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| 7 | <META name="Author" content="Herve Bronnimann"> |
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| 8 | <META name="Description" content="Small library to propose minmax_element algorithm."> |
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| 9 | <title>Boost Minmax library</title> |
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| 10 | </HEAD> |
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| 11 | <body text="#000000" bgcolor="#FFFFFF" link="#0000EE" vlink="#551A8B" alink="#FF0000"> |
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| 12 | |
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| 13 | <h2><img src="../../../boost.png" WIDTH="276" HEIGHT="86">Header <<A |
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| 14 | HREF="../../../boost/algorithm/minmax.hpp">boost/algorithm/minmax.hpp</A>> </H2> |
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| 15 | |
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| 16 | <quote> |
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| 17 | <b> |
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| 18 | <a href="#minmax_element">Motivation</a><br> |
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| 19 | <a href="#synopsis">Synopsis</a><br> |
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| 20 | <a href="#description">Function templates description</a><br> |
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| 21 | <a href="#definition">Definition</a><br> |
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| 22 | <a href="#reqs">Requirements on types</a><br> |
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| 23 | <a href="#precond">Preconditions</a><br> |
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| 24 | <a href="#postcond">Postconditions</a><br> |
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| 25 | <a href="#complexity">Complexity</a><br> |
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| 26 | <a href="#example">Example</a><br> |
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| 27 | <a href="#notes">Notes</a><br> |
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| 28 | <a href="#rationale">Rationale</a><br> |
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| 29 | <a href="#perf">Note about performance</a><br> |
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| 30 | <a href="#acks">Acknowledgements</a> |
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| 31 | </b> |
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| 32 | </quote> |
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| 33 | |
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| 34 | |
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| 35 | <a name="minmax_element"> |
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| 36 | <h3> |
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| 37 | Motivation</h3> |
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| 38 | |
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| 39 | <p>The minmax library is composed of two headers, <a |
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| 40 | href="../../../boost/algorithm/minmax.hpp"><boost/algorithm/minmax.hpp></a> |
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| 41 | and <a |
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| 42 | href="../../../boost/algorithm/minmax_element.hpp"><boost/algorithm/minmax_element.hpp></a>. |
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| 43 | (See <a href="#two_headers">rationale</a>.) |
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| 44 | The problem solved by this library is that simultaneous min and max |
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| 45 | computation requires |
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| 46 | only one comparison, but using <tt>std::min</tt> and <tt>std::max</tt> |
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| 47 | forces two comparisons. Worse, to compute the minimum and |
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| 48 | maximum elements of a range of <tt>n</tt> elements requires only |
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| 49 | <tt>3n/2+1</tt> comparisons, instead of the <tt>2n</tt> (in two passes) |
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| 50 | forced by <tt>std::min_element</tt> and <tt>std::max_element</tt>. |
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| 51 | I always thought it is a waste to have to call two functions to compute the |
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| 52 | extent of a range, performing two passes over the input, when one should |
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| 53 | be enough. The present library solves both problems.</p> |
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| 54 | |
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| 55 | <p>The first file implements the function templates |
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| 56 | <tt>minmax</tt> |
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| 57 | as straightforward extensions of the C++ |
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| 58 | standard. As it returns a pair of <tt>const&</tt>, we must use the <a |
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| 59 | href=:../../../../tuple/index.html>Boost.tuple</a> library to construct such |
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| 60 | pairs. (Please note: the intent is not to fix the known defaults of |
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| 61 | <tt>std::min</tt> |
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| 62 | and <tt>std::max</tt>, but to add one more algorithms that combines both; see the |
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| 63 | <a href="#no-fix">rationale</a>.)</p> |
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| 64 | |
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| 65 | <p>The second file implements the function templates |
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| 66 | <tt>minmax_element</tt>. In a |
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| 67 | second part, it also proposes variants that can usually not be computed by |
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| 68 | the minmax algorithm, and which are more flexible in case some elements are equal. |
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| 69 | Those variants could have been also provided with policy-based design, |
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| 70 | but I ruled against that (see <a href="#no-policy">rationale</a>). |
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| 71 | </p> |
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| 72 | |
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| 73 | <p>If you are interested about |
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| 74 | <a href="doc/minmax_benchs.html">performance</a>, |
