def _int_keys(self, data):
		t = self.cls()
		self.assertNode(t)
		self.assertIsNone(t._root)
		i = len(data)
		for j in range(i):
			self.assertEqual(j, len(t))
			self.assertNotIn(j, t)
			self.assertEqual(list(range(j)), list(t))
			t._set(j, data[j])
			self.assertNode(t)
			if j > 1:
				self.assertLess(t._root.height(), 2.0 * _log(j + 1, 2))
			else:
				self.assertEqual(t._root.height(), j + 1)
			self.assertEqual(data[j], t.get(j))
			self.assertIn(j, t)
		self.assertEqual(i, len(t))
		self.assertEqual(list(range(len(data))), list(t))
		self.assertIsNone(t.get(i + 1))
		# Deletion
		for j in range(i - 1, -1, -1):
			self.assertEqual(j + 1, len(t))
			self.assertIn(j, t)
			t._delete(j)
			self.assertNode(t)
			if j > 1:
				self.assertLessEqual(t._root.height(), 2.0 * _log(j, 2))
			elif j == 1:
				self.assertEqual(t._root.height(), j)
			else:
				self.assertIsNone(t._root)
			self.assertNotIn(j, t)
			self.assertEqual(list(range(j)), list(t))
		self.assertIsNone(t._root)

    def gammavariate(self, alpha, beta):
        """Gamma distribution.  Not the gamma function!
        
        Conditions on the parameters are alpha > 0 and beta > 0.
        
        The probability distribution function is:
        
                    x ** (alpha - 1) * math.exp(-x / beta)
          pdf(x) =  --------------------------------------
                      math.gamma(alpha) * beta ** alpha
        
        """
        if alpha <= 0.0 or beta <= 0.0:
            raise ValueError, "gammavariate: alpha and beta must be > 0.0"
        random = self.random
        if alpha > 1.0:
            ainv = _sqrt(2.0 * alpha - 1.0)
            bbb = alpha - LOG4
            ccc = alpha + ainv
            while 1:
                u1 = random()
                if not 1e-07 < u1 < 0.9999999:
                    continue
                u2 = 1.0 - random()
                v = _log(u1 / (1.0 - u1)) / ainv
                x = alpha * _exp(v)
                z = u1 * u1 * u2
                r = bbb + ccc * v - x
                if r + SG_MAGICCONST - 4.5 * z >= 0.0 or r >= _log(z):
                    return x * beta

        else:
            if alpha == 1.0:
                u = random()
                while u <= 1e-07:
                    u = random()

                return -_log(u) * beta
            while 1:
                u = random()
                b = (_e + alpha) / _e
                p = b * u
                if p <= 1.0:
                    x = p ** (1.0 / alpha)
                else:
                    x = -_log((b - p) / alpha)
                u1 = random()
                if p > 1.0:
                    if u1 <= x ** (alpha - 1.0):
                        break
                elif u1 <= _exp(-x):
                    break

            return x * beta

    def projCoordinate(self, point):
        """ Convert from Point object in EPSG:900913 to a Coordinate object
        """
        # the zoom at which we're dealing with meters on the ground
        diameter = 2 * _pi * 6378137
        zoom = _log(diameter) / _log(2)

        # global offsets
        coord = Coordinate(point.y, point.x, zoom)
        coord.column = coord.column + diameter/2
        coord.row = diameter/2 - coord.row
        
        return coord

    def stdgamma(self, alpha, ainv, bbb, ccc):
        # ainv = sqrt(2 * alpha - 1)
        # bbb = alpha - log(4)
        # ccc = alpha + ainv

        random = self.random
        if alpha <= 0.0:
            raise ValueError, 'stdgamma: alpha must be > 0.0'

        if alpha > 1.0:

            # Uses R.C.H. Cheng, "The generation of Gamma
            # variables with non-integral shape parameters",
            # Applied Statistics, (1977), 26, No. 1, p71-74

            while 1:
                u1 = random()
                u2 = random()
                v = _log(u1/(1.0-u1))/ainv
                x = alpha*_exp(v)
                z = u1*u1*u2
                r = bbb+ccc*v-x
                if r + SG_MAGICCONST - 4.5*z >= 0.0 or r >= _log(z):
                    return x

        elif alpha == 1.0:
            # expovariate(1)
            u = random()
            while u <= 1e-7:
                u = random()
            return -_log(u)

        else:   # alpha is between 0 and 1 (exclusive)

            # Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle

            while 1:
                u = random()
                b = (_e + alpha)/_e
                p = b*u
                if p <= 1.0:
                    x = pow(p, 1.0/alpha)
                else:
                    # p > 1
                    x = -_log((b-p)/alpha)
                u1 = random()
                if not (((p <= 1.0) and (u1 > _exp(-x))) or
                          ((p > 1)  and  (u1 > pow(x, alpha - 1.0)))):
                    break
            return x

    def coordinateProj(self, coord):
        """ Convert from Coordinate object to a Point object in EPSG:900913
        """
        # the zoom at which we're dealing with meters on the ground
        diameter = 2 * _pi * 6378137
        zoom = _log(diameter) / _log(2)
        coord = coord.zoomTo(zoom)
        
        # global offsets
        point = Point(coord.column, coord.row)
        point.x = point.x - diameter/2
        point.y = diameter/2 - point.y

        return point

	def sample(self, population, k, generator=None):
		# This function exactly parallels the code in Random.py.
		# Comments are therefore omitted, to save space.
		
		n = len(population)
		if not 0 <= k <= n:
			raise ValueError('sample larger than population')
		randrange = self.randrange
		result = [None] * k
		setsize = 21
		if k > 5:
			setsize += 4 ** _ceil(_log(k * 3, 4))
		if n <= setsize or hasattr(population, 'keys'):
			pool = list(population)
			for i in xrange(k):
				j = randrange(n-i, generator=generator)
				result[i] = pool[j]
				pool[j] = pool[n-i-1]
		else:
			try:
				selected = set()
				selected_add = selected.add
				for i in xrange(k):
					j = randrange(n, generator=generator)
					while j in selected:
						j = randrange(n, generator=generator)
					selected_add(j)
					result[i] = population[j]
			except (TypeError, KeyError):
				if isinstance(population, list):
					raise
				return self.sample(tuple(population), k, generator)
		return result

 def sample(self, population, k):
    """Chooses k unique random elements from a population sequence."""
    # https://github.com/python/cpython/blob/2.7/Lib/random.py#L275
    n = len(population)
    if not 0 <= k <= n:
       raise ValueError("sample larger than population")
    random = self.random
    _int = int
    result = [None] * k
    setsize = 21      # size of a small set minus size of an empty list
    if k > 5:
       setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
    if n <= setsize or hasattr(population, "keys"):
       # An n-length list is smaller than a k-length set, or this is a
       # mapping type so the other algorithm wouldn't work.
       pool = list(population)
       for i in xrange(k):       # invariant:  non-selected at [0,n-i)
          j = _int(random() * (n-i))
          result[i] = pool[j]
          pool[j] = pool[n-i-1]   # move non-selected item into vacancy
    else:
       try:
          selected = set()
          selected_add = selected.add
          for i in xrange(k):
             j = _int(random() * n)
             while j in selected:
                j = _int(random() * n)
             selected_add(j)
             result[i] = population[j]
       except (TypeError, KeyError):   # handle (at least) sets
          if isinstance(population, list):
             raise
          return self.sample(tuple(population), k)
    return result

def dup_revert(f, n, K):
    """
    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.

