def solve(n, k):
tree = FastRBTree()
tree.insert(n, 1)
ls = rs = n
for i in range(k):
key, val = tree.max_item()
tree.remove(key)
if val > 1:
tree.insert(key, val - 1)
if key % 2 == 1:
key //= 2
ls = rs = key
update_tree(tree, key)
update_tree(tree, key)
else:
key //= 2
ls = key
rs = key - 1
update_tree(tree, ls)
update_tree(tree, rs)
return str(ls) + " " + str(rs)
class Tdigest(object):
def __init__(self, delta=0.01, K=25, CX=1.1):
self.delta = delta
self.K = K
self.CX = CX
self.centroids = RBTree()
self.nreset = 0
self.reset()
def reset(self):
self.centroids.clear()
self.n = 0
self.nreset += 1
self.last_cumulate = 0
self.compressing = False
def push(self, x, n=1):
if not isinstance(x, list):
x = [x]
for item in x:
self._digest(item, n)
def percentile(self, p):
if self.size() == 0:
return None
self._cumulate(True)
cumn = self.n * p
lower = self.centroids.min_item()[1]
upper = self.centroids.max_item()[1]
for c in self.centroids.values():
if c.cumn <= cumn:
lower = c
else:
upper = c
break
if lower == upper:
return lower.mean
return lower.mean + (cumn - lower.cumn) * (upper.mean - lower.mean) / \
(upper.cumn - lower.cumn)
def serialize(self):
result = '%s~%s~%s~' % (self.delta, self.K, self.size())
if self.size() == 0:
return result
self._cumulate(True)
means = []
counts = []
for c in self.centroids.values():
means.append(str(c.mean))
counts.append(str(c.n))
return '%s%s~%s' % (result, '~'.join(means), '~'.join(counts))
@classmethod
def deserialize(cls, serialized_str):
if not isinstance(serialized_str, basestring):
raise Exception(u'serialized_str must be str')
data = serialized_str.split('~')
t = Tdigest(delta=float(data[0]), K=int(data[1]))
size = int(data[2])
for i in xrange(size):
t.push(float(data[i + 3]), int(data[size + i + 3]))
t._cumulate(True)
return t
def _digest(self, x, n):
if self.size() == 0:
self._new_centroid(x, n, 0)
else:
_min = self.centroids.min_item()[1]
_max = self.centroids.max_item()[1]
nearest = self.find_nearest(x)
if nearest and nearest.mean == x:
self._addweight(nearest, x, n)
elif nearest == _min:
self._new_centroid(x, n, 0)
elif nearest == _max:
self._new_centroid(x, n, self.n)
else:
p = (nearest.cumn + nearest.n / 2.0) / self.n
max_n = int(4 * self.n * self.delta * p * (1 - p))
if max_n >= nearest.n + n:
self._addweight(nearest, x, n)
else:
self._new_centroid(x, n, nearest.cumn)
self._cumulate(False)
if self.K and self.size() > self.K / self.delta:
self.compress()
def find_nearest(self, x):
if self.size() == 0:
return None
try:
lower = self.centroids.ceiling_item(x)[1]
except KeyError:
lower = None
if lower and lower.mean == x:
return lower
try:
prev = self.centroids.floor_item(x)[1]
except KeyError:
prev = None
if not lower:
return prev
if not prev:
return lower
if abs(prev.mean - x) < abs(lower.mean - x):
return prev
else:
return lower
def size(self):
return len(self.centroids)
def compress(self):
if self.compressing:
return
points = self.toList()
self.reset()
self.compressing = True
for point in sorted(points, key=lambda x: random()):
self.push(point['mean'], point['n'])
self._cumulate(True)
self.compressing = False
def _cumulate(self, exact):
if self.n == self.last_cumulate:
return
if not exact and self.CX and self.last_cumulate and \
self.CX > (self.n / self.last_cumulate):
return
cumn = 0
for c in self.centroids.values():
cumn = c.cumn = cumn + c.n
self.n = self.last_cumulate = cumn
def toList(self):
return [dict(mean=c.mean, n=c.n, cumn=c.cumn) for
c in self.centroids.values()]
def _addweight(self, nearest, x, n):
if x != nearest.mean:
nearest.mean += n * (x - nearest.mean) / (nearest.n + n)
nearest.cumn += n
nearest.n += n
self.n += n
def _new_centroid(self, x, n, cumn):
c = Centroid(x, n, cumn)
self.centroids.insert(x, c)
self.n += n
return c