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| 75 | you will see that <tt>minmax_element</tt> is just slightly less efficient |
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| 76 | than a single <tt>min_element</tt> or <tt>max_element</tt>, and thus |
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| 77 | twice as efficient as two separate calls to <tt>min_element</tt> and |
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| 78 | <tt>max_element</tt>. From a |
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| 79 | theoretical standpoint, |
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| 80 | all the <tt>minmax_element</tt> functions perform at most |
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| 81 | <tt>3n/2+1</tt> |
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| 82 | comparisons and exactly n increments of the |
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| 83 | <tt>ForwardIterator</tt>.</p> |
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| 84 | </a> |
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| 85 | |
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| 86 | <a name="synopsis"> |
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| 87 | <h3> |
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| 88 | Synopsis of <tt><boost/algorithm/minmax.hpp></tt></h3> |
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| 89 | |
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| 90 | <pre>#include <boost/tuple/tuple.hpp> |
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| 91 | |
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| 92 | namespace boost { |
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| 93 | |
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| 94 | template <class T> |
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| 95 | tuple<T const&, T const&> > |
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| 96 | minmax(const T& a, const T& b); |
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| 97 | |
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| 98 | template <class T, class <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">BinaryPredicate</a>> |
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| 99 | tuple<T const&, T const&> > |
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| 100 | minmax(const T& a, const T& b, BinaryPredicate comp); |
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| 101 | |
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| 102 | } |
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| 103 | </pre> |
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| 104 | |
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| 105 | <h3> |
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| 106 | Synopsis of <tt><boost/algorithm/minmax_element.hpp></tt></h3> |
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| 107 | |
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| 108 | <pre>#include <utility> // for std::pair |
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| 109 | |
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| 110 | namespace boost { |
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| 111 | |
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| 112 | template <class <a href="http://www.sgi.com/tech/stl/ForwardIterator.html">ForwardIterator</a>> |
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| 113 | std::pair<ForwardIterator,ForwardIterator> |
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| 114 | minmax_element(ForwardIterator first, ForwardIterator last); |
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| 115 | |
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| 116 | template <class <a href="http://www.sgi.com/tech/stl/ForwardIterator.html">ForwardIterator</a>, class <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">BinaryPredicate</a>> |
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| 117 | std::pair<ForwardIterator,ForwardIterator> |
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| 118 | minmax_element(ForwardIterator first, ForwardIterator last, |
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| 119 | BinaryPredicate comp); |
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| 120 | |
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| 121 | } |
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| 122 | </pre> |
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| 123 | |
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| 124 | In addition, there are a bunch of extensions which specify |
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| 125 | which element(s) you want to pick in case of equal elements. They are: |
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| 126 | <ul> |
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| 127 | <li><tt>first_min_element</tt> and <tt>last_min_element</tt></li> |
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| 128 | <li><tt>first_max_element</tt> and <tt>last_max_element</tt></li> |
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| 129 | <li><tt>first_min_first_max_element</tt>, |
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| 130 | <tt>first_min_last_max_element</tt>, |
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| 131 | <tt>last_min_first_max_element</tt>, and |
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| 132 | <tt>last_min_last_max_element</tt></li> |
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| 133 | </ul> |
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| 134 | I won't bore you with the complete synopsis, they have exactly the same |
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| 135 | declaration as their corresponding <tt>_element</tt> function. Still, |
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| 136 | you can find the complete synopsis <a href="doc/minmax_synopsis.html">here</a>. |
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| 137 | </a> |
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| 138 | |
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| 139 | <a name="description"> |
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| 140 | <h3> |
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| 141 | Function templates description</h3> |
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| 142 | The <tt>minmax</tt> algorithm returns a pair <tt>p</tt> containing either |
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| 143 | <i>(a,b)</i> |
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| 144 | or <i>(b,a)</i>, such that <tt>p.first<p.second</tt> in the first version, |
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| 145 | or <tt>comp(p.first,p.second)</tt> in the second version. If the elements |
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| 146 | are equivalent, the pair <i>(a,b) </i>is returned. <a href="#Note1">[1]</a> |