    This function computes first ``2**n`` terms of a polynomial that
    is a result of inversion of a polynomial modulo ``x**n``. This is
    useful to efficiently compute series expansion of ``1/f``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1

    >>> R.dup_revert(f, 8)
    61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1

    """
    g = [K.revert(dup_TC(f, K))]
    h = [K.one, K.zero, K.zero]

    N = int(_ceil(_log(n, 2)))

    for i in range(1, N + 1):
        a = dup_mul_ground(g, K(2), K)
        b = dup_mul(f, dup_sqr(g, K), K)
        g = dup_rem(dup_sub(a, b, K), h, K)
        h = dup_lshift(h, dup_degree(h), K)

    return g

 def sample(self, population, k):
     if isinstance(population, _Set):
         population = tuple(population)
     if not isinstance(population, _Sequence):
         raise TypeError('Population must be a sequence or set.  For dicts, use list(d).')
     randbelow = self._randbelow
     n = len(population)
     if not 0 <= k <= n:
         raise ValueError('Sample larger than population')
     result = [None]*k
     setsize = 21
     if k > 5:
         setsize += 4**_ceil(_log(k*3, 4))
     if n <= setsize:
         pool = list(population)
         for i in range(k):
             j = randbelow(n - i)
             result[i] = pool[j]
             pool[j] = pool[n - i - 1]
     else:
         selected = set()
         selected_add = selected.add
         for i in range(k):
             j = randbelow(n)
             while j in selected:
                 j = randbelow(n)
             selected_add(j)
             result[i] = population[j]
     return result

    def expovariate(self, lambd):

        """Exponential distribution.


        lambd is 1.0 divided by the desired mean.  It should be

        nonzero.  (The parameter would be called "lambda", but that is

        a reserved word in Python.)  Returned values range from 0 to

        positive infinity if lambd is positive, and from negative

        infinity to 0 if lambd is negative.


        """

        # lambd: rate lambd = 1/mean

        # ('lambda' is a Python reserved word)


        # we use 1-random() instead of random() to preclude the

        # possibility of taking the log of zero.

        return -_log(1.0 - self.random())/lambd

def integer_nthroot(y, n):
    """
    Return a tuple containing x = floor(y**(1/n))
    and a boolean indicating whether the result is exact (that is,
    whether x**n == y).

    >>> from sympy import integer_nthroot
    >>> integer_nthroot(16,2)
    (4, True)
    >>> integer_nthroot(26,2)
    (5, False)

    """
    y, n = int(y), int(n)
    if y < 0:
        raise ValueError("y must be nonnegative")
    if n < 1:
        raise ValueError("n must be positive")
    if y in (0, 1):
        return y, True
    if n == 1:
        return y, True
    if n == 2:
        x, rem = mpmath_sqrtrem(y)
        return int(x), not rem
    if n > y:
        return 1, False
    # Get initial estimate for Newton's method. Care must be taken to
    # avoid overflow
    try:
        guess = int(y**(1./n) + 0.5)
    except OverflowError:
        exp = _log(y, 2)/n
        if exp > 53:
            shift = int(exp - 53)
            guess = int(2.0**(exp - shift) + 1) << shift
        else:
            guess = int(2.0**exp)
    #print n
    if guess > 2**50:
        # Newton iteration
        xprev, x = -1, guess
        while 1:
            t = x**(n - 1)
            #xprev, x = x, x - (t*x-y)//(n*t)
            xprev, x = x, ((n - 1)*x + y//t)//n
            #print n, x-xprev, abs(x-xprev) < 2
            if abs(x - xprev) < 2:
                break
    else:
        x = guess
    # Compensate
    t = x**n
    while t < y:
        x += 1
        t = x**n
    while t > y:
        x -= 1
        t = x**n
    return x, t == y

    def _randbelow(self, n, _log=_log, _int=int, _maxwidth=1<<BPF,
                   _Method=_MethodType, _BuiltinMethod=_BuiltinMethodType):
        """Return a random int in the range [0,n)

        Handles the case where n has more bits than returned
        by a single call to the underlying generator.
        """

        try:
            getrandbits = self.getrandbits
        except AttributeError:
            pass
        else:
            # Only call self.getrandbits if the original random() builtin method
            # has not been overridden or if a new getrandbits() was supplied.
            # This assures that the two methods correspond.
            if type(self.random) is _BuiltinMethod or type(getrandbits) is _Method:
                k = _int(1.00001 + _log(n-1, 2.0))   # 2**k > n-1 > 2**(k-2)
                r = getrandbits(k)
                while r >= n:
                    r = getrandbits(k)
                return r
        if n >= _maxwidth:
            _warn("Underlying random() generator does not supply \n"
                "enough bits to choose from a population range this large")
        return _int(self.random() * n)

    def sample(self, population, k):
        """Chooses k unique random elements from a population sequence.

