons and exactly n increments of the
<tt>ForwardIterator</tt>.</p>
</a>

<a name="synopsis">
<h3>
Synopsis of <tt>&lt;boost/algorithm/minmax.hpp></tt></h3>

<pre>#include &lt;boost/tuple/tuple.hpp>

namespace boost {

  template &lt;class T>
  tuple&lt;T const&amp;, T const&amp;>
  minmax(const T&amp; a, const T&amp; b);

  template &lt;class T, class <a href="https://www.boost.org/sgi/stl/BinaryPredicate.html">BinaryPredicate</a>>
  tuple&lt;T const&amp;, T const&amp;>
  minmax(const T&amp; a, const T&amp; b, BinaryPredicate comp);

}
</pre>

<h3>
Synopsis of <tt>&lt;boost/algorithm/minmax_element.hpp></tt></h3>

<pre>#include &lt;utility> // for std::pair

namespace boost {

  template &lt;class <a href="https://www.boost.org/sgi/stl/ForwardIterator.html">ForwardIterator</a>>
  std::pair&lt;ForwardIterator,ForwardIterator>
  minmax_element(ForwardIterator first, ForwardIterator last);

  template &lt;class <a href="https://www.boost.org/sgi/stl/ForwardIterator.html">ForwardIterator</a>, class <a href="https://www.boost.org/sgi/stl/BinaryPredicate.html">BinaryPredicate</a>>
  std::pair&lt;ForwardIterator,ForwardIterator>
  minmax_element(ForwardIterator first, ForwardIterator last,
                 BinaryPredicate comp);

}
</pre>

In addition, there are a bunch of extensions which specify
which element(s) you want to pick in case of equal elements. They are:
<ul>
<li><tt>first_min_element</tt> and <tt>last_min_element</tt></li>
<li><tt>first_max_element</tt> and <tt>last_max_element</tt></li>
<li><tt>first_min_first_max_element</tt>,
<tt>first_min_last_max_element</tt>,
<tt>last_min_first_max_element</tt>, and
<tt>last_min_last_max_element</tt></li>
</ul>
I won't bore you with the complete synopsis, they have exactly the same
declaration as their corresponding <tt>_element</tt> function. Still,
you can find the complete synopsis <a href="doc/minmax_synopsis.html">here</a>.
</a>

<a name="description">
<h3>
Function templates description</h3>
The <tt>minmax</tt> algorithm returns a pair <tt>p</tt> containing either
<i>(a,b)</i>
or <i>(b,a)</i>, such that <tt>p.first&lt;p.second</tt> in the first version,
or <tt>comp(p.first,p.second)</tt> in the second version. If the elements
are equivalent, the pair <i>(a,b) </i>is returned. <a href="#Note1">[1]</a>
<p>The <tt>minmax_element </tt>is semantically equivalent to <tt>first_min_first_max_element</tt>.
<p><tt>First_min_element</tt> and <tt>first_max_element</tt> find the smallest
and largest elements in the range <tt>[first, last)</tt>. If there are
several instance of these elements, the first one is returned. They are
identical to
<tt>std::min_element</tt> and <tt>std::max_element</tt>and
are only included in this library for symmetry.
<p><tt>Last_min_element</tt> and <tt>last_max_element</tt> find the smallest
and largest elements in the range <tt>[first, last)</tt>. They are almost
identical to
<tt>std::min_element</tt> and <tt>std::max_element</tt>, except
that they return the last instance of the largest element (and not the
first, as <tt>first_min_element</tt> and <tt>last_max_element</tt> would).
<p>The family of algorithms comprising <tt>first_min_first_max_element</tt>,
<tt>first_min_last_max_element</tt>,
<tt>last_min_first_max_element</tt>,
and <tt>last_min_last_max_element</tt> can be described generically as
follows (using <i><tt>which</tt></i> and
<i><tt>what</tt></i> for <tt>first</tt>
or <tt>last</tt>): <tt><i>which</i>_min_<i>what</i>_max_element</tt> finds
the (first or last, according to <i><tt>which</tt></i>) smallest element
and the (first or last, according to <i><tt>what</tt></i>) largest element
in the range
<tt>[first, last)</tt>. The first version is semantically
equivalent to:
<pre><tt>  std::make_pair(boost::<i>which</i>_min_element(first,last),
                 boost::<i>what</i>_max_element(first,last))</tt>,</pre>
and the second version to:
<pre><tt>  std::make_pair(boost::<i>which</i>_min_element(first,last,comp),
                 boost::<i>what</i>_max_element(first,last,comp))</tt>.</pre>