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| 147 | <p>The <tt>minmax_element </tt>is semantically equivalent to <tt>first_min_first_max_element</tt>. |
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| 148 | <p><tt>First_min_element</tt> and <tt>first_max_element</tt> find the smallest |
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| 149 | and largest elements in the range <tt>[first, last)</tt>. If there are |
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| 150 | several instance of these elements, the first one is returned. They are |
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| 151 | identical to |
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| 152 | <tt>std::min_element</tt> and <tt>std::max_element</tt>and |
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| 153 | are only included in this library for symmetry. |
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| 154 | <p><tt>Last_min_element</tt> and <tt>last_max_element</tt> find the smallest |
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| 155 | and largest elements in the range <tt>[first, last)</tt>. They are almost |
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| 156 | identical to |
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| 157 | <tt>std::min_element</tt> and <tt>std::max_element</tt>, except |
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| 158 | that they return the last instance of the largest element (and not the |
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| 159 | first, as <tt>first_min_element</tt> and <tt>last_max_element</tt> would). |
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| 160 | <p>The family of algorithms comprising <tt>first_min_first_max_element</tt>, |
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| 161 | <tt>first_min_first_max_element</tt>, |
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| 162 | <tt>first_min_first_max_element</tt>, |
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| 163 | and <tt>first_min_first_max_element</tt> can be described generically as |
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| 164 | follows (using <i><tt>which</tt></i> and |
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| 165 | <i><tt>what</tt></i> for <tt>first</tt> |
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| 166 | or <tt>last</tt>): <tt><i>which</i>_min_<i>what</i>_max_element</tt> finds |
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| 167 | the (first or last, according to <i><tt>which</tt></i>) smallest element |
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| 168 | and the (first or last, according to <i><tt>what</tt></i>) largest element |
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| 169 | in the range |
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| 170 | <tt>[first, last)</tt>. The first version is semantically |
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| 171 | equivalent to: |
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| 172 | <pre><tt> std::make_pair(boost::<i>which</i>_min_element(first,last), |
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| 173 | boost::<i>what</i>_max_element(first,last))</tt>,</pre> |
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| 174 | and the second version to: |
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| 175 | <pre><tt> std::make_pair(boost::<i>which</i>_min_element(first,last,comp), |
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| 176 | boost::<i>what</i>_max_element(first,last,comp))</tt>.</pre> |
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| 177 | |
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| 178 | <p><br><b><i>Note</i></b>: the <tt>first_min_last_max_element</tt> can also be described |
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| 179 | as finding the first and last elements in the range if it were stably sorted. |
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| 180 | </a> |
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| 181 | |
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| 182 | <a name="definition"> |
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| 183 | <h3> |
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| 184 | Definition</h3> |
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| 185 | Defined in <a href="../../../boost/algorithm/minmax.hpp">minmax.hpp</a> |
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| 186 | and |
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| 187 | in <a href="../../../boost/algorithm/minmax_element.hpp">minmax_element.hpp</a>. |
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| 188 | </a> |
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| 189 | |
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| 190 | <a name="reqs"> |
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| 191 | <h3> |
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| 192 | Requirements on types</h3> |
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| 193 | For minmax, <tt>T</tt> must be a model of <a href="http://www.sgi.com/tech/stl/LessThanComparable.html">LessThan |
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| 194 | Comparable</a>. |
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| 195 | <p>For all the other function templates, versions with two template parameters: |
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| 196 | <ul> |
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| 197 | <li> |
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| 198 | <tt>ForwardIterator</tt> is a model of <a href="http://www.sgi.com/tech/stl/ForwardIterator.html">Forward |
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| 199 | Iterator</a>.</li> |
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| 200 | |
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| 201 | <li> |
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| 202 | <tt>ForwardIterator</tt>'s value type is <a href="http://www.sgi.com/tech/stl/LessThanComparable.html">LessThan |
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| 203 | Comparable</a>.</li> |
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| 204 | </ul> |
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| 205 | For the versions with three template parameters: |
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| 206 | <ul> |
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| 207 | <li> |
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| 208 | <tt>ForwardIterator</tt> is a model of <a href="http://www.sgi.com/tech/stl/ForwardIterator.html">Forward |
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| 209 | Iterator</a>.</li> |
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| 210 | |
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| 211 | <li> |
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| 212 | <tt>BinaryPredicate</tt> is a model of <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">Binary |
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| 213 | Predicate</a>.</li> |
