Boost Minmax library
Author: Herve Bronnimann
Description: Small library to propose minmax_element algorithm.

## Header <boost/algorithm/minmax.hpp>

Motivation
Synopsis
Function templates description
Definition
Requirements on types
Preconditions
Postconditions
Complexity
Example
Notes
Rationale
Note about performance
Acknowledgements

### Motivation

The minmax library is composed of two headers, <boost/algorithm/minmax.hpp> and <boost/algorithm/minmax_element.hpp>. (See rationale.) The problem solved by this library is that simultaneous min and max computation requires only one comparison, but using `std::min` and `std::max` forces two comparisons. Worse, to compute the minimum and maximum elements of a range of `n` elements requires only `3n/2+1` comparisons, instead of the `2n` (in two passes) forced by `std::min_element` and `std::max_element`. 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.

The first file implements the function templates `minmax` as straightforward extensions of the C++ standard. As it returns a pair of `const&`, we must use the Boost.tuple library to construct such pairs. (Please note: the intent is not to fix the known defaults of `std::min` and `std::max`, but to add one more algorithms that combines both; see the rationale.)

The second file implements the function templates `minmax_element`. 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 rationale).

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

### Synopsis of `<boost/algorithm/minmax.hpp>`

    #include <boost/tuple/tuple.hpp>

    namespace boost {

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

      template <class T, class BinaryPredicate>
      tuple<T const&, T const&>
      minmax(const T& a, const T& b, BinaryPredicate comp);

    }

### Synopsis of `<boost/algorithm/minmax_element.hpp>`

    #include <utility> // for std::pair

    namespace boost {

      template <class ForwardIterator>
      std::pair<ForwardIterator,ForwardIterator>
      minmax_element(ForwardIterator first, ForwardIterator last);

      template <class ForwardIterator, class BinaryPredicate>
      std::pair<ForwardIterator,ForwardIterator>
      minmax_element(ForwardIterator first, ForwardIterator last,
                     BinaryPredicate comp);

    }

In addition, there are a bunch of extensions which specify which element(s) you want to pick in case of equal elements. They are:

- `first_min_element` and `last_min_element`
- `first_max_element` and `last_max_element`
- `first_min_first_max_element`, `first_min_last_max_element`, `last_min_first_max_element`, and `last_min_last_max_element`

I won't bore you with the complete synopsis, they have exactly the same declaration as their corresponding `_element` function. Still, you can find the complete synopsis here.

### Function templates description

The `minmax` algorithm returns a pair `p` containing either (a,b) or (b,a), such that `p.first<p.second` in the first version, or `comp(p.first,p.second)` in the second version. If the elements are equivalent, the pair (a,b) is returned. [1]

The `minmax_element` is semantically equivalent to `first_min_first_max_element`.

`First_min_element` and `first_max_element` find the smallest and largest elements in the range `[first, last)`. If there are several instance of these elements, the first one is returned. They are identical to `std::min_element` and `std::max_element`and are only included in this library for symmetry.

`Last_min_element` and `last_max_element` find the smallest and largest elements in the range `[first, last)`. They are almost identical to `std::min_element` and `std::max_element`, except that they return the last instance of the largest element (and not the first, as `first_min_element` and `last_max_element` would).

The family of algorithms comprising `first_min_first_max_element`, `first_min_last_max_element`, `last_min_first_max_element`, and `last_min_last_max_element` can be described generically as follows (using `which` and `what` for `first` or `last`): `which_min_what_max_element` finds the (first or last, according to `which`) smallest element and the (first or last, according to `what`) largest element in the range `[first, last)`. The first version is semantically equivalent to:

      std::make_pair(boost::which_min_element(first,last),
                     boost::what_max_element(first,last)),

and the second version to:

      std::make_pair(boost::which_min_element(first,last,comp),
                     boost::what_max_element(first,last,comp)).

Note: the `first_min_last_max_element` can also be described as finding the first and last elements in the range if it were stably sorted.

### Definition

Defined in minmax.hpp and in minmax_element.hpp.

### Requirements on types

For minmax, `T` must be a model of LessThan Comparable.

For all the other function templates, versions with two template parameters:

- `ForwardIterator` is a model of Forward Iterator.
- `ForwardIterator`'s value type is LessThan Comparable.

For the versions with three template parameters:

- `ForwardIterator` is a model of Forward Iterator.
- `BinaryPredicate` is a model of Binary Predicate.
- `ForwardIterator`'s value type is convertible to `BinaryPredicate`'s first argument type and second argument type.

### Preconditions

- `[first, last)` is a valid range.

### Postconditions

In addition to the semantic description above. for `minmax_element` and all the `which_min_what_max_element` variants, the return value is `last` or `std::make_pair(last,last)` if and only if `[first, last)` is an empty range. Otherwise, the return value or both members of the resulting pair are iterators in the range `[first, last)`.

### Complexity

Minmax performs a single comparison and is otherwise of constant complexity. The use of `boost::tuple<T const&>` prevents copy constructors in case the arguments are passed by reference.