<p><br><b><i>Note</i></b>: the <tt>first_min_last_max_element</tt> can also be described
as finding the first and last elements in the range if it were stably sorted.
</a>

<a name="definition">
<h3>
Definition</h3>
Defined in <a href="../../../boost/algorithm/minmax.hpp">minmax.hpp</a>
and
in <a href="../../../boost/algorithm/minmax_element.hpp">minmax_element.hpp</a>.
</a>

<a name="reqs">
<h3>
Requirements on types</h3>
For minmax, <tt>T</tt> must be a model of <a href="https://www.boost.org/sgi/stl/LessThanComparable.html">LessThan
Comparable</a>.
<p>For all the other function templates, versions with two template parameters:
<ul>
<li>
<tt>ForwardIterator</tt> is a model of <a href="https://www.boost.org/sgi/stl/ForwardIterator.html">Forward
Iterator</a>.</li>

<li>
<tt>ForwardIterator</tt>'s value type is <a href="https://www.boost.org/sgi/stl/LessThanComparable.html">LessThan
Comparable</a>.</li>
</ul>
For the versions with three template parameters:
<ul>
<li>
<tt>ForwardIterator</tt> is a model of <a href="https://www.boost.org/sgi/stl/ForwardIterator.html">Forward
Iterator</a>.</li>

<li>
<tt>BinaryPredicate</tt> is a model of <a href="https://www.boost.org/sgi/stl/BinaryPredicate.html">Binary
Predicate</a>.</li>

<li>
<tt>ForwardIterator</tt>'s value type is convertible to <tt>BinaryPredicate</tt>'s
first argument type and second argument type.</li>
</ul>
</a>

<a name="precond">
<h3>
Preconditions</h3>

<ul>
<li>
<tt>[first, last)</tt> is a valid range.</li>
</ul>
</a>

<a name="postcond">
<h3>
Postconditions</h3>
In addition to the semantic description above. for <tt>minmax_element</tt>
and all the <tt><i>which</i>_min_<i>what</i>_max_element</tt>
variants, the return value is
<tt>last</tt> or <tt>std::make_pair(last,last)</tt>
if and only if <tt>[first, last)</tt> is an empty range. Otherwise, the
return value or both members of the resulting pair are iterators in the
range
<tt>[first, last)</tt>.
</a>

<a name="complexity">
<h3>
Complexity</h3>
Minmax performs a single comparison and is otherwise of constant complexity.
The use of <tt>boost::tuple&lt;T const&amp;></tt> prevents copy
constructors in case the arguments are passed by reference.
<p>The complexity of all the other algorithms is linear. They all perform
exactly n increment operations, and zero comparisons if <tt>[first,last)</tt>
is empty, otherwise :
<ul>
<li>
all the <tt>min_element</tt> and <tt>max_element</tt> variants perform
exactly<tt>(n-1)</tt> comparisons,</li>

<li>
<tt>minmax_element</tt> , <tt>first_min_first_max_element</tt>, and <tt>last_min_last_max_element</tt>
perform at most <tt>3(n/2)-1</tt> comparisons if <tt>n</tt> is even and
non-zero, and at most <tt>3(n/2)+1</tt> if <tt>n</tt> is odd,
<a href="#Note2">[2]</a></li>

<li>
<tt>first_min_last_max_element</tt>, and <tt>last_min_first_max_element</tt>
perform exactly <tt>3n/2-2</tt> comparisons if n is even and non-zero,
and at most <tt>3(n/2)</tt> if <tt>n</tt> is odd,
<a href="#Note1">[3]</a></li>
</ul>
where <tt>n</tt> is the number of elements in <tt>[first,last)</tt>.
</a>

<a name="example">
<h3>
Example</h3>
This example is included in the distribution in the examples section of
the library under
<a href="example/minmax_ex.cpp">minmax_ex.cpp</a>.