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| 214 | |
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| 215 | <li> |
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| 216 | <tt>ForwardIterator</tt>'s value type is convertible to <tt>BinaryPredicate</tt>'s |
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| 217 | first argument type and second argument type.</li> |
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| 218 | </ul> |
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| 219 | </a> |
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| 220 | |
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| 221 | <a name="precond"> |
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| 222 | <h3> |
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| 223 | Preconditions</h3> |
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| 224 | |
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| 225 | <ul> |
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| 226 | <li> |
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| 227 | <tt>[first, last)</tt> is a valid range.</li> |
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| 228 | </ul> |
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| 229 | </a> |
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| 230 | |
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| 231 | <a name="postcond"> |
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| 232 | <h3> |
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| 233 | Postconditions</h3> |
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| 234 | In addition to the semantic description above. for <tt>minmax_element</tt> |
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| 235 | and all the <tt><i>which</i>_min_<i>what</i>_max_element</tt> |
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| 236 | variants, the return value is |
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| 237 | <tt>last</tt> or <tt>std::make_pair(last,last)</tt> |
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| 238 | if and only if <tt>[first, last)</tt> is an empty range. Otherwise, the |
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| 239 | return value or both members of the resulting pair are iterators in the |
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| 240 | range |
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| 241 | <tt>[first, last)</tt>. |
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| 242 | </a> |
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| 243 | |
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| 244 | <a name="complexity"> |
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| 245 | <h3> |
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| 246 | <a NAME="Complexity"></a>Complexity</h3> |
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| 247 | Minmax performs a single comparison and is otherwise of constant complexity. |
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| 248 | The use of <tt>boost::tuple<T const&></tt> prevents copy |
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| 249 | constructors in case the arguments are passed by reference. |
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| 250 | <p>The complexity of all the other algorithms is linear. They all perform |
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| 251 | exactly n increment operations, and zero comparisons if <tt>[first,last)</tt> |
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| 252 | is empty, otherwise : |
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| 253 | <ul> |
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| 254 | <li> |
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| 255 | all the <tt>min_element</tt> and <tt>max_element</tt> variants perform |
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| 256 | exactly<tt>(n-1)</tt> comparisons,</li> |
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| 257 | |
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| 258 | <li> |
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| 259 | <tt>minmax_element</tt> , <tt>first_min_first_max_element</tt>, and <tt>last_min_last_max_element</tt> |
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| 260 | perform at most <tt>3(n/2)-1</tt> comparisons if <tt>n</tt> is even and |
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| 261 | non-zero, and at most <tt>3(n/2)+1</tt> if <tt>n</tt> is odd, |
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| 262 | <a href="#Note2">[2]</a></li> |
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| 263 | |
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| 264 | <li> |
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| 265 | <tt>first_min_last_max_element</tt>, and <tt>last_min_first_max_element</tt> |
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| 266 | perform exactly <tt>3n/2-2</tt> comparisons if n is even and non-zero, |
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| 267 | and at most <tt>3(n/2)</tt> if <tt>n</tt> is odd, |
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| 268 | <a href="#Note1">[3]</a></li> |
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| 269 | </ul> |
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| 270 | where <tt>n</tt> is the number of elements in <tt>[first,last)</tt>. |
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| 271 | </a> |
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| 272 | |
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| 273 | <a name="example"> |
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| 274 | <h3> |
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| 275 | Example</h3> |
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| 276 | This example is included in the distribution in the examples section of |
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| 277 | the library under |
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| 278 | <a href="example/minmax_ex.cpp">minmax_ex.cpp</a>. |
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| 279 | |
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| 280 | <pre>int main() |
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| 281 | { |
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| 282 | using namespace std; |
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| 283 | boost::tuple<int const&, int const&> result1 = boost::minmax(1, 0); |
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| 284 | |
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| 285 | assert( result1.get<0>() == 0 ); |
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| 286 | assert( result1.get<1>() == 1 ); |
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| 287 | |
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| 288 | <a href="http://www.sgi.com/tech/stl/List.html">list</a><int> L; |
