Thursday, May 22, 2014

Using The Pigeonhole Principle in C++ Metaprogramming

The Pigeonhole Principle is one of the most obvious fundamentals in mathematics. It is so obvious that you may be surprised that there is even a name for it. It states that:

"If n items are put into m containers, with n > m, then at least one container must contain more than one item."


"If there are n items and m containers, with n > m, and only one item can fit in a container, then at least one item must remain out."

For those who prefer visuals and really hate math:

Even though the principle is simple it has been used to prove many complex mathematical theorems and lemmas. Here is one I find quite interesting:

"Incompressible strings of every length exist."


"There is a file of every size that your favorite zip program can't compress."

The solution is left to the reader as an exercise.

So, does the Pigeonhole Principle show up in programming. Of course it does. That's why std::vector must allocate memory when its capacity is full. OK, but does it manifest in more interesting ways? As it turns out, it has been used in compile-time meta-programming to achieve interesting results. It manifests in preprocessor meta-programming and in template meta-programming in two distinct flavors.

The Pigeonhole Principle in C++ Preprocessor Meta-programming

Check out the following example. Also available here. The original author of this trick is unknown to me.
#include <iostream>

#define COUNT_ARGS(...)     PP_NARG_IMPL(__VA_ARGS__,PP_RSEQ_N()) 
#define PP_NARG_IMPL(...)   PP_ARG_N(__VA_ARGS__) 
#define PP_ARG_N( _1, _2, _3, _4, _5, _6, _7, _8, _9, _10, N, ...) N 
#define PP_RSEQ_N() 10,9,8,7,6,5,4,3,2,1,0 

int main()
  std::cout << COUNT_ARGS(a,b,c,d); // prints 4
COUNT_ARGS is a "simple" macro that counts the number of variadic arguments it is called with. It does that by using a preprocessing programming trick based on the Pigeonhole principle. Here is how the macro expands:
  1. The COUNT_ARGS macro substitutes the arguments (a,b,c,d) in the __VA_ARGS__ part before calling PP_NARG_IMPL. The PP_RSEQ_N macro is a list of integers from 10 to 0, which is substituted in the PP_NARG_IMPL. Therefore, the PP_NARG_IMPL macro is "called" with actual arguments = a,b,c,d,10,9,8,7,6,5,4,3,2,1,0
  2. The PP_NARG_IMPL macro simply forwards its arguments to the PP_ARG_N macro.
  3. The PP_ARG_N macro is where the Pigeonhole Principle comes in to play. It has 11 named arguments: From _1, _2, _3, etc. and N. Note that _1, _2, etc. are not special. They are just macro arguments with an underscore at the beginning. You may want to rename them as one, two, three, four, etc. It won't make a difference. The PP_ARG_N always expands to its 11th argument because of N.
  4. The original argument list has 15 arguments but there are only 11 arguments to the PP_ARG_N macro. Obviously, not all are going to fit. The PP_ARG_N macro only "picks up" the first actual argument that does not get a slot (i.e., 11th)
  5. As N always coincides with the 11th actual argument, the PP_ARG_N results in that value producing the count.
Needless to say, that's clever! Now let's proceed with template meta-programming.

The Pigeonhole Principle in C++ Template Meta-programming

Check out the following example. Also available here.
int main()
 auto x = ::nth<7>(0,"1",'2',3,"4",'5',6,"7");
 std::cerr << x << std::endl;
The goal is to access the N-th element in a variadic function argument list. The output of the above program should be 7.

There are many ways to go about implementing it, most using recursion of some sort. However, there is one implementation I came across, which I find particularly interesting. Why? You guessed it ... It uses the Pigeonhole Principle to avoid recursion.