class TDigest(object):
def __init__(self, delta=0.01, K=25):
self.C = RBTree()
self.n = 0
self.delta = delta
self.K = K
def __add__(self, other_digest):
C1 = list(self.C.values())
C2 = list(other_digest.C.values())
shuffle(C1)
shuffle(C2)
data = C1 + C2
new_digest = TDigest(self.delta, self.K)
for c in data:
new_digest.update(c.mean, c.count)
return new_digest
def __len__(self):
return len(self.C)
def __repr__(self):
return """<T-Digest: n=%d, centroids=%d>""" % (self.n, len(self))
def _add_centroid(self, centroid):
if centroid.mean not in self.C:
self.C.insert(centroid.mean, centroid)
else:
self.C[centroid.mean].update(centroid.mean, centroid.count)
def _compute_centroid_quantile(self, centroid):
denom = self.n
cumulative_sum = sum(
c_i.count for c_i in self.C.value_slice(-float('Inf'), centroid.mean))
return (centroid.count / 2. + cumulative_sum) / denom
def _update_centroid(self, centroid, x, w):
self.C.pop(centroid.mean)
centroid.update(x, w)
self._add_centroid(centroid)
def _find_closest_centroids(self, x):
try:
ceil_key = self.C.ceiling_key(x)
except KeyError:
floor_key = self.C.floor_key(x)
return [self.C[floor_key]]
try:
floor_key = self.C.floor_key(x)
except KeyError:
ceil_key = self.C.ceiling_key(x)
return [self.C[ceil_key]]
if abs(floor_key - x) < abs(ceil_key - x):
return [self.C[floor_key]]
elif abs(floor_key - x) == abs(ceil_key - x) and (ceil_key != floor_key):
return [self.C[ceil_key], self.C[floor_key]]
else:
return [self.C[ceil_key]]
def _theshold(self, q):
return 4 * self.n * self.delta * q * (1 - q)
def update(self, x, w=1):
"""
Update the t-digest with value x and weight w.
"""
self.n += w
if len(self) == 0:
self._add_centroid(Centroid(x, w))
return
S = self._find_closest_centroids(x)
while len(S) != 0 and w > 0:
j = choice(list(range(len(S))))
c_j = S[j]
q = self._compute_centroid_quantile(c_j)
# This filters the out centroids that do not satisfy the second part
# of the definition of S. See original paper by Dunning.
if c_j.count + w > self._theshold(q):
S.pop(j)
continue
delta_w = min(self._theshold(q) - c_j.count, w)
self._update_centroid(c_j, x, delta_w)
w -= delta_w
S.pop(j)
if w > 0:
self._add_centroid(Centroid(x, w))
if len(self) > self.K / self.delta:
self.compress()
return
def batch_update(self, values, w=1):
"""
Update the t-digest with an iterable of values. This assumes all points have the
same weight.
"""
for x in values:
self.update(x, w)
self.compress()
return
def compress(self):
T = TDigest(self.delta, self.K)
C = list(self.C.values())
shuffle(C)
for c_i in C:
T.update(c_i.mean, c_i.count)
self.C = T.C
def percentile(self, q):
"""
Computes the percentile of a specific value in [0,1], ie. computes F^{-1}(q) where F^{-1} denotes
the inverse CDF of the distribution.
"""
if not (0 <= q <= 1):
raise ValueError("q must be between 0 and 1, inclusive.")
t = 0
q *= self.n
for i, key in enumerate(self.C.keys()):
c_i = self.C[key]
k = c_i.count
if q < t + k:
if i == 0:
return c_i.mean
elif i == len(self) - 1:
return c_i.mean
else:
delta = (self.C.succ_item(key)[1].mean - self.C.prev_item(key)[1].mean) / 2.
return c_i.mean + ((q - t) / k - 0.5) * delta
t += k
return self.C.max_item()[1].mean
def quantile(self, q):
"""
Computes the quantile of a specific value, ie. computes F(q) where F denotes
the CDF of the distribution.
"""
t = 0
N = float(self.n)
for i, key in enumerate(self.C.keys()):
c_i = self.C[key]
if i == len(self) - 1:
delta = (c_i.mean - self.C.prev_item(key)[1].mean) / 2.
else:
delta = (self.C.succ_item(key)[1].mean - c_i.mean) / 2.
z = max(-1, (q - c_i.mean) / delta)
if z < 1:
return t / N + c_i.count / N * (z + 1) / 2
t += c_i.count
return 1
def trimmed_mean(self, q1, q2):
"""
Computes the mean of the distribution between the two percentiles q1 and q2.