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| 289 | <a href="http://www.sgi.com/tech/stl/generate_n.html">generate_n</a>(<a href="http://www.sgi.com/tech/stl/front_insert_iterator.html">front_inserter</a>(L), 1000, rand); |
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| 290 | |
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| 291 | typedef list<int>::const_iterator iterator; |
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| 292 | pair< iterator, iterator > result2 = boost::minmax_element(L.begin(), L.end()); |
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| 293 | cout << "The smallest element is " << *(result2.first) << endl; |
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| 294 | cout << "The largest element is " << *(result2.second) << endl; |
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| 295 | |
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| 296 | assert( result2.first == std::min_element(L.begin(), L.end()); |
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| 297 | assert( result2.second == std::max_element(L.begin(), L.end()); |
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| 298 | }</pre> |
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| 299 | </a> |
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| 300 | |
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| 301 | <a name="notes"> |
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| 302 | <h3> |
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| 303 | Notes</h3> |
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| 304 | <a NAME="Note1"></a><a href="#Note1">[1]</a> We do not support |
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| 305 | idioms such as <tt><a href="../../tuple/doc/tuple_users_guide.html#tiers">tie</a>(a,b)=minmax(a,b)</tt> |
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| 306 | to order two elements <tt>a</tt>, <tt>b</tt>, although this would have |
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| 307 | the desired effect if we returned a reference instead of a constant |
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| 308 | reference. The reason is that two unnecessary assignments are |
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| 309 | performed if a and b are in order. It is better to stick to <tt>if (b<a) |
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| 310 | swap(a,b)</tt> to achieve that effect. |
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| 311 | <p><a NAME="Note2"></a><a href="#Note2">[2]</a> These algorithms always |
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| 312 | perform at least <tt>3n/2-2</tt> comparisons, which is a lower bound on |
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| 313 | the number of comparisons in any case (Cormen, Leiserson, Rivest: "Introduction |
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| 314 | to Algorithms", section 9.1, Exercise 9.1-). The algorithms essentially compare |
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| 315 | the elements in pairs, performing 1 comparison for the first two elements, |
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| 316 | then 3 comparisons for each remaining pair of elements (one to order the |
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| 317 | elements and one for updating each the minimum and and the maximum). When |
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| 318 | the number of elements is odd, the last one needs to be compared to the |
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| 319 | current minimum and also to the current maximum. In addition, for <tt>minmax</tt>, |
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| 320 | in cases where equality of the two members in the pair could occur, and |
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| 321 | the update stores the second, we save the first to check at the end if |
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| 322 | the update should have stored the first (in case of equality). It's hard |
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| 323 | to predict if the last comparison is performed or not, hence the at most |
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| 324 | in both cases. |
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| 325 | <p><a NAME="Note3"></a><a href="#Note3">[3]</a> These algorithms always |
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| 326 | perform at least <tt>3n/2-2</tt> comparisons, which is a lower bound on |
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| 327 | the number of comparisons in any case. The method is the same as in note |
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| 328 | <a href="#Note2">[2]</a> |
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| 329 | above, and like above, when the number of elements is odd, the last one |
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| 330 | needs to be compared to the current minimum and also to the current maximum. |
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| 331 | We can avoid the latter comparison if the former is successful, hence the |
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| 332 | <i>at |
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| 333 | most</i> instead of <i>exactly</i> in the odd case. |
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| 334 | </a> |
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| 335 | |
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| 336 | <a name="rationale"> |
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| 337 | <h3> |
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| 338 | <b>Rationale:</b></h3> |
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| 339 | |
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| 340 | <a name="two_headers"> |
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| 341 | <h4><b>Why not a single header <tt>&boost/algorithm/minmax.hpp></tt>?</b></h4> |
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| 342 | <p>This was the design originally proposed and approved in the formal |
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| 343 | review. As the need for Boost.tuple became clear (due to the limitations |
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| 344 | of <tt>std::pair</tt>), it became also annoying to require another |
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| 345 | library for <tt>minmax_element</tt> which does not need it. Hence the |
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| 346 | separation into two header files.</p> |
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| 347 | |
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| 348 | <a name="no-fix"> |
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| 349 | <h4><b>Your minmax suffers from the same problems as std::min and |
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| 350 | std::max.</b></h4> |
