    /**************************************************************************\

    MODULE: vector

    SUMMARY:

    Template class for dynamic-sized vectors.

    The declaration

       Vec<T> v;

    creates a zero-length vector.  To grow this vector to length n,
    execute

       v.SetLength(n)

    This causes space to be allocated for (at least) n elements, and also causes
    the default constructor for T to be called to initialize these elements.

    The current length of a vector is available as v.length().

    Let n = v.length().  Calling v.SetLength(m) with m <= n sets the current length
    of v to m (but does not call any destructors or free any space).  Calling
    v.SetLength(m) with m > n will allocate space and initialize as necessary, but
    will leave the values of the already allocated elements unchanged (although
    their addresses may change).  If T has a user-defined default constructor, that
    is invoked.  Otherwise, the new memory locations are "default initialized".  In
    particular, this means that POD types may be uninitialized.

    v.MaxLength() is the largest value of n for which v.SetLength(n) was invoked,
    and is equal to the number of entries that have been initialized.
    v.SetMaxLength(n) will allocate space for and initialize up to n elements,
    without changing v.length().

    When v's destructor is called, all constructed elements will be destructed, and
    all space will be relinquished.

    v.allocated() is the number of elements which have been allocated,
    which may be more than the number elements initialized.
    Note that if n <= v.allocated(), then v.SetLength(n) is guaranteed
    not to cause any memory allocation, or movement of objects.

    Space is managed using malloc, realloc, and free.  When a vector is grown, a
    bit more space may be allocated than was requested for efficiency reasons.

    Note that when a vector is grown, the space is reallocated using realloc, and
    thus the addresses of vector elements may change, possibly creating dangling
    references to vector elements.  One has to be especially careful of this when
    using vectors passed as reference parameters that may alias one another.

    Because realloc is used to grow a vector, the objects stored in a vector should
    be "relocatable"---that is, they shouldn't care what their actual address is,
    which may change over time.  Most reasonable objects satisfy this constraint.

    NOTE: as of NTL 10.4, you may compile NTL in "safe vector" mode by setting the
    configuration flag NTL_SAFE_VECTORS=on.  In this mode, a safer approach for
    resizing vectors is used, which uses copy/move constructors instead of realloc,
    unless the type T is explicitly declared as "relocatable".  See below for more
    details.

    ----------------------
    IMPLEMENTATION DETAILS
    ----------------------

    A Vec<T> object is just a pointer to the first element of the array.  There is
    a control block immediately before the first element that keeps track of
    several parameters:
       len    -- the logical length of the array (returned by length())
       init   -- the number of elements constructed (returned ny MaxLength())
       alloc  -- the number of elements for which space has been allocated
                 (returned by allocated())
       fixed  -- flag that indicates that the length is fixed
                 (returned by fixed())

    Note that 0 <= len <= init <- alloc

    ---------------
    SPECIALIZATIONS
    ---------------

    NTL currently defines an optimized specialization Vec<GF2>.  There are a few
    quirks to be aware of.  See the documentation file for vec_GF2 for more
    details.

    ---------------------
    RANGE-BASED FOR LOOPS
    ---------------------

    NTL's vectors provide sufficient functionality to enable range-based for-loops
    (in C++11).  The safest ways to use this functionality are to write:

       for (auto&& item : vec) { ... } // for read-only or read/write access

    or

       for (T item : vec) { ... } // for access via a copy

    This is especially true if vec may be of type Vec<GF2>.  Again, see the
    documentation file for vec_GF2 for more details.

    --------------
    OTHER FEATURES
    --------------

    The i-th vector element (counting from 0) is accessed as v[i].  If the macro
    NTL_RANGE_CHECK is defined, code is emitted to test if 0 <= i < v.length().
    This check is not performed by default.

    For old-time FORTRAN programmers, the i-th vector element (counting from 1) is
    accessed as v(i).