The code was originally written by Richard Smith. I found it through a post by Roland Bock on the boost developers mailing list. If you prefer more comments, please see the same example with comments by LJEvans.
#include <utility>
#include <iostream>

namespace detail
  struct any { template<typename T> any(T &&) {} };

  template<typename T, typename U> struct first { typedef T type; };

  template<typename ...Ts>
  struct select_impl 
    template<typename U, typename ...Vs>
 static U &&select(typename first<any, Ts>::type..., U &&u, Vs &&...) 
    return static_cast<U&&>(u);

  template<std::size_t... Idx, typename... Ts>
  static auto select(const std::index_sequence<Idx...>&, Ts&&... ts)
     return select_impl<decltype(Idx)...>::select(static_cast<Ts&&>(ts)...);

template<std::size_t N, typename ...Ts>
auto nth(Ts &&...ts)
  return detail::select(std::make_index_sequence<N>(), static_cast<Ts&&>(ts)...);

int main()
 auto x = ::nth<7>(0,"1",'2',3,"4",'5',6,"7"); // prints 7
 std::cerr << x << std::endl;
Here is how the nth<7>(...) function works in the example above.
  1. N is 7 and Ts is a variadic parameter pack of integers, character strings, and plain characters.
  2. The std::make_index_sequence is a new addition in C++14 that produces an instance of std::index_sequence given a compile-time integral constant. Here, it produces std::index_sequence<0,1,2,3,4,5,6>.
  3. The formal arguments to the nth function (captured in the parameter pack ts) are forwarded to detail::select using a static_cast. This function must return the nth argument among the forwarded arguments.
  4. In detail::select, the Idx parameter pack represents the indices from 0 to 6. Its deduced by the compiler looking at the type of the index_sequence instance.
  5. The select_impl class template is instantiated with the decltype of each member in the Idx parameter pack. decltype(ts)... expands in to a list of types for every member in Ids. In this case, it is just 'int, int, int,... 7 times. The remaining arguments to select_impl::select are just being forwarded as before.
  6. The select_impl::select has access to Ts parameter pack, which is at the class-template level. Recall that it is 'int,int,int,....'. The list of formal arguments to select_impl::select is broken down in to 3 parts: a variadic piece of N-1 arguments at the beginning, U&& in the middle, and everything else in Vs.
  7. The first N-1 arguments to select_impl::select are "absorbed" using the detail::any class. The detail::any has a single argument constructor that converts argument of any type to any. The first N-1 arguments are thus converted to any. In our example, all the arguments from 0 to 6 are converted to any. The conversion is achieved using an in place parameter pack expansion 'typename first::type...'. For every argument in the Ts parameter pack, the 'first' meta-function is applied, which results into the 'any' type every time.
  8. As the first N-1 arguments are out of the way, U&& necessarily fits the N-th argument. This is where the Pigeonhole Principle springs back in to action.
  9. The remaining argument after the N-th (if any) are left unused in the Vs parameter pack.

So, there it is: returning the N-th argument in a argument list without using recursion. In practice, however, std::make_index_sequence is implemented using recursion. So, the above code is not truly recursion-free.

OK ... So you read it all! I'm sure you found the use of the Pigeonhole Principle in processing variadics in C++ very interesting.

Wednesday, May 07, 2014

A tale of noexcept swap for user-defined classes in C++11

C++11 has taken another stab at function exception specifications for the masses. The shiny new ‘noexcept’ keyword is phasing out its zombie cousin ‘throw’. The well accepted guideline for the old exception specification is: "don’t use them!" That’s good for us because now we don’t have to bother (the reasons for not using throw specification aren’t for the faint hearted anyways.) The noexcept feature does not appear to be a walk in a park either. So hold on tight!

noexcept is meta-programmable! a.k.a conditional noexcept. It is possible to conditionally specify functions to throw any exceptions. The noexcept specification of functions can be inspected at compile-time and functions can derive their own noexcept specification based on exception specifications found elsewhere in the program. Meta-programs are not off-limits here. An excellent introduction of the noexcept feature and its history can be found in June’11 Overload Journal and on Andrzej's C++ blog. I won’t repeat that here but a quick motivation is in order.

Compile-time knowledge about a move-constructor that it does not throw (i.e., annotated with noexcept(true)) can be used for improved run-time performance. For example, std::vector<T>::resize, std::vector<T>::reserve automatically use T’s move-constructor instead of a copy-constructor if T's move-constructor does not throw. Moving of T objects instead of copying would likely achieve higher performance. The standard library provides std::move_if_noexcept function that does move only if it is guaranteed to not fail. Otherwise, it simply resorts to copy-construction. More on this can be found here. I'll return to the topic of writing noexcept move-ctor towards the end of this post.