        Returns a new list containing elements from the population while
        leaving the original population unchanged.  The resulting list is
        in selection order so that all sub-slices will also be valid random
        samples.  This allows raffle winners (the sample) to be partitioned
        into grand prize and second place winners (the subslices).

        Members of the population need not be hashable or unique.  If the
        population contains repeats, then each occurrence is a possible
        selection in the sample.

        To choose a sample in a range of integers, use xrange as an argument.
        This is especially fast and space efficient for sampling from a
        large population:   sample(xrange(10000000), 60)
        """

        # Sampling without replacement entails tracking either potential
        # selections (the pool) in a list or previous selections in a set.

        # When the number of selections is small compared to the
        # population, then tracking selections is efficient, requiring
        # only a small set and an occasional reselection.  For
        # a larger number of selections, the pool tracking method is
        # preferred since the list takes less space than the
        # set and it doesn't suffer from frequent reselections.

        n = len(population)
        if not 0 <= k <= n:
            raise ValueError("sample larger than population")
        random = self.random
        _int = int
        result = [None] * k
        setsize = 21        # size of a small set minus size of an empty list
        if k > 5:
            setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
        if n <= setsize or hasattr(population, "keys"):
            # An n-length list is smaller than a k-length set, or this is a
            # mapping type so the other algorithm wouldn't work.
            pool = list(population)
            for i in xrange(k):         # invariant:  non-selected at [0,n-i)
                j = _int(random() * (n-i))
                result[i] = pool[j]
                pool[j] = pool[n-i-1]   # move non-selected item into vacancy
        else:
            try:
                selected = set()
                selected_add = selected.add
                for i in xrange(k):
                    j = _int(random() * n)
                    while j in selected:
                        j = _int(random() * n)
                    selected_add(j)
                    result[i] = population[j]
            except (TypeError, KeyError):   # handle (at least) sets
                if isinstance(population, list):
                    raise
                return self.sample(tuple(population), k)
        return result

    def gauss(self, mu, sigma):
        # """Gaussian distribution.
        # mu is the mean, and sigma is the standard deviation.  This is
        # slightly faster than the normalvariate() function.
        # Not thread-safe without a lock around calls.
        # """

        # When x and y are two variables from [0, 1), uniformly
        # distributed, then
        #
        #    cos(2*pi*x)*sqrt(-2*log(1-y))
        #    sin(2*pi*x)*sqrt(-2*log(1-y))
        #
        # are two *independent* variables with normal distribution
        # (mu = 0, sigma = 1).
        # (Lambert Meertens)
        # (corrected version; bug discovered by Mike Miller, fixed by LM)

        # Multithreading note: When two threads call this function
        # simultaneously, it is possible that they will receive the
        # same return value.  The window is very small though.  To
        # avoid this, you have to use a lock around all calls.  (I
        # didn't want to slow this down in the serial case by using a
        # lock here.)

        __random = self.random
        z = self.gauss_next
        self.gauss_next = None
        if z is None:
            x2pi = __random() * TWOPI
            g2rad = _sqrt(-2.0 * _log(1.0 - __random()))
            z = _cos(x2pi) * g2rad
            self.gauss_next = _sin(x2pi) * g2rad

        return mu + z*sigma

def dup_revert(f, n, K):
    """
    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.

    This function computes first ``2**n`` terms of a polynomial that
    is a result of inversion of a polynomial modulo ``x**n``. This is
    useful to efficiently compute series expansion of ``1/f``.

