#
# Licensed to the Apache Software Foundation (ASF) under one or more
# contributor license agreements. See the NOTICE file distributed with
# this work for additional information regarding copyright ownership.
# The ASF licenses this file to You under the Apache License, Version 2.0
# (the "License"); you may not use this file except in compliance with
# the License. You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#
"""This module has all statistic related transforms.
This ApproximateUnique class will be deprecated [1]. PLease look into using
HLLCount in the zetasketch extension module [2].
[1] https://lists.apache.org/thread.html/501605df5027567099b81f18c080469661fb426
4a002615fa1510502%40%3Cdev.beam.apache.org%3E
[2] https://beam.apache.org/releases/javadoc/2.16.0/org/apache/beam/sdk/extensio
ns/zetasketch/HllCount.html
"""
# pytype: skip-file
from __future__ import absolute_import
from __future__ import division
import hashlib
import heapq
import itertools
import logging
import math
import sys
import typing
from builtins import round
from apache_beam import coders
from apache_beam import typehints
from apache_beam.transforms.core import *
from apache_beam.transforms.display import DisplayDataItem
from apache_beam.transforms.ptransform import PTransform
__all__ = [
'ApproximateQuantiles',
'ApproximateUnique',
]
# Type variables
T = typing.TypeVar('T')
K = typing.TypeVar('K')
V = typing.TypeVar('V')
def _get_default_hash_fn():
"""Returns either murmurhash or md5 based on installation."""
try:
import mmh3 # pylint: disable=import-error
def _mmh3_hash(value):
# mmh3.hash64 returns two 64-bit unsigned integers
return mmh3.hash64(value, seed=0, signed=False)[0]
return _mmh3_hash
except ImportError:
logging.warning(
'Couldn\'t find murmurhash. Install mmh3 for a faster implementation of'
'ApproximateUnique.')
def _md5_hash(value):
# md5 is a 128-bit hash, so we truncate the hexdigest (string of 32
# hexadecimal digits) to 16 digits and convert to int to get the 64-bit
# integer fingerprint.
return int(hashlib.md5(value).hexdigest()[:16], 16)
return _md5_hash
[docs]class ApproximateUnique(object):
"""
Hashes input elements and uses those to extrapolate the size of the entire
set of hash values by assuming the rest of the hash values are as densely
distributed as the sample space.
"""
_NO_VALUE_ERR_MSG = 'Either size or error should be set. Received {}.'
_MULTI_VALUE_ERR_MSG = 'Either size or error should be set. ' \
'Received {size = %s, error = %s}.'
_INPUT_SIZE_ERR_MSG = 'ApproximateUnique needs a size >= 16 for an error ' \
'<= 0.50. In general, the estimation error is about ' \
'2 / sqrt(sample_size). Received {size = %s}.'
_INPUT_ERROR_ERR_MSG = 'ApproximateUnique needs an estimation error ' \
'between 0.01 and 0.50. Received {error = %s}.'
@staticmethod
def _get_sample_size_from_est_error(est_err):
"""
:return: sample size
Calculate sample size from estimation error
"""
#math.ceil in python2.7 returns a float, while it returns an int in python3.
return int(math.ceil(4.0 / math.pow(est_err, 2.0)))
[docs] @typehints.with_input_types(T)
@typehints.with_output_types(int)
class Globally(PTransform):
""" Approximate.Globally approximate number of unique values"""
def __init__(self, size=None, error=None):
self._sample_size = ApproximateUnique.parse_input_params(size, error)
[docs] def expand(self, pcoll):
coder = coders.registry.get_coder(pcoll)
return pcoll \
| 'CountGlobalUniqueValues' \
>> (CombineGlobally(ApproximateUniqueCombineFn(self._sample_size,
coder)))
[docs] @typehints.with_input_types(typing.Tuple[K, V])
@typehints.with_output_types(typing.Tuple[K, int])
class PerKey(PTransform):
""" Approximate.PerKey approximate number of unique values per key"""
def __init__(self, size=None, error=None):
self._sample_size = ApproximateUnique.parse_input_params(size, error)
[docs] def expand(self, pcoll):
coder = coders.registry.get_coder(pcoll)
return pcoll \
| 'CountPerKeyUniqueValues' \
>> (CombinePerKey(ApproximateUniqueCombineFn(self._sample_size,
coder)))
class _LargestUnique(object):
"""
An object to keep samples and calculate sample hash space. It is an
accumulator of a combine function.
"""
# We use unsigned 64-bit integer hashes.
_HASH_SPACE_SIZE = 2.0**64
def __init__(self, sample_size):
self._sample_size = sample_size
self._min_hash = 2.0**64
self._sample_heap = []
self._sample_set = set()
def add(self, element):
"""
:param an element from pcoll.
:return: boolean type whether the value is in the heap
Adds a value to the heap, returning whether the value is (large enough to
be) in the heap.
"""
if len(self._sample_heap) >= self._sample_size and element < self._min_hash:
return False
if element not in self._sample_set:
self._sample_set.add(element)
heapq.heappush(self._sample_heap, element)
if len(self._sample_heap) > self._sample_size:
temp = heapq.heappop(self._sample_heap)
self._sample_set.remove(temp)
self._min_hash = self._sample_heap[0]
elif element < self._min_hash:
self._min_hash = element
return True
def get_estimate(self):
"""
:return: estimation count of unique values
If heap size is smaller than sample size, just return heap size.