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| 351 | <p>I am aware of the problems with std::min and |
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| 352 | std::max, and all the debate that has been going on (please consult |
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| 353 | <a href="http://www.cuj.com/experts/1904/alexandr.htm?topic=experts&topic=experts">Alexandrescu's |
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| 354 | paper</a> and the links therein). But I don't see the purpose of this |
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| 355 | library as fixing something that is part of the C++ standard. I humbly |
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| 356 | think it's beyond the scope of this library. Rather, I am |
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| 357 | following the way of the standard in simply providing one more function |
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| 358 | of the same family. If someone else wants to fix std::min, their fix |
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| 359 | would probably apply to boost::minmax as well.</p> |
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| 360 | </a> |
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| 361 | |
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| 362 | <h4><b>Why no <tt>min/max_element_if</tt>?</b></h4> |
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| 363 | <p>In a first version of the library, I proposed <tt>_if</tt> versions of |
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| 364 | all the algorithms (well, not all, because that would be too much). |
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| 365 | However, there is simply no reason to do so, and all the versions I had |
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| 366 | were just as fast implemented using the excellent |
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| 367 | <tt><boost/iterator_adaptors.hpp></tt> library. Namely, a call to |
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| 368 | <tt>min_element_if(first, last, pred)</tt> would be just as well |
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| 369 | implemented by: |
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| 370 | <pre> |
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| 371 | // equivalent to min_element_if(first, last, pred) |
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| 372 | min_element(boost::make_filter_iterator(first, last, pred), |
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| 373 | boost::make_filter_iterator(last, last, pred)); |
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| 374 | </pre> |
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| 375 | Arguably, the <tt>min_element_if</tt> version is somewhat shorter, but |
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| 376 | the overhead of iterator adaptors is not large, and they get rid of a |
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| 377 | lot of code (think of all the combinations between first/last and |
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| 378 | doubling them with _if variants!).</p> |
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| 379 | |
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| 380 | <h4><b>Discussion: about std::max_element</b></h4> |
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| 381 | <p>This rationale is somewhat historical, but explains why there are all |
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| 382 | these <tt>first/last_min/max_element</tt> functions.</p> |
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| 383 | <p>The C++ standard mandates that <tt>std::min_element</tt> and <tt>std::max_element</tt> |
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| 384 | return the first instance of the smallest and largest elements (as opposed |
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| 385 | to, say, the last). This arbitrary choice has some consistency: In the |
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| 386 | case of v of type vector<int>, for instance, it is true that <tt>std::min_element(v.begin(),v.end(),std::less<int>()) |
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| 387 | == std::max_element(v.begin(),v.end(),std::greater<int>())</tt>. |
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| 388 | <p>There is of course nothing wrong with this: it's simply a matter of |
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| 389 | choice. Yet another way to specify min_element and max_element is to define |
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| 390 | them as the first and the last elements if the range was stably sorted. |
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| 391 | (The <i>stable</i> sort is necessary to disambiguate between iterators |
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| 392 | that have the same value.) In that case, min should return the first instance |
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| 393 | and max should return the last. Then, both functions are related by |
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| 394 | <tt>reverse_iterator(std::first_min_element(v.begin(),v.end(),std::less<int>())) |
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| 395 | == |
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| 396 | std::last_max_element(v.rbegin(),v.rend(),std::greater<int>())</tt>. |
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| 397 | This definition is subtly different from the previous one.</p> |
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| 398 | <p>The definition problem surfaces when one tries to design a minmax_element, |
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| 399 | using the procedure proposed in (Cormen, Leiserson, Rivest: "Introduction |
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| 400 | to Algorithms", section 9.1). It <i>should</i> be possible to derive an |
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| 401 | algorithm using only <tt>3n/2</tt> comparisons if <tt>[first,last) </tt>has |
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| 402 | <tt>n</tt> |
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| 403 | elements, but if one tries to write a function called <tt>first_min_first_max_element()</tt> |
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| 404 | which returns both <tt>std::min_element</tt> and <tt>std::max_element</tt> |
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| 405 | in a pair, the trivial implementation does not work. The problem, rather |
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| 406 | subtly, is about equal elements: I had to think for a while to find a |
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| 407 | way to perform only three |
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| 408 | comparisons per pair and return the first min and first max elements. |