    -------------------------
    COMPARISON TO STL VECTORS
    -------------------------

    When the length of an NTL vector is reduced, no objects are destroyed.
    In contrast, when the length of an STL vector is reduced, objects are
    destroyed (effectively, maintaining the invariant len == init).

    If not using NTL's safe vector mode (see below), when the length of an NTL
    vector is increased, and the new value of len exceeds the current value of
    alloc, the underying array of objects is resized using realloc.  This implies
    that existing objects are moved using a bit-wise copy.  As mentioned above,
    this means that objects should be "relocatable", in the sense that they do not
    care what their actual address is.  Most reasonable objects satisfy this
    constraint.  An example of an object that does not is one that stores in one
    data member a pointer to another data member within the same object.

    In contrast, when the length of an STL vector is increased, a new array is
    allocated, and objects from the old array are copied to the new array, and then
    destroyed in the old array.  This obviously is much more expensive that NTL's
    strategy.  However, the new "move semantics", introduced in C++11, mitigate
    this issue somewhat.

    Because of NTL's relocatability requirement, it is not recommended to use NTL
    vectors over classes coming from the standard library, which may not satisfy
    the requirement.  In those cases, you could either use an STL vector, or use an
    NTL vector and wrap the suspect classes in an NTL smart pointer of some kind
    (e.g., SmartPtr or CopiedPtr).

    Note also that Facebook's open source "folly" library also provides a vector
    class that uses realloc in a manner very similar to NTL's vector class.  See
    https://github.com/facebook/folly/blob/master/folly/docs/FBVector.md

    See also http://open-std.org/JTC1/SC22/WG21/docs/papers/2016/p0023r0.pdf
    for more on this issue.

    ----------------
    SAFE VECTOR MODE
    ----------------

    As of NTL 10.4, you can configure NTL in "safe vector mode" by configuring NTL
    with NTL_SAFE_VECTORS=on.  If you do this, then by default, NTL will *not* use
    the above realloc strategy for resizing a vector.  Rather, it will use the same
    strategy used by STL vectors: using a nothrow move constructor if available,
    and a copy constructor otherwise.  However, one can also explicitly declare
    that a class T is relocatable.  The easiest way to do this is to put the
    following line of code immediately after the definition of class T:

       NTL_DECLARE_RELOCATABLE((T*))

    Notice the double parentheses and the "*" --- these are unfortunately
    necessary.  This macro expands as

       constexpr bool DeclareRelocatableType(T*) { return true; }

    Inside the class Vec<T>, this function is invoked with ((T*) 0) as the
    argument.  By declaring relocatability via a function definition in this way,
    the Vec<T> class will always find (via "Koenig lookup", or ADL) the declaration
    for T in the namespace in which T is defined (assuming you put the
    relocatability declaration there as well).

    You can also declare a template class to be relocatable, as in:

       template<class X, class Y>
       NTL_DECLARE_RELOCATABLE((T<X,Y>*))

    There is also a macro NTL_DECLARE_RELOCATABLE_WHEN, which leaves off the
    function body, if you want something other than { return true; }

    NTL also defines template functions:

       template<class T>
       constexpr bool DeclareRelocatableType(T*)
       { return ... ;  }

    where "..." is an expression that returns true if T is a simple type that is
    safe to relocate, as determined by C++11 type traits.  Such simple types
    include scalar types and simple classes with trivial copy constructor and
    destructor.

    Thus, even in "safe vector mode", NTL will continue to use the realloc strategy
    on such simple types.  NTL also declares most of the important classes in the
    library to be relocatable.  Some of these classes do not (in general) have
    nothrow move constructors, yet it is still safe to relocate via realloc in the
    context of NTL Vec's.

    Note that safe vector mode requires C++11 features.  When not in safe vector
    mode, you can still leave in the above declarations and things will be fine,
    even without C++11 (the constexpr will become an inline, to keep the compiler
    happy).