    Examples
    ========

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.densetools import dup_revert

    >>> f = [-QQ(1,720), QQ(0), QQ(1,24), QQ(0), -QQ(1,2), QQ(0), QQ(1)]

    >>> dup_revert(f, 8, QQ)
    [61/720, 0/1, 5/24, 0/1, 1/2, 0/1, 1/1]

    """
    g = [K.revert(dup_TC(f, K))]
    h = [K.one, K.zero, K.zero]

    N = int(_ceil(_log(n, 2)))

    for i in xrange(1, N + 1):
        a = dup_mul_ground(g, K(2), K)
        b = dup_mul(f, dup_sqr(g, K), K)
        g = dup_rem(dup_sub(a, b, K), h, K)
        h = dup_lshift(h, dup_degree(h), K)

    return g

    def expovariate(self, lambd):
        random = self.random
        u = random()
        while u <= 1e-07:
            u = random()

        return -_log(u) / lambd

    def sample(self, population, k):
        n = len(population)
        if not 0 <= k <= n:
            raise ValueError, 'sample larger than population'
        random = self.random
        _int = int
        result = [None] * k
        setsize = 21
        if k > 5:
            setsize += 4 ** _ceil(_log(k * 3, 4))
        if n <= setsize or hasattr(population, 'keys'):
            pool = list(population)
            for i in xrange(k):
                j = _int(random() * (n - i))
                result[i] = pool[j]
                pool[j] = pool[n - i - 1]

        else:
            try:
                selected = set()
                selected_add = selected.add
                for i in xrange(k):
                    j = _int(random() * n)
                    while j in selected:
                        j = _int(random() * n)

                    selected_add(j)
                    result[i] = population[j]

            except (TypeError, KeyError):
                if isinstance(population, list):
                    raise
                return self.sample(tuple(population), k)

        return result

    def sample(self, population: Iterable[T], k: int) -> List[T]:
        """Chooses k unique random elements from a population sequence or set.

        Returns a new list containing elements from the population while
        leaving the original population unchanged.  The resulting list is
        in selection order so that all sub-slices will also be valid random
        samples.  This allows raffle winners (the sample) to be partitioned
        into grand prize and second place winners (the subslices).

        Members of the population need not be hashable or unique.  If the
        population contains repeats, then each occurrence is a possible
        selection in the sample.

        To choose a sample in a range of integers, use range as an argument.
        This is especially fast and space efficient for sampling from a
        large population:   sample(range(10000000), 60)
        """

        # Sampling without replacement entails tracking either potential
        # selections (the pool) in a list or previous selections in a set.

        # When the number of selections is small compared to the
        # population, then tracking selections is efficient, requiring
        # only a small set and an occasional reselection.  For
        # a larger number of selections, the pool tracking method is
        # preferred since the list takes less space than the
        # set and it doesn't suffer from frequent reselections.

        if isinstance(population, _Sequence):
            populationseq = population
        elif isinstance(population, _Set):
            populationseq = list(population)
        else:
            raise TypeError("Population must be a sequence or set.  For dicts, use list(d).")
        randbelow = self._randbelow
        n = len(populationseq)
        if not (0 <= k and k <= n):
            raise ValueError("Sample larger than population")
        result = [cast(T, None)] * k
        setsize = 21        # size of a small set minus size of an empty list
        if k > 5:
            setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
        if n <= setsize:
            # An n-length list is smaller than a k-length set
            pool = list(populationseq)
            for i in range(k):         # invariant:  non-selected at [0,n-i)
                j = randbelow(n-i)
                result[i] = pool[j]
                pool[j] = pool[n-i-1]   # move non-selected item into vacancy
        else:
            selected = Set[int]()
            selected_add = selected.add
            for i in range(k):
                j = randbelow(n)
                while j in selected:
                    j = randbelow(n)
                selected_add(j)
                result[i] = populationseq[j]
        return result

    def weibullvariate(self, alpha, beta):
        # """Weibull distribution.
        # alpha is the scale parameter and beta is the shape parameter.
        # """
        # Jain, pg. 499; bug fix courtesy Bill Arms

        u = 1.0 - self.random()
        return alpha * pow(-_log(u), 1.0/beta)

 def weibullvariate(self, alpha, beta):
     """Weibull distribution.
     
     alpha is the scale parameter and beta is the shape parameter.
     
     """
     u = 1.0 - self.random()
     return alpha * pow(-_log(u), 1.0 / beta)