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| 409 | For a long time, it seemed any |
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| 410 | attempts at doing so would consume four comparisons per pair in the worst |
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| 411 | case. This implementation achieves three.</p> |
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| 412 | <p>It is not possible (or even desirable) to change the meaning of |
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| 413 | <tt>max_element</tt>, |
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| 414 | but it is still beneficial to provide a function called <tt>minmax_element</tt>, |
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| 415 | which returns a pair of <tt>min_element</tt> and <tt>max_element</tt>. |
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| 416 | Although it is easy enough to call <tt>min_element</tt> and <tt>max_element</tt>, |
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| 417 | this performs |
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| 418 | <tt>2(n-1)</tt> comparisons, and necessitates <b>two</b> |
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| 419 | passes over the input. In contrast, |
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| 420 | <tt>minmax_element</tt> will perform |
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| 421 | the fewer comparisons and perform a <b>single</b> pass over the input. |
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| 422 | The savings can be significant when the iterator type is not a raw pointer, |
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| 423 | or even is just a model of the InputIterator concept (although in that |
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| 424 | case the interface would have to be |
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| 425 | changed, as the return type could not be copied, so one could e.g. |
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| 426 | return a value).</p> |
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| 427 | <p>In order to benefit from all the variants of the algorithm, I propose |
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| 428 | to introduce both <tt>first_min_element</tt> and <tt>last_min_element</tt>, |
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| 429 | and their counterparts <tt>first_max_element</tt> and <tt>last_max_element</tt>. |
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| 430 | Then I also propose all the variants algorithms: <tt>first_min_last_max_element</tt> |
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| 431 | and <tt>last_min_first_max_element</tt>, which perform only at most <tt>3n/2</tt> |
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| 432 | comparisons, and only a single pass on the input. In fact, it can be proven |
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| 433 | that computing minmax requires at least <tt>3(n/2)-2</tt> comparisons in |
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| 434 | any instance of the problem (Cormen, Leiserson, Rivest, 2nd edition, section |
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| 435 | 9.1). The implementation I give does not perform unnecessary comparisons |
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| 436 | (whose result could have been computed by transitivity from previous |
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| 437 | comparisons).</p> |
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| 438 | <p>It appears that <tt>first_min_last_max_element</tt> may be just a tad |
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| 439 | slower than |
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| 440 | <tt>first_min_element</tt> alone, still much less than <tt>first_min_element</tt> |
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| 441 | and |
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| 442 | <tt>last_max_element</tt> called separately. <a href="#Performance">[2]</a> |
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| 443 | |
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| 444 | <h4><b>Why algorithms and not accumulators?</b></h4> |
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| 445 | <p>The minmax algorithms are useful in computing the extent of a range. |
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| 446 | In computer graphics, we need a bounding box of a set of objects. |
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| 447 | In that case the need for a single pass is even more stringent |
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| 448 | as all three directions must be done at once. Food for thoughts: there |
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| 449 | is matter for a nice generic programming library with stackable <tt>update_min</tt> |
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| 450 | and <tt>update_max</tt> function objects which store a reference to the |
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| 451 | <tt>min_result</tt>and |
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| 452 | <tt>max_result</tt> variables, in conjunction with the <tt>for_each</tt> |
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| 453 | algorithm).</p> |
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| 454 | <p>I believe many standard sequential algorithms could be reformulated |
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| 455 | with accumulators (and many others, such as in statistics, expectation / |
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| 456 | variance / etc.). It seems that there is room for another library, but I |
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| 457 | do not see it competing with minmax, rather extending several algorithms |
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| 458 | (including minmax) to the accumulator framework. However, I felt it is |
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| 459 | beyond the scope of this library to provide such accumulators.</p> |
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| 460 | |
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| 461 | <a NAME="no-policy"> |
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| 462 | <h4><b>This first/last is a perfect application for a policy-based |
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| 463 | design.</b></h4> |
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| 464 | <p>True, and I could have gone that way, with the default policy for |
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| 465 | <tt>min_element</tt> and <tt>max_element</tt> to pick the first |
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| 466 | occurence of the result. This would have thinned the number of |
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| 467 | combinations of the minmax_element variants. But it would also have |