This is a modified algorithm than the one presented in the original t-Digest paper.
"""
if not (q1 < q2):
raise ValueError("q must be between 0 and 1, inclusive.")
s = k = t = 0
q1 *= self.n
q2 *= self.n
for i, key in enumerate(self.C.keys()):
c_i = self.C[key]
k_i = c_i.count
if q1 < t + k_i:
if i == 0:
delta = self.C.succ_item(key)[1].mean - c_i.mean
elif i == len(self) - 1:
delta = c_i.mean - self.C.prev_item(key)[1].mean
else:
delta = (self.C.succ_item(key)[1].mean - self.C.prev_item(key)[1].mean) / 2.
nu = ((q1 - t) / k_i - 0.5) * delta
s += nu * k_i * c_i.mean
k += nu * k_i
if q2 < t + k_i:
return s/k
t += k_i
return s/k
from bintrees import FastRBTree
T = int(sys.stdin.readline().strip())
def split(n):
n -= 1
return n // 2, n // 2 + (n % 2)
assert (0, 0) == split(1)
assert (0, 1) == split(2)
assert (1, 1) == split(3)
for t in range(1, T + 1):
N, K = map(int, sys.stdin.readline().strip().split())
holes = FastRBTree({N: 1})
while K > 0:
size, count = holes.max_item()
del holes[size]
for s in split(size):
holes[s] = holes.get(s, 0) + count
K -= count
minD, maxD = split(size)
print("Case #%d: %d %d" % (t, maxD, minD))
class SparseArray(object):
def __init__(self):
self.tree = FastRBTree()
def __len__(self):
try:
k, v = self.tree.max_item()
except KeyError:
return 0
return k + len(v)
def __getitem__(self, ndx):
try:
base, chunk = self.tree.floor_item(ndx)
except KeyError:
return None
offset = ndx - base
if offset < len(chunk):
return chunk[offset]
else:
return None
def __setitem__(self, ndx, item):
try:
base, chunk = self.tree.floor_item(ndx)
except KeyError:
try:
base, chunk = self.tree.ceiling_item(ndx)
except KeyError:
self.tree[ndx] = [item]
return
if ndx + 1 == base:
chunk.insert(0, item)
del self.tree[base]
self.tree[ndx] = chunk
return
if base > ndx:
self.tree[ndx] = [item]
return
offset = ndx - base
if offset < len(chunk):
chunk[offset] = item
else:
nextbase, nextchunk = (None, None)
try:
nextbase, nextchunk = self.tree.succ_item(base)
except KeyError:
pass
if offset == len(chunk):
chunk.append(item)
if offset + 1 == nextbase:
chunk += nextchunk
del self.tree[nextbase]
elif offset + 1 == nextbase:
nextchunk.insert(0, item)
del self.tree[nextbase]
self.tree[ndx] = nextchunk
else:
self.tree[ndx] = [item]
def __delitem__(self, ndx):
base, chunk = self.tree.floor_item(ndx)
offset = ndx - base
if offset < len(chunk):
before = chunk[:offset]
after = chunk[offset + 1:]
if len(before):
self.tree[base] = before
else:
del self.tree[base]
if len(after):
self.tree[ndx + 1] = after
def items(self):
for k, vs in self.tree.items():
for n, v in enumerate(vs):
yield (k + n, v)
def runs(self):
return self.tree.items()
def run_count(self):
return len(self.tree)
def __repr__(self):
arep = []
for k, v in self.tree.items():
arep.append('[%r]=%s' % (k, ', '.join([repr(item) for item in v])))
return 'SparseArray(%s)' % ', '.join(arep)
class ExclusiveRangeDict(object):
"""A class like dict whose key is a range [begin, end) of integers.
It has an attribute for each range of integers, for example:
[10, 20) => Attribute(0),
[20, 40) => Attribute(1),
[40, 50) => Attribute(2),
...
An instance of this class is accessed only via iter_range(begin, end).
The instance is accessed as follows:
1) If the given range [begin, end) is not covered by the instance,
the range is newly created and iterated.
2) If the given range [begin, end) exactly covers ranges in the instance,
the ranges are iterated.
(See test_set() in tests/range_dict_tests.py.)
3) If the given range [begin, end) starts at and/or ends at a mid-point of
an existing range, the existing range is split by the given range, and
ranges in the given range are iterated. For example, consider a case that
[25, 45) is given to an instance of [20, 30), [30, 40), [40, 50). In this
case, [20, 30) is split into [20, 25) and [25, 30), and [40, 50) into
[40, 45) and [45, 50). Then, [25, 30), [30, 40), [40, 45) are iterated.