    The nice thing about safe vector mode is that you can freely use any type T to
    form a Vec<T>, without worrying about relocatability.  If there is a
    performance issue, you can consider declaring T to be relocatable.

    ----------------------
    HISTORICAL PERSPECTIVE
    ----------------------

    When I first started working on NTL around 1990, C++ was a much simpler
    language.  Among other things, there were no templates and no STL, so the Vec
    class started out (and remained, for many years) a set of macros (indeed, this
    approach was advocated in the first edition of Stroustrup's book).

    My thinking at that time was very much "C oriented".  The idea of resizing a
    vector without using realloc never occured to me, and all of the classes I
    wanted to put in vectors were relocatable.  Why would you ever bother making
    copies of vector elements and destroy the originals when you could just
    realloc?  It wasn't until many years later that I even realized this was
    somehow questionable practice.  Indeed, resizing via malloc is technically
    undefined behavior, but it's been working for me for over 25 years without
    problems.

    Furthermore, because of the poor performance of malloc in those days (which is
    much better now), I designed the Vec<ZZ_p> class (and a few others) to allocate
    the underlying ZZ's in blocks.  This not only reduces the number of calls to
    malloc, but it also gives better locality of reference.  Experiments in the
    last couple of years show that this is still benefit to doing this.

    With all of these design decisions baked into NTL's Vec class, transitioning to
    STL vectors would be problematic, and is unlikely to ever happen.  But as
    things evolve, NTL Vec's are starting to offer many of the same convenience and
    safety features as STL vector's.

    \**************************************************************************/

    // EXCEPTIONS: all functions below do not throw any exceptions,
    //   except as noted

    template<class T>
    class Vec {
    public:

       Vec();  // initially length 0

       Vec(const Vec& a);
       // copy constructor;  uses the assignment operator of T
       // for copying into locations that have already been initialized,
       // and uses the copy constructor for T for initializing new locations.

       // EXCEPTIONS: may throw

       Vec& operator=(const Vec& a);
       // assignment;  uses the assignment operator of T
       // for copying into locations that have already been initialized,
       // and uses the copy constructor for T for initializing new locations.

       // EXCEPTIONS: weak ES (but if it throws, neither length nor MaxLength
       //    will change, although some previously initialized elements
       //    may have been assigned new values).

       Vec(Vec&& a);
       // move constructor (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set
       // will revert to copy constructor if a is fixed

       Vec& operator=(Vec&& a);
       // move assignment (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set
       // will revert to copy assignment if *this or a is fixed

       // NOTES: If neither source nor destination are fixed, these operations will
       // be fast pointer moves and no exceptions will be thrown.  If either are
       // fixed (say, the row in a Mat<T>), these operations revert to copy
       // operations, which may either throw an exception (if NTL_EXCEPTIONS is on)
       // or abort the program (if NTL_EXCEPTIONS is off).

       ~Vec();
       // destructor: calls T's destructor for all initialized
       // elements in the vector, and then frees the vector itself

       void SetLength(long n);
       // set current length to n, growing vector if necessary
       // new objects are initialized using the default contructor for T

       // EXCEPTIONS: strong ES (but the vector may have been
       //    reallocated)

       void SetLength(long n, const T& a);
       // set current length to n, growing vector if necessary
       // new objects are initialized using the copy contructor for T

       // EXCEPTIONS: strong ES (but the vector may have been
       //    reallocated)

       template<class F>
       void SetLengthAndApply(long n, F f);
       // set current length to n, growing vector if necessary
       // any new objects are initialized using defauly constructor
       // for T, and after that, f is applied to each new object x
       // as f(x).

       // EXCEPTIONS: strong ES (but the vector may have been
       //    reallocated)

       long length() const;
       // current length

       T& operator[](long i);
       const T& operator[](long i) const;
       // indexing operation, starting from 0.
       // The first version is applied to non-const Vec<T>,
       // and returns a non-const reference to a T, while the second version
       // is applied to a const Vec<T> and returns a const reference to a T.