<pre>int main()
{
  using namespace std;
  boost::tuple&lt;int const&amp;, int const&amp;> result1 = boost::minmax(1, 0);

  assert( result1.get<0>() == 0 );
  assert( result1.get<1>() == 1 );

  <a href="https://www.boost.org/sgi/stl/List.html">list</a>&lt;int> L;
  <a href="https://www.boost.org/sgi/stl/generate_n.html">generate_n</a>(<a href="https://www.boost.org/sgi/stl/front_insert_iterator.html">front_inserter</a>(L), 1000, rand);

  typedef list&lt;int>::const_iterator iterator;
  pair&lt; iterator, iterator > result2 = boost::minmax_element(L.begin(), L.end());
  cout &lt;&lt; "The smallest element is " &lt;&lt; *(result2.first) &lt;&lt; endl;
  cout &lt;&lt; "The largest element is  " &lt;&lt; *(result2.second) &lt;&lt; endl;

  assert( result2.first  == std::min_element(L.begin(), L.end());
  assert( result2.second == std::max_element(L.begin(), L.end());
}</pre>
</a>

<a name="notes">
<h3>
Notes</h3>
<a NAME="Note1"></a><a href="#Note1">[1]</a> We do not support
idioms such as <tt><a href="../../tuple/doc/tuple_users_guide.html#tiers">tie</a>(a,b)=minmax(a,b)</tt>
to order two elements <tt>a</tt>, <tt>b</tt>, although this would have
the desired effect if we returned a reference instead of a constant
reference. The reason is that two unnecessary assignments are
performed if a and b are in order. It is better to stick to <tt>if (b&lt;a)
swap(a,b)</tt> to achieve that effect.
<p><a NAME="Note2"></a><a href="#Note2">[2]</a> These algorithms always
perform at least <tt>3n/2-2</tt> comparisons, which is a lower bound on
the number of comparisons in any case (Cormen, Leiserson, Rivest: "Introduction
to Algorithms", section 9.1, Exercise 9.1-). The algorithms essentially compare
the elements in pairs, performing 1 comparison for the first two elements,
then 3 comparisons for each remaining pair of elements (one to order the
elements and one for updating each the minimum and and the maximum). When
the number of elements is odd, the last one needs to be compared to the
current minimum and also to the current maximum. In addition, for <tt>minmax</tt>,
in cases where equality of the two members in the pair could occur, and
the update stores the second, we save the first to check at the end if
the update should have stored the first (in case of equality). It's hard
to predict if the last comparison is performed or not, hence the at most
in both cases.
<p><a NAME="Note3"></a><a href="#Note3">[3]</a> These algorithms always
perform at least <tt>3n/2-2</tt> comparisons, which is a lower bound on
the number of comparisons in any case. The method is the same as in note
<a href="#Note2">[2]</a>
above, and like above, when the number of elements is odd, the last one
needs to be compared to the current minimum and also to the current maximum.
We can avoid the latter comparison if the former is successful, hence the
<i>at
most</i> instead of <i>exactly</i> in the odd case.
</a>

<a name="rationale">
<h3>
<b>Rationale:</b></h3>

<a name="two_headers">
<h4><b>Why not a single header <tt>&lt;boost/algorithm/minmax.hpp></tt>?</b></h4>
<p>This was the design originally proposed and approved in the formal
review. As the need for Boost.tuple became clear (due to the limitations
of <tt>std::pair</tt>), it became also annoying to require another
library for <tt>minmax_element</tt> which does not need it. Hence the
separation into two header files.</p>

<a name="no-fix">
<h4><b>Your minmax suffers from the same problems as std::min and
std::max.</b></h4>
<p>I am aware of the problems with std::min and
std::max, and all the debate that has been going on (please consult
<a href="#Alexandrescu">Alexandrescu's paper</a> and the links therein). But I don't see the purpose of this
library as fixing something that is part of the C++ standard. I humbly
think it's beyond the scope of this library. Rather, I am
following the way of the standard in simply providing one more function
of the same family. If someone else wants to fix std::min, their fix
would probably apply to boost::minmax as well.</p>
</a>

<h4><b>Why no <tt>min/max_element_if</tt>?</b></h4>
<p>In a first version of the library, I proposed <tt>_if</tt> versions of
all the algorithms (well, not all, because that would be too much).
However, there is simply no reason to do so, and all the versions I had
were just as fast implemented using the excellent
<tt>&lt;boost/iterator_adaptors.hpp></tt> library. Namely, a call to
<tt>min_element_if(first, last, pred)</tt> would be just as well
implemented by:
<pre>
     // equivalent to min_element_if(first, last, pred)
     min_element(boost::make_filter_iterator(first, last, pred),
                 boost::make_filter_iterator(last, last, pred));
</pre>
Arguably, the <tt>min_element_if</tt> version is somewhat shorter, but
the overhead of iterator adaptors is not large, and they get rid of a
lot of code (think of all the combinations between first/last and
doubling them with _if variants!).</p>