(See test_split() in tests/range_dict_tests.py.)
4) If the given range [begin, end) includes non-existing ranges in an
instance, the gaps are filled with new ranges, and all ranges are iterated.
For example, consider a case that [25, 50) is given to an instance of
[30, 35) and [40, 45). In this case, [25, 30), [35, 40) and [45, 50) are
created in the instance, and then [25, 30), [30, 35), [35, 40), [40, 45)
and [45, 50) are iterated.
(See test_fill() in tests/range_dict_tests.py.)
"""
class RangeAttribute(object):
def __init__(self):
pass
def __str__(self):
return '<RangeAttribute>'
def __repr__(self):
return '<RangeAttribute>'
def copy(self): # pylint: disable=R0201
return ExclusiveRangeDict.RangeAttribute()
def __init__(self, attr=RangeAttribute):
self._tree = FastRBTree()
self._attr = attr
def iter_range(self, begin=None, end=None):
if not begin:
begin = self._tree.min_key()
if not end:
end = self._tree.max_item()[1][0]
# Assume that self._tree has at least one element.
if self._tree.is_empty():
self._tree[begin] = (end, self._attr())
# Create a beginning range (border)
try:
bound_begin, bound_value = self._tree.floor_item(begin)
bound_end = bound_value[0]
if begin >= bound_end:
# Create a blank range.
try:
new_end, _ = self._tree.succ_item(bound_begin)
except KeyError:
new_end = end
self._tree[begin] = (min(end, new_end), self._attr())
elif bound_begin < begin and begin < bound_end:
# Split the existing range.
new_end = bound_value[0]
new_value = bound_value[1]
self._tree[bound_begin] = (begin, new_value.copy())
self._tree[begin] = (new_end, new_value.copy())
else: # bound_begin == begin
# Do nothing (just saying it clearly since this part is confusing)
pass
except KeyError: # begin is less than the smallest element.
# Create a blank range.
# Note that we can assume self._tree has at least one element.
self._tree[begin] = (min(end, self._tree.min_key()), self._attr())
# Create an ending range (border)
try:
bound_begin, bound_value = self._tree.floor_item(end)
bound_end = bound_value[0]
if end > bound_end:
# Create a blank range.
new_begin = bound_end
self._tree[new_begin] = (end, self._attr())
elif bound_begin < end and end < bound_end:
# Split the existing range.
new_end = bound_value[0]
new_value = bound_value[1]
self._tree[bound_begin] = (end, new_value.copy())
self._tree[end] = (new_end, new_value.copy())
else: # bound_begin == begin
# Do nothing (just saying it clearly since this part is confusing)
pass
except KeyError: # end is less than the smallest element.
# It must not happen. A blank range [begin,end) has already been created
# even if [begin,end) is less than the smallest range.
# Do nothing (just saying it clearly since this part is confusing)
raise
missing_ranges = []
prev_end = None
for range_begin, range_value in self._tree.itemslice(begin, end):
range_end = range_value[0]
# Note that we can assume that we have a range beginning with |begin|
# and a range ending with |end| (they may be the same range).
if prev_end and prev_end != range_begin:
missing_ranges.append((prev_end, range_begin))
prev_end = range_end
for missing_begin, missing_end in missing_ranges:
self._tree[missing_begin] = (missing_end, self._attr())
for range_begin, range_value in self._tree.itemslice(begin, end):
yield range_begin, range_value[0], range_value[1]
def __str__(self):
return str(self._tree)
class ExclusiveRangeDict(object):
"""A class like dict whose key is a range [begin, end) of integers.
It has an attribute for each range of integers, for example:
[10, 20) => Attribute(0),
[20, 40) => Attribute(1),
[40, 50) => Attribute(2),
...
An instance of this class is accessed only via iter_range(begin, end).
The instance is accessed as follows:
1) If the given range [begin, end) is not covered by the instance,
the range is newly created and iterated.
2) If the given range [begin, end) exactly covers ranges in the instance,
the ranges are iterated.
(See test_set() in tests/range_dict_tests.py.)
3) If the given range [begin, end) starts at and/or ends at a mid-point of
an existing range, the existing range is split by the given range, and
ranges in the given range are iterated. For example, consider a case that
[25, 45) is given to an instance of [20, 30), [30, 40), [40, 50). In this
case, [20, 30) is split into [20, 25) and [25, 30), and [40, 50) into
[40, 45) and [45, 50). Then, [25, 30), [30, 40), [40, 45) are iterated.