The complexity of all the other algorithms is linear. They all perform exactly n increment operations, and zero comparisons if `[first,last)` is empty, otherwise :

- all the `min_element` and `max_element` variants perform exactly`(n-1)` comparisons,
- `minmax_element` , `first_min_first_max_element`, and `last_min_last_max_element` perform at most `3(n/2)-1` comparisons if `n` is even and non-zero, and at most `3(n/2)+1` if `n` is odd, [2]
- `first_min_last_max_element`, and `last_min_first_max_element` perform exactly `3n/2-2` comparisons if n is even and non-zero, and at most `3(n/2)` if `n` is odd, [3]

where `n` is the number of elements in `[first,last)`.

### Example

This example is included in the distribution in the examples section of the library under minmax_ex.cpp.

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

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

      list<int> L;
      generate_n(front_inserter(L), 1000, rand);

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

      assert( result2.first  == std::min_element(L.begin(), L.end());
      assert( result2.second == std::max_element(L.begin(), L.end());
    }

### Notes

[1] We do not support idioms such as `tie(a,b)=minmax(a,b)` to order two elements `a`, `b`, 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 `if (b<a) swap(a,b)` to achieve that effect.

[2] These algorithms always perform at least `3n/2-2` 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 `minmax`, 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.

[3] These algorithms always perform at least `3n/2-2` comparisons, which is a lower bound on the number of comparisons in any case. The method is the same as in note [2] 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 at most instead of exactly in the odd case.

### Rationale:

#### Why not a single header `<boost/algorithm/minmax.hpp>`?

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 `std::pair`), it became also annoying to require another library for `minmax_element` which does not need it. Hence the separation into two header files.

#### Your minmax suffers from the same problems as std::min and std::max.

I am aware of the problems with std::min and std::max, and all the debate that has been going on (please consult Alexandrescu's paper 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.

#### Why no `min/max_element_if`?

In a first version of the library, I proposed `_if` 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 `<boost/iterator_adaptors.hpp>` library. Namely, a call to `min_element_if(first, last, pred)` would be just as well implemented by:

         // equivalent to min_element_if(first, last, pred)
         min_element(boost::make_filter_iterator(first, last, pred),
                     boost::make_filter_iterator(last, last, pred));

Arguably, the `min_element_if` 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!).

#### Discussion: about std::max_element

This rationale is somewhat historical, but explains why there are all these `first/last_min/max_element` functions.

The C++ standard mandates that `std::min_element` and `std::max_element` 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<int>, for instance, it is true that `std::min_element(v.begin(),v.end(),std::less<int>()) == std::max_element(v.begin(),v.end(),std::greater<int>())`.

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 stable 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 `reverse_iterator(std::first_min_element(v.begin(),v.end(),std::less<int>())) == std::last_max_element(v.rbegin(),v.rend(),std::greater<int>())`. This definition is subtly different from the previous one.

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 should be possible to derive an algorithm using only `3n/2` comparisons if `[first,last)` has `n` elements, but if one tries to write a function called `first_min_first_max_element()` which returns both `std::min_element` and `std::max_element` 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.

It is not possible (or even desirable) to change the meaning of `max_element`, but it is still beneficial to provide a function called `minmax_element`, which returns a pair of `min_element` and `max_element`. Although it is easy enough to call `min_element` and `max_element`, this performs `2(n-1)` comparisons, and necessitates two passes over the input. In contrast, `minmax_element` will perform the fewer comparisons and perform a single 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).

In order to benefit from all the variants of the algorithm, I propose to introduce both `first_min_element` and `last_min_element`, and their counterparts `first_max_element` and `last_max_element`. Then I also propose all the variants algorithms: `first_min_last_max_element` and `last_min_first_max_element`, which perform only at most `3n/2` comparisons, and only a single pass on the input. In fact, it can be proven that computing minmax requires at least `3(n/2)-2` 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).

It appears that `first_min_last_max_element` may be just a tad slower than `first_min_element` alone, still much less than `first_min_element` and `last_max_element` called separately. [2]

#### Why algorithms and not accumulators?

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 `update_min` and `update_max` function objects which store a reference to the `min_result`and `max_result` variables, in conjunction with the `for_each` algorithm).

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.

#### This first/last is a perfect application for a policy-based design.

True, and I could have gone that way, with the default policy for `min_element` and `max_element` 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 `boost::minmax_element`. One of the goals of the `minmax_element` algorithm is its eventual addition to the C++ standard, in connection with `std::min_element` and `std::max_element` (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.

### About performance

### Acknowledgements

Generic: Min and Max Redivivus, by Andrei Alexandrescu Dr. Dobbs, April 2001

My students in CS903 (Polytechnic Univ., http://photon.poly.edu/~hbr/cs903/) who had `minmax_element` as an assignment helped clarify the issues, and also come up with the optimum number of comparisons for `first_min_last_max_element`. The identification of the issue surrounding `max_element` is solely my own.

One `minmax_element` implementation, which performs `3(n/2)+O(log n)` comparisons on the average when the elements are `random_shuffle`d, was suggested by my student Marc Glisse. The current one, which performs `3(n/2)+1` comparisons in the worst case, was suggested by John Iacono.

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 `std::pair` for `minmax` and suggested to use Boost.tuple instead. All my thanks for the excellent advice and reviews from all.

### See also

`min`, `max`, `min_element`, `max_element`, `LessThan Comparable`, `sort`, `nth_element` .

Last modified 2012-12-10

© Copyright Hervé Brönnimann, Polytechnic University, 2002--2004. Use, modification, and distribution is subject to the Boost Software License, Version 1.0. (See accompanying file License_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)