But the post is about swapping, so lets talk about that. Here is how the std::pair::swap is declared in C++11.

void pair::swap(pair& p)
  noexcept(noexcept(swap(first, p.first)) &&
           noexcept(swap(second, p.second)));

It basically says that pair's swap will throw if either swapping of first or second member throws. The swapping of first and second possibly resolve to their own namespace-level swap via ADL or else they simply pick up std::swap (because pair::swap is declared in std namespace). This declaration seems to indicate a few things.

  1. For a user-defined type it is now much more preferable to have its own namespace-level swap overload instead of using vanilla std::swap. The namespace-level swap function, being an extension of the type’s interface, is vastly more efficient and often can be written in way that it does not throw. The std::swap function, on the other hand, may throw because it uses a copy-constructor and copy-assignment operator while swapping two objects. If you are like me, your member copy-assignment operator will use copy-and-swap idiom for strong exception safety. That means std::swap will create and destroy 3 copies. What a waste! Moreover, that increases the chance of throwing. Bottom line: If T has a member swap, a swap function should be added in the same namespace as of T.

  2. There are some assumptions buried in the above reasoning: (1) move-constructor is not available for a user-defined T, and (2) it is possible to implement member swap of T that does not throw and you can actually say so.

  3. As far as move-constructor being not available is concerned, I think, there are large number of C++03 libraries, which will acquire move-semantics slowly than you might desire. Till then std::swap will, unfortunately, use copy-ctor and copy-assign. A legacy class with a copy-ctor won’t get an automatic move-ctor because it would do wrong things.

  4. The noexcept specification of the user-defined swap better be accurate. Can we really write a nothrow member swap and say so confidently? Obviously, it depends on the members of the user-defined type that we’re swapping. To me, it’s all uphill from here.

Lets look at how standard library uses noexcept for some of its own classes. We already saw std::pair before. How about std::tuple, which is a generalization of a struct and that’s why may be user-defined classes should follow the same basic principle. Tuple’s member swap is declared as

void tuple::swap(tuple& rhs) noexcept(see below)
// The expression inside noexcept is equivalent to the
// logical AND of the following expressions:

noexcept(swap(declval<T&>(), declval<T&>()))
// where Ti is ith type in the variadic list of types.

Tuple’s member swap inspects the noexcept specifications of its members’ swap functions and derives its own specification from that. That makes sense. If swapping of any of the member throws, the tuple’s swap throws. The declval function belongs to same clique as std::move, std::forward, etc. It is a way to obtain rvalue reference of a type without using a constructor. It suffices to say that declval<T> returns "T&&" and declval<T&> returns "T & &&", which is the same as "T &."

Consider a typical C++03 user-defined legacy class that manages its own resources.

namespace L 
struct legacy 
  legacy();                             // Does not throw
  legacy(const legacy &);               // This may throw
  legacy & operator = (const legacy &); // This may throw (strong exception safe)
  ~legacy() throw();                    // Does not throw
  void swap(legacy &) throw();          // Does not throw
} // namespace L

The above legacy class is well-designed except for the fact that it does not have any namespace-level swap. To make it a better citizen in C++11, it needs a namespace-level swap. Interestingly enough, there is substantial code out there that has a specialization of the std::swap in the std namespace. Considering the popularity of this anti-pattern, it probably makes sense to add a swap in two places. It is not too much work in the first place and that may help people who always use a qualified swap (std::swap) habitually.

namespace L 
  void swap(legacy &, legacy &) noexcept;

namespace std 
  template <>
  void swap(L::legacy &, L::legacy &) noexcept;

Let’s try to use this modernized legacy class in our C++11 class: Modern.

namespace M 
struct Modern 
  int count;
  std::string str;
  std::array<L::legacy, 5> data;
  /// default-ctor, copy-ctor, copy-assign, move-ctor, move-assign are defined = default.
  void swap(Modern & p) 
    noexcept(noexcept(swap(std::declval<int &>(),
                           std::declval<int &>())) &&
             noexcept(swap(std::declval<std::string &>(),
                           std::declval<std::string &>())) &&
             noexcept(swap(std::declval<std::array<L::legacy, 5> &>(),
                           std::declval<std::array<L::legacy, 5> &>())));
} // namespace M

That’s one awful looking swap. My eyes bleed when I look at it.