(See test_split() in tests/range_dict_tests.py.)
4) If the given range [begin, end) includes non-existing ranges in an
instance, the gaps are filled with new ranges, and all ranges are iterated.
For example, consider a case that [25, 50) is given to an instance of
[30, 35) and [40, 45). In this case, [25, 30), [35, 40) and [45, 50) are
created in the instance, and then [25, 30), [30, 35), [35, 40), [40, 45)
and [45, 50) are iterated.
(See test_fill() in tests/range_dict_tests.py.)
"""
class RangeAttribute(object):
def __init__(self):
pass
def __str__(self):
return '<RangeAttribute>'
def __repr__(self):
return '<RangeAttribute>'
def copy(self): # pylint: disable=R0201
return ExclusiveRangeDict.RangeAttribute()
def __init__(self, attr=RangeAttribute):
self._tree = FastRBTree()
self._attr = attr
def iter_range(self, begin=None, end=None):
if not begin:
begin = self._tree.min_key()
if not end:
end = self._tree.max_item()[1][0]
# Assume that self._tree has at least one element.
if self._tree.is_empty():
self._tree[begin] = (end, self._attr())
# Create a beginning range (border)
try:
bound_begin, bound_value = self._tree.floor_item(begin)
bound_end = bound_value[0]
if begin >= bound_end:
# Create a blank range.
try:
new_end, _ = self._tree.succ_item(bound_begin)
except KeyError:
new_end = end
self._tree[begin] = (min(end, new_end), self._attr())
elif bound_begin < begin and begin < bound_end:
# Split the existing range.
new_end = bound_value[0]
new_value = bound_value[1]
self._tree[bound_begin] = (begin, new_value.copy())
self._tree[begin] = (new_end, new_value.copy())
else: # bound_begin == begin
# Do nothing (just saying it clearly since this part is confusing)
pass
except KeyError: # begin is less than the smallest element.
# Create a blank range.
# Note that we can assume self._tree has at least one element.
self._tree[begin] = (min(end, self._tree.min_key()), self._attr())
# Create an ending range (border)
try:
bound_begin, bound_value = self._tree.floor_item(end)
bound_end = bound_value[0]
if end > bound_end:
# Create a blank range.
new_begin = bound_end
self._tree[new_begin] = (end, self._attr())
elif bound_begin < end and end < bound_end:
# Split the existing range.
new_end = bound_value[0]
new_value = bound_value[1]
self._tree[bound_begin] = (end, new_value.copy())
self._tree[end] = (new_end, new_value.copy())
else: # bound_begin == begin
# Do nothing (just saying it clearly since this part is confusing)
pass
except KeyError: # end is less than the smallest element.
# It must not happen. A blank range [begin,end) has already been created
# even if [begin,end) is less than the smallest range.
# Do nothing (just saying it clearly since this part is confusing)
raise
missing_ranges = []
prev_end = None
for range_begin, range_value in self._tree.itemslice(begin, end):
range_end = range_value[0]
# Note that we can assume that we have a range beginning with |begin|
# and a range ending with |end| (they may be the same range).
if prev_end and prev_end != range_begin:
missing_ranges.append((prev_end, range_begin))
prev_end = range_end
for missing_begin, missing_end in missing_ranges:
self._tree[missing_begin] = (missing_end, self._attr())
for range_begin, range_value in self._tree.itemslice(begin, end):
yield range_begin, range_value[0], range_value[1]
def __str__(self):
return str(self._tree)
class TDigest(object):
def __init__(self, delta=0.01, K=25):
self.C = RBTree()
self.n = 0
self.delta = delta
self.K = K
def __add__(self, other_digest):
data = list(chain(self.C.values(), other_digest.C.values()))
new_digest = TDigest(self.delta, self.K)
if len(data) > 0:
for c in pyudorandom.items(data):
new_digest.update(c.mean, c.count)
return new_digest
def __len__(self):
return len(self.C)
def __repr__(self):
return """<T-Digest: n=%d, centroids=%d>""" % (self.n, len(self))
def _add_centroid(self, centroid):
if centroid.mean not in self.C:
self.C.insert(centroid.mean, centroid)
else:
self.C[centroid.mean].update(centroid.mean, centroid.count)
def _compute_centroid_quantile(self, centroid):
denom = self.n
cumulative_sum = sum(
c_i.count
for c_i in self.C.value_slice(-float('Inf'), centroid.mean))
return (centroid.count / 2. + cumulative_sum) / denom
def _update_centroid(self, centroid, x, w):
self.C.pop(centroid.mean)
centroid.update(x, w)
self._add_centroid(centroid)
def _find_closest_centroids(self, x):
try:
ceil_key = self.C.ceiling_key(x)
except KeyError:
floor_key = self.C.floor_key(x)
return [self.C[floor_key]]
try:
floor_key = self.C.floor_key(x)
except KeyError:
ceil_key = self.C.ceiling_key(x)
return [self.C[ceil_key]]
if abs(floor_key - x) < abs(ceil_key - x):
return [self.C[floor_key]]
elif abs(floor_key - x) == abs(ceil_key -
x) and (ceil_key != floor_key):
return [self.C[ceil_key], self.C[floor_key]]
else:
return [self.C[ceil_key]]
def _theshold(self, q):
return 4 * self.n * self.delta * q * (1 - q)
def update(self, x, w=1):
"""
Update the t-digest with value x and weight w.
