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

    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 delault constructor for T to be called to initialize these
    elements.

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

    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).

    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.

    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.

    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.

    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

    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).

    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 malloc.  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, an 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

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

    // 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<T>& 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<T>& 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();
       // 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<T>& y);
       // swap with y (fast: just swaps pointers)

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

       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<T>& 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 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.

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