<h4><b>Discussion: about std::max_element</b></h4>
<p>This rationale is somewhat historical, but explains why there are all
these <tt>first/last_min/max_element</tt> functions.</p>
<p>The C++ standard mandates that <tt>std::min_element</tt> and <tt>std::max_element</tt>
return the first instance of the smallest and largest elements (as opposed
to, say, the last). This arbitrary choice has some consistency: In the
case of v of type vector&lt;int>, for instance, it is true that <tt>std::min_element(v.begin(),v.end(),std::less&lt;int>())
== std::max_element(v.begin(),v.end(),std::greater&lt;int>())</tt>.
<p>There is of course nothing wrong with this: it's simply a matter of
choice. Yet another way to specify min_element and max_element is to define
them as the first and the last elements if the range was stably sorted.
(The <i>stable</i> sort is necessary to disambiguate between iterators
that have the same value.) In that case, min should return the first instance
and max should return the last. Then, both functions are related by
<tt>reverse_iterator(std::first_min_element(v.begin(),v.end(),std::less&lt;int>()))
==
std::last_max_element(v.rbegin(),v.rend(),std::greater&lt;int>())</tt>.
This definition is subtly different from the previous one.</p>
<p>The definition problem surfaces when one tries to design a minmax_element,
using the procedure proposed in (Cormen, Leiserson, Rivest: "Introduction
to Algorithms", section 9.1). It <i>should</i> be possible to derive an
algorithm using only <tt>3n/2</tt> comparisons if <tt>[first,last) </tt>has
<tt>n</tt>
elements, but if one tries to write a function called <tt>first_min_first_max_element()</tt>
which returns both <tt>std::min_element</tt> and <tt>std::max_element</tt>
in a pair, the trivial implementation does not work. The problem, rather
subtly, is about equal elements: I had to think for a while to find a
way to perform only three
comparisons per pair and return the first min and first max elements.
For a long time, it seemed any
attempts at doing so would consume four comparisons per pair in the worst
case. This implementation achieves three.</p>
<p>It is not possible (or even desirable) to change the meaning of
<tt>max_element</tt>,
but it is still beneficial to provide a function called <tt>minmax_element</tt>,
which returns a pair of <tt>min_element</tt> and <tt>max_element</tt>.
Although it is easy enough to call <tt>min_element</tt> and <tt>max_element</tt>,
this performs
<tt>2(n-1)</tt> comparisons, and necessitates <b>two</b>
passes over the input. In contrast,
<tt>minmax_element</tt> will perform
the fewer comparisons and perform a <b>single</b> pass over the input.
The savings can be significant when the iterator type is not a raw pointer,
or even is just a model of the InputIterator concept (although in that
case the interface would have to be
changed, as the return type could not be copied, so one could e.g.
return a value).</p>
<p>In order to benefit from all the variants of the algorithm, I propose
to introduce both <tt>first_min_element</tt> and <tt>last_min_element</tt>,
and their counterparts <tt>first_max_element</tt> and <tt>last_max_element</tt>.
Then I also propose all the variants algorithms: <tt>first_min_last_max_element</tt>
and <tt>last_min_first_max_element</tt>, which perform only at most <tt>3n/2</tt>
comparisons, and only a single pass on the input. In fact, it can be proven
that computing minmax requires at least <tt>3(n/2)-2</tt> comparisons in
any instance of the problem (Cormen, Leiserson, Rivest, 2nd edition, section
9.1). The implementation I give does not perform unnecessary comparisons
(whose result could have been computed by transitivity from previous
comparisons).</p>
<p>It appears that <tt>first_min_last_max_element</tt> may be just a tad
slower than
<tt>first_min_element</tt> alone, still much less than <tt>first_min_element</tt>
and
<tt>last_max_element</tt> called separately.  <a href="#Note2">[2]</a>