       // EXCEPTIONS: may throw if range checking turned on, strong ES

       T& operator()(long i);
       const T& operator()(long i) const;
       // indexing operation, starting from 1
       // The first version is applied to non-const Vec<T>,
       // and returns a non-const reference to a T, while the second version
       // is applied to a const Vec<T> and returns a const reference to a T.

       // EXCEPTIONS: may throw if range checking turned on, strong ES

       T* elts();
       const T* elts() const;
       // returns address of first vector element (or 0 if no space has been
       // allocated for this vector).  If a vector potentially has length 0, it is
       // safer to write v.elts() instead of &v[0]: the latter is not well defined
       // by the C++ standard (although this is likely an academic concern).
       //
       // The first version is applied to non-const Vec<T>, and returns a non-const
       // pointer to a T, while the second version is applied to a const Vec<T> and
       // returns a const reference to a T.

       void swap(Vec& y);
       // swap with y (fast: just swaps pointers)

       // EXCEPTIONS: throws if vectors are fixed and lengths do not match, strong ES

       void move(Vec& y);
       // move y into *this, killing y (fast: just moves pointers)

       // EXCEPTIONS: strong ES, raises an error if
       // &y != this and either y or *this are fixed

       void append(const T& a);
       // append a to end of vector; uses the assignment operator of T
       // for copying into locations that have already been initialized,
       // and uses the copy constructor for T for initializing new locations.

       // EXCEPTIONS: strong ES if initializing a new element (and in any
       //    case, if an exception throws, length and MaxLength remain
       //    unchanged).

       void append(const Vec& w);
       // append w to end of vector; uses the assignment operator of T
       // for copying into locations that have already been initialized,
       // and uses the copy constructor for T for initializing new locations.

       // EXCEPTIONS: strong ES if initializing new elements (and in any
       //    case, if an exception throws, length and MaxLength remain
       //    unchanged).

    // Alternative access interface

       const T& get(long i) const;
       // v.get(i) returns v[i]

       void put(long i, const T& a);
       // v.put(i, a) equivalent to v[i] = q

    // Some STL compatibility

       typedef T value_type;
       typedef value_type& reference;
       typedef const value_type& const_reference;
       typedef value_type *iterator;
       typedef const value_type *const_iterator;

       T* data();
       const T* data() const;
       // v.data() same as v.elts()

       T* begin();
       const T* begin() const;
       // v.begin() same as v.elts()

       T* end();
       const T* end() const;
       // pointer to (one past) last element (or NULL)

       T& at(long i);
       const T& at(long i) const;
       // indexing with range checking

    // the remaining member functions are a bit esoteric (skip on first
    // reading)

       Vec(INIT_SIZE_TYPE, long n);
       // Vec(INIT_SIZE, n) initializes vector with an intial length of n.
       // new objects are initialized using the default contructor for T

       // EXCEPTIONS: may throw

       Vec(INIT_SIZE_TYPE, long n, const T& a);
       // Vec(INIT_SIZE, n, a) initializes vector with an intial length of n.
       // new objects are initialized using the copy contructor for T

       // EXCEPTIONS: may throw

       void kill();
       // release space and set to length 0

       void SetMaxLength(long n);
       // allocates space and initializes up to n elements. Does not change
       // current length

       // EXCEPTIONS: may throw, strong ES

       void FixLength(long n);
       // sets length to n and prohibits all future length changes.
       // FixLength may only be invoked immediately after the default
       // construction or kill.

       // The kill operation is also subsequently prohibited, and swap is
       // allowed on fixed length vectors of the same length.

       // FixLength is provided mainly to implement Mat<T>, to enforce
       // the restriction that all rows have the same length.

       // EXCEPTIONS: may throw, strong ES

       void FixAtCurrentLength();
       // fixes the length at the cuurent length and prohibits
       // all future length changes.

       // It is required that length() == MaxLength() when called.