"""
self.n += w
if len(self) == 0:
self._add_centroid(Centroid(x, w))
return
S = self._find_closest_centroids(x)
while len(S) != 0 and w > 0:
j = choice(list(range(len(S))))
c_j = S[j]
q = self._compute_centroid_quantile(c_j)
# This filters the out centroids that do not satisfy the second part
# of the definition of S. See original paper by Dunning.
if c_j.count + w > self._theshold(q):
S.pop(j)
continue
delta_w = min(self._theshold(q) - c_j.count, w)
self._update_centroid(c_j, x, delta_w)
w -= delta_w
S.pop(j)
if w > 0:
self._add_centroid(Centroid(x, w))
if len(self) > self.K / self.delta:
self.compress()
return
def batch_update(self, values, w=1):
"""
Update the t-digest with an iterable of values. This assumes all points have the
same weight.
"""
for x in values:
self.update(x, w)
self.compress()
return
def compress(self):
T = TDigest(self.delta, self.K)
C = list(self.C.values())
for c_i in pyudorandom.items(C):
T.update(c_i.mean, c_i.count)
self.C = T.C
def percentile(self, p):
"""
Computes the percentile of a specific value in [0,100].
"""
if not (0 <= p <= 100):
raise ValueError("p must be between 0 and 100, inclusive.")
t = 0
p = float(p) / 100.
p *= self.n
for i, key in enumerate(self.C.keys()):
c_i = self.C[key]
k = c_i.count
if p < t + k:
if i == 0:
return c_i.mean
elif i == len(self) - 1:
return c_i.mean
else:
delta = (self.C.succ_item(key)[1].mean -
self.C.prev_item(key)[1].mean) / 2.
return c_i.mean + ((p - t) / k - 0.5) * delta
t += k
return self.C.max_item()[1].mean
def quantile(self, q):
"""
Computes the quantile of a specific value, ie. computes F(q) where F denotes
the CDF of the distribution.
"""
t = 0
N = float(self.n)
for i, key in enumerate(self.C.keys()):
c_i = self.C[key]
if i == len(self) - 1:
delta = (c_i.mean - self.C.prev_item(key)[1].mean) / 2.
else:
delta = (self.C.succ_item(key)[1].mean - c_i.mean) / 2.
z = max(-1, (q - c_i.mean) / delta)
if z < 1:
return t / N + c_i.count / N * (z + 1) / 2
t += c_i.count
return 1
def trimmed_mean(self, p1, p2):
"""
Computes the mean of the distribution between the two percentiles p1 and p2.
This is a modified algorithm than the one presented in the original t-Digest paper.
"""
if not (p1 < p2):
raise ValueError("p1 must be between 0 and 100 and less than p2.")
s = k = t = 0
p1 /= 100.
p2 /= 100.
p1 *= self.n
p2 *= self.n
for i, key in enumerate(self.C.keys()):
c_i = self.C[key]
k_i = c_i.count
if p1 < t + k_i:
if i == 0:
delta = self.C.succ_item(key)[1].mean - c_i.mean
elif i == len(self) - 1:
delta = c_i.mean - self.C.prev_item(key)[1].mean
else:
delta = (self.C.succ_item(key)[1].mean -
self.C.prev_item(key)[1].mean) / 2.
nu = ((p1 - t) / k_i - 0.5) * delta
s += nu * k_i * c_i.mean
k += nu * k_i
if p2 < t + k_i:
return s / k
t += k_i
return s / k