<h4><b>Why algorithms and not accumulators?</b></h4>
<p>The minmax algorithms are useful in computing the extent of a range.
In computer graphics, we need a bounding box of a set of objects.
In that case the need for a single pass is even more stringent
as all three directions must be done at once. Food for thoughts: there
is matter for a nice generic programming library with stackable <tt>update_min</tt>
and <tt>update_max</tt> function objects which store a reference to the
<tt>min_result</tt>and
<tt>max_result</tt> variables, in conjunction with the <tt>for_each</tt>
algorithm).</p>
<p>I believe many standard sequential algorithms could be reformulated
with accumulators (and many others, such as in statistics, expectation /
variance / etc.). It seems that there is room for another library, but I
do not see it competing with minmax, rather extending several algorithms
(including minmax) to the accumulator framework. However, I felt it is
beyond the scope of this library to provide such accumulators.</p>

<a NAME="no-policy">
<h4><b>This first/last is a perfect application for a policy-based
design.</b></h4>
<p>True, and I could have gone that way, with the default policy for
<tt>min_element</tt> and <tt>max_element</tt> to pick the first
occurence of the result. This would have thinned the number of
combinations of the minmax_element variants. But it would also have
meant to change the interface of <tt>boost::minmax_element</tt>.
One of the goals of the <tt>minmax_element</tt> algorithm is its
eventual addition to the C++ standard, in connection with
<tt>std::min_element</tt> and <tt>std::max_element</tt>
(and I feel that it would be quite natural
given the shortness of the implementation, and the not quite trivial
detail which is needed to get it right). So changing the interface by
adding policies would have meant unfortunately to depart from the
standard and created an obstacle towards that goal. Besides, the code
remains rather readable and simple without policies. So I am quite happy
to keep it like this.
</p></a>
</a>

<a name="perf">
<a href="doc/minmax_benchs.html"><h3><b>About performance</b></h3></a>
</a>

<a name="acks">
<h3>
Acknowledgements</h3>

<a name="Alexandrescu">
<a href="http://www.drdobbs.com/generic-min-and-max-redivivus/184403774">Generic: Min and Max Redivivus, by Andrei Alexandrescu</a>
Dr. Dobbs, April 2001

<p>My students in CS903 (Polytechnic Univ., <a href="http://photon.poly.edu/~hbr/cs903/">http://photon.poly.edu/~hbr/cs903/</a>)
who had <tt>minmax_element</tt> as an assignment helped clarify the issues,
and also come up with the optimum number of comparisons for <tt>first_min_last_max_element</tt>.
The identification of the issue surrounding <tt>max_element</tt> is solely
my own.
<p>One <tt>minmax_element</tt> implementation, which performs <tt>3(n/2)+O(log
n)</tt> comparisons on the average when the elements are <tt>random_shuffle</tt>d,
was suggested by my student Marc Glisse. The current one, which performs
<tt>3(n/2)+1</tt>
comparisons in the worst case, was suggested by John Iacono.<p>
<p>Finally, Matthew Wilson and Jeremy Siek contributed pre-review
comments, while Gennadiy Rozental, John Maddock, Craig Henderson, Gary
Powell participated in the review of the library, managed by Thomas
Witt. In particular, Gennadiy suggested a factorization of the code;
while I haven't followed it all the way, his suggestions do make the
code more readable and still work with older compilers. 
Late after the review, as I finally scrounged to add the library for a
release, Eric Niebler noted the bad behavior of <tt>std::pair</tt> for
<tt>minmax</tt> and suggested to use Boost.tuple instead. 
All my thanks for the excellent advice and reviews from all.
<h3>
See also</h3>
<tt><a href="https://www.boost.org/sgi/stl/min.html">min</a></tt>, <tt><a href="https://www.boost.org/sgi/stl/max.html">max</a></tt>,
<tt><a href="https://www.boost.org/sgi/stl/min_element.html">min_element</a></tt>,
<tt><a href="https://www.boost.org/sgi/stl/max_element.html">max_element</a></tt>,
<tt><a href="https://www.boost.org/sgi/stl/LessThanComparable.html">LessThan
Comparable</a></tt>,
<tt><a href="https://www.boost.org/sgi/stl/sort.html">sort</a></tt>,
<tt><a href="https://www.boost.org/sgi/stl/nth_element.html">nth_element</a></tt>
.
<hr SIZE="6">
<br>Last modified 2012-12-10
<p><font face="Arial,Helvetica"><font size=-1>&copy; Copyright Herv&eacute;
Br&ouml;nnimann, Polytechnic University, 2002--2004. 
Use, modification, and distribution is subject to the Boost Software
License, Version 1.0. (See accompanying file <a href="../../../LICENSE_1_0.txt">License_1_0.txt</a> or copy at
<a href="http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)
</font></font>
</body>
</html>