       // EXCEPTIONS: if length() != MaxLength() and error is raised;
       // if length() == 0, a memory allocation error may be raised.
       // Strong ES.

       long fixed() const;
       // test if length has been fixed by FixLength() or FixAtCurrentLength()

       long MaxLength() const;
       // maximum length, i.e., number of allocated and initialized elements

       long allocated() const;
       // the number of objects for which space has been allocated, but not
       // necessarily initialized;  this may be larger than MaxLength().

       T& RawGet(long i);
       const T& RawGet(long i) const;
       // indexing with no range checking

       long position(const T& a) const;
       // returns position of a in the vector, or -1 if it is not there.
       // The search is conducted from position 0 to allocated()-1 the vector,
       // and an error is raised if the object is found at position MaxLength()
       // or higher (in which case a references an uninitialized object).
       // Note that if NTL_CLEAN_PTR flag is set, this routine takes
       // linear time, and otherwise, it takes constant time.

       // EXCEPTIONS: may throw (as indicated above)

       long position1(const T& a) const;
       // returns position of a in the vector, or -1 if it is not there.
       // The search is conducted from position 0 to length()-1 of the vector.
       // Note that if NTL_CLEAN_PTR flag is set, this routine takes
       // linear time, and otherwise, it takes constant time.

    };

    /**************************************************************************\

                           Some utility routines

    \**************************************************************************/

    template<class T>
    void swap(Vec<T>& x, Vec<T>& y);
    // swaps x & y; same as x.swap(y)

    // EXCEPTIONS: same as for swap member function

    template<class T>
    void append(Vec<T>& v, const T& a);
    // appends a to the end of v; same as v.append(a)

    // EXCEPTIONS: same as for append member function

    template<class T>
    void append(Vec<T>& v, const Vec<T>& w);
    // appends w to the end of v; same as v.append(w)

    // EXCEPTIONS: same as for append member function

    /**************************************************************************\

                                 Input/Output

    The I/O format for a vector v with n elements is:

       [v[0] v[1] ... v[n-1]]

    Uses corresponding I/O operators for T

    \**************************************************************************/

    template<class T>
    istream& operator>>(istream&, Vec<T>&);

    // EXCEPTIONS: may throw, weak ES

    template<class T>
    ostream& operator<<(ostream&, const Vec<T>&);

    // EXCEPTIONS: may throw, weak ES

    /**************************************************************************\

                                  Equality Testing

    \**************************************************************************/

    template<class T>
    long operator==(const Vec<T>& a, const Vec<T>& b);

    template<class T>
    long operator!=(const Vec<T>& a, const Vec<T>& b);

    /**************************************************************************\

                      Customized Constructors and Destructors

    Esoteric: skip on first reading...also these interfaces are subject to change

    When new elements in a vector need to be constructed, one of the
    following routines is called:

       void BlockConstruct(T* p, long n);
       // invokes T() to initialize p[i] for i = 0..n-1

       void BlockConstructFromVec(T* p, long n, const T* q);
       // invokes T(q[i]) to initialize p[i] for i = 0..n-1;
       // q points to elements from a Vec<T>

       void BlockConstructFromObj(T* p, long n, const T& q);
       // invokes T(q) to initialize p[i] for i = 0..n-1

    When a vector is destroyed, the following routine is called:

       void BlockDestroy(T* p, long n);
       // invokes ~T() on p[i] for i = 0..n-1

    The default behavior of these routines may be modified by
    overloading these functions with a custom implementation.

    EXCEPTIONS:
    In order to provide exception safe code, the Construct routines
    should provide strong ES; in particular, if any constructor
    throws, all newly constructed objects should be destroyed.
    Moreover, the BlockDestroy routine should not throw at all.

    In NTL, these routines are overridden for the ZZ_p and GF2E classes,
    so that many vector entries will be packed into contiguous storage
    locations.  This reduces the number of invocations of malloc, and
    increases locality of reference.

    \**************************************************************************/