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| 468 | meant to change the interface of <tt>boost::minmax_element</tt>. |
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| 469 | One of the goals of the <tt>minmax_element</tt> algorithm is its |
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| 470 | eventual addition to the C++ standard, in connection with |
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| 471 | <tt>std::min_element</tt> and <tt>std::max_element</tt> |
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| 472 | (and I feel that it would be quite natural |
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| 473 | given the shortness of the implementation, and the not quite trivial |
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| 474 | detail which is needed to get it right). So changing the interface by |
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| 475 | adding policies would have meant unfortunately to depart from the |
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| 476 | standard and created an obstacle towards that goal. Besides, the code |
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| 477 | remains rather readable and simple without policies. So I am quite happy |
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| 478 | to keep it like this. |
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| 479 | </p></a> |
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| 480 | </a> |
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| 481 | |
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| 482 | <a name="perf"> |
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| 483 | <a href="doc/minmax_benchs.html"><h3><b>About performance</b></h3></a> |
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| 484 | </a> |
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| 485 | |
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| 486 | <a name="acks"> |
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| 487 | <h3> |
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| 488 | Acknowledgements</h3> |
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| 489 | My students in CS903 (Polytechnic Univ., <a href="http://photon.poly.edu/~hbr/cs903/">http://photon.poly.edu/~hbr/cs903/</a>) |
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| 490 | who had <tt>minmax_element</tt> as an assignment helped clarify the issues, |
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| 491 | and also come up with the optimum number of comparisons for <tt>first_min_last_max_element</tt>. |
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| 492 | The identification of the issue surrounding <tt>max_element</tt> is solely |
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| 493 | my own. |
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| 494 | <p>One <tt>minmax_element</tt> implementation, which performs <tt>3(n/2)+O(log |
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| 495 | n)</tt> comparisons on the average when the elements are <tt>random_shuffle</tt>d, |
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| 496 | was suggested by my student Marc Glisse. The current one, which performs |
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| 497 | <tt>3(n/2)+1</tt> |
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| 498 | comparisons in the worst case, was suggested by John Iacono.<p> |
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| 499 | <p>Finally, Matthew Wilson and Jeremy Siek contributed pre-review |
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| 500 | comments, while Gennadiy Rozental, John Maddock, Craig Henderson, Gary |
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| 501 | Powell participated in the review of the library, managed by Thomas |
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| 502 | Witt. In particular, Gennadiy suggested a factorization of the code; |
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| 503 | while I haven't followed it all the way, his suggestions do make the |
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| 504 | code more readable and still work with older compilers. |
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| 505 | Late after the review, as I finally scrounged to add the library for a |
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| 506 | release, Eric Niebler noted the bad behavior of <tt>std::pair</tt> for |
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| 507 | <tt>minmax</tt> and suggested to use Boost.tuple instead. |
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| 508 | All my thanks for the excellent advice and reviews from all. |
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| 509 | <h3> |
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| 510 | See also</h3> |
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| 511 | <tt><a href="http://www.sgi.com/tech/stl/min.html">min</a></tt>, <tt><a href="http://www.sgi.com/tech/stl/max.html">max</a></tt>, |
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| 512 | <tt><a href="http://www.sgi.com/tech/stl/min_element.html">min_element</a></tt>, |
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| 513 | <tt><a href="http://www.sgi.com/tech/stl/max_element.html">max_element</a></tt>, |
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| 514 | <tt><a href="http://www.sgi.com/tech/stl/LessThanComparable.html">LessThan |
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| 515 | Comparable</a></tt>, |
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| 516 | <tt><a href="http://www.sgi.com/tech/stl/sort.html">sort</a></tt>, |
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| 517 | <tt><a href="http://www.sgi.com/tech/stl/nth_element.html">nth_element</a></tt> |
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| 518 | . |
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| 519 | <hr SIZE="6"> |
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| 520 | <br>Last modified 2004-07-01 |
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| 521 | <p><font face="Arial,Helvetica"><font size=-1>© Copyright Hervé |
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| 522 | Brönnimann, Polytechnic University, 2002--2004. |
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| 523 | Use, modification, and distribution is subject to the Boost Software |
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| 524 | License, Version 1.0. (See accompanying file <a href="../../../LICENSE_1_0.txt">License_1_0.txt</a> or copy at |
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| 525 | <a href="http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>) |
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| 526 | </font></font> |
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| 527 | </body> |
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| 528 | </html> |
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