It is doing exactly the same thing as std::tuple. It simply checks whether swapping two integers, two strings, and two arrays throw or not. If it does, then member swap of Modern throws. You can skip checks for integer swapping but what’s left is not pretty at all.

Note: The text in italics below is not accurate anymore as noted in the comment below. C++11 std::swap uses move-construction and move-assignment internally. As a result, std::swap in C++11 is generally very efficient IF the move-operations are defined and they are really more efficient than their copy counterparts. In such cases, there is little need to write a custom namespace-level swap. However, member swap may still be useful. See the "default-construct-and-swap" technique for implementing a move-constructor below.

Couple of important things to note here:
  1. Because we added namespace-level swap for the L::legacy class, we dodged several problems.If we did not have free swap, the compiler will instantiate std::swap if it is in the scope. Note, however, that vanilla std::swap may throw defeating the purpose of Modern::swap. If no swap is to be found, compiler issues an error.

  2. We hope that swap of std::string and std::array do not throw. As mentioned in [container.requirements.general], std::string::swap does not throw but std::array<T> swap may throw if an unqualified call to swap(T &, T &) throws. Once again, the namespace-level swap will be chosen here, if available. If that does not exist, a specialization of std::swap will be chosen. If none of them exist, std::swap will be instantiated giving our Modern::swap a nothrow(false) specification.

I like my strong exception safety in my little copy-assignment operator so I don’t want my swap to throw but I don’t want my program to std::terminate() if an exception is really thrown. With all that in mind, I would rewrite the swap as follows.

void Modern::swap(Modern &) noexcept 
  static_assert(noexcept(swap(std::declval<int &>(),
                              std::declval<int &>())) &&
                noexcept(swap(std::declval<std::string &>(),
                              std::declval<std::string &>())) &&
                noexcept(swap(std::declval<std::array<L::legacy, 5> &>(),
                              std::declval<std::array<L::legacy, 5> &>())), 
                "throwing swap");
  //.... remaining swap code

This is not unlike what’s proposed by others but there’s more. The static assert looks awful and looks redundant. There is already a standard library utility that does the same thing: std::tuple. As mentioned before, std::tuple’s swap throws if any member swap throws. We use it here to make our job a lot easier.

void Modern::swap(Modern & that) noexcept 
  typedef std::tuple<int, std::string, std::array <L::legacy, 5> > MemberTuple;
  static MemberTuple *t = 0;
  static_assert(noexcept(t->swap(*t)), "throwing swap");
  // .... remaining swap code

If you an enthusiastically paranoid C++ programmer, you may want to use conditional noexcept with a little meta-function (is_nothrow_swappable_all) to save typing.

template<typename... T>
struct is_nothrow_swappable_all
  static std::tuple<T...> *t = 0;
  enum { value = noexcept(t->swap(*t)) };

void Modern::swap(Modern & that) 
   noexcept(is_nothrow_swappable_all<int, std::string, std::array<L::legacy, 5> >::value);

The "remaining swap code" section above is also rather important. The recommended (by Dave and Howard) way to write it is to use unqualified swap but bring std::swap in the scope:

void Modern::swap(Modern & that) 
  noexcept(is_nothrow_swappable_all<int, std::string, std::array<L::legacy, 5> >::value);
  using std::swap;
  swap(this->count, that->count);
  swap(this->str, that->str);
  swap(this->data, that->data);

The appropriate swap (namespace-level via ADL or std::swap) will be chosen depending upon availability. That’s exactly what tuple’s swap noexcept checker is going to do for us.

Back to the move-constructor, as promised! As mentioned before, noexcept specification of a move-ctor may have performance implications. So you want to get it right. For the M::Modern class we could have defaulted the move-ctor (= default). That will give it a noexcept(false) specification automatically because internally it uses L::legacy’s copy constructor, which throws. As a consequence, std::vector::resize is not as fast as it can be. We can do better.

Implementing a move-ctor using the default-construct-and-swap idiom turns out be quite handy. Default-construct-and-swap does what it says by first delegating to a noexcept default constructor followed by a swap. To implement a noexcept move-ctor this way, you really need to make sure that swap does not throw. As always, you can rely on a conditional noexcept. I'm using std::is_nothrow_constructible type trait to test the obvious!

Modern::Modern(Modern && m) 
  noexcept(std::is_nothrow_constructible<Modern>::value &&
  : Modern()           

As long as Modern's default constructor and member swap are noexcept, the move-ctor is noexcept.

Finally, the move-assign operator can be simply implemented as a single swap operation but for some classes that may not be accurate.

Modern::operator = (Modern && m) noexcept

Acknowledgements: I want to thank Howard Hinnant (former Library Working Group chair) for reviewing this article and providing me valuable feedback. Any inaccuracies and mistakes left in the article are strictly mine.

Sunday, May 04, 2014

Fun with Lambdas: C++14 Style (part 2)

Look at some interesting examples of C++11/14 lambdas and how they interact with other language features and libraries. I hope to find some time to add some explanations. See part 1 if you missed it.

  • Associative containers and lambdas
    std::set<int, std::function<bool(int, int)>> 
      numbers([](int i, int j) { return i < j; });
  • Recursive Lambdas (see Creating recursive lambdas and returning them too!)
    auto make_fibo() 
      return [](int n) {
        std::function<int(int)> recurse;
        recurse = [&](int n){ 
           return (n<=2)? 1 : recurse(n-1) + recurse(n-2); 
        return recurse(n);
  • Composable list manipulation (e.g., cpplinq, narl, LEESA)
    Box boxes[] = { ... };
    int sum_of_weights = 
      >> where([](const Box & box) { 
           return box.color == Color.RED;
      >> select([](const Box & box) {
           return box.get_weight();
      >> sum();
  • Overloaded Lambdas
    template <class... F>
    struct overload : F... {
      overload(F... f) : F(f)... {}  
    template <class... F>
    auto make_overload(F... f) {
      return overload<F...>(f...);   
    auto f = 
        make_overload([](int i) { /* print */ },
                      [](double d) { /* print */ });
    f(10); // int 
    f(9.99); // double
  • Type Switch (simple pattern matching) (see type_switch.cpp and this paper)
    struct Base { 
      virtual ~Base() {} 
    struct Derived : Base {};
    template <class Poly>
    void test(Poly& p) {  
        [](int i)             { cout << "int";       },
        [](std::string &)     { cout << "string";    },
        [](Derived &)         { cout << "Derived";   },     
        [](Base &)            { cout << "Base";      },    
        otherwise([](auto x)  { cout << "Otherwise"; })
    Derived d;
    Base &b = d;
    std::string cpptruths = "C++ Truths";
    boost::any something = cpptruths;
    test(10);        // int
    test(cpptruths); // string
    test(something); // string
    test(b);         // Derived
    test(9.99);      // Otherwise
  • Converting shared_ptr between boost and std (see StackOverflow)
    template <typename T>
    make_shared_ptr(std::shared_ptr<T> ptr) 
      return boost::shared_ptr<T>(ptr.get(), 
        [ptr](T*) mutable { ptr.reset(); });
    template <typename T>
    make_shared_ptr(boost::shared_ptr<T> ptr)
      return std::shared_ptr<T>(ptr.get(), 
        [ptr](T*) mutable { ptr.reset(); });
  • In-place parameter pack expansion 
    template <class... T>
    void foreach(T... args) 
      bool b[] = { [=](){ 
        std::cout << args << "\n"; 
        return true; 
      }()... }; 
    foreach(10, 20.2, true);
  • Memoization (see original)
    template <typename ReturnType, 
              typename... Args>
    auto memoize(ReturnType (*func)(Args...))
        std::map<std::tuple<Args...>, ReturnType> cache;
        return ([=](Args... args) mutable  
            std::tuple<Args...> t(args...);
            if (cache.find(t) == cache.end())                
              std::cout << "not found\n";
              cache[t] = func(args...);
            return cache[t];
  • Finally, slides