Otherwise, takes into account the possibility of hash collisions,
which become more likely than not for 2^32 distinct elements.
Note that log(1+x) ~ x for small x, so for sampleSize << maxHash
log(1 - sample_size/sample_space) / log(1 - 1/sample_space) ~ sample_size
and hence estimate ~ sample_size * hash_space / sample_space
as one would expect.
Given sample_size / sample_space = est / hash_space
est = sample_size * hash_space / sample_space
Given above sample_size approximate,
est = log1p(-sample_size/sample_space) / log1p(-1/sample_space)
* hash_space / sample_space
"""
if len(self._sample_heap) < self._sample_size:
return len(self._sample_heap)
else:
sample_space_size = self._HASH_SPACE_SIZE - 1.0 * self._min_hash
est = (
math.log1p(-self._sample_size / sample_space_size) /
math.log1p(-1 / sample_space_size) * self._HASH_SPACE_SIZE /
sample_space_size)
return round(est)
class ApproximateUniqueCombineFn(CombineFn):
"""
ApproximateUniqueCombineFn computes an estimate of the number of
unique values that were combined.
"""
def __init__(self, sample_size, coder):
self._sample_size = sample_size
coder = coders.typecoders.registry.verify_deterministic(
coder, 'ApproximateUniqueCombineFn')
self._coder = coder
self._hash_fn = _get_default_hash_fn()
def create_accumulator(self, *args, **kwargs):
return _LargestUnique(self._sample_size)
def add_input(self, accumulator, element, *args, **kwargs):
try:
hashed_value = self._hash_fn(self._coder.encode(element))
accumulator.add(hashed_value)
return accumulator
except Exception as e:
raise RuntimeError("Runtime exception: %s" % e)
# created an issue https://issues.apache.org/jira/browse/BEAM-7285 to speed up
# merge process.
def merge_accumulators(self, accumulators, *args, **kwargs):
merged_accumulator = self.create_accumulator()
for accumulator in accumulators:
for i in accumulator._sample_heap:
merged_accumulator.add(i)
return merged_accumulator
@staticmethod
def extract_output(accumulator):
return accumulator.get_estimate()
def display_data(self):
return {'sample_size': self._sample_size}
[docs]class ApproximateQuantiles(object):
"""
PTransform for getting the idea of data distribution using approximate N-tile
(e.g. quartiles, percentiles etc.) either globally or per-key.
Examples:
in: list(range(101)), num_quantiles=5
out: [0, 25, 50, 75, 100]
in: [(i, 1 if i<10 else 1e-5) for i in range(101)], num_quantiles=5,
weighted=True
out: [0, 2, 5, 7, 100]
"""
@staticmethod
def _display_data(num_quantiles, key, reverse, weighted):
return {
'num_quantiles': DisplayDataItem(num_quantiles, label='Quantile Count'),
'key': DisplayDataItem(
key.__name__
if hasattr(key, '__name__') else key.__class__.__name__,
label='Record Comparer Key'),
'reverse': DisplayDataItem(str(reverse), label='Is Reversed'),
'weighted': DisplayDataItem(str(weighted), label='Is Weighted')
}
[docs] @typehints.with_input_types(
typehints.Union[typing.Sequence[T], typing.Tuple[T, float]])
@typehints.with_output_types(typing.List[T])
class Globally(PTransform):
"""
PTransform takes PCollection and returns a list whose single value is
approximate N-tiles of the input collection globally.
Args:
num_quantiles: number of elements in the resulting quantiles values list.
key: (optional) Key is a mapping of elements to a comparable key, similar
to the key argument of Python's sorting methods.
reverse: (optional) whether to order things smallest to largest, rather
than largest to smallest.
weighted: (optional) if set to True, the transform returns weighted
quantiles. The input PCollection is then expected to contain tuples of
input values with the corresponding weight.
"""
def __init__(self, num_quantiles, key=None, reverse=False, weighted=False):
self._num_quantiles = num_quantiles
self._key = key
self._reverse = reverse
self._weighted = weighted
[docs] def expand(self, pcoll):
return pcoll | CombineGlobally(
ApproximateQuantilesCombineFn.create(
num_quantiles=self._num_quantiles,
key=self._key,
reverse=self._reverse,
weighted=self._weighted))
[docs] def display_data(self):
return ApproximateQuantiles._display_data(
num_quantiles=self._num_quantiles,
key=self._key,
reverse=self._reverse,
weighted=self._weighted)
[docs] @typehints.with_input_types(
typehints.Union[typing.Tuple[K, V],
typing.Tuple[K, typing.Tuple[V, float]]])
@typehints.with_output_types(typing.Tuple[K, typing.List[V]])
class PerKey(PTransform):
"""
PTransform takes PCollection of KV and returns a list based on each key
whose single value is list of approximate N-tiles of the input element of
the key.
Args:
num_quantiles: number of elements in the resulting quantiles values list.
key: (optional) Key is a mapping of elements to a comparable key, similar
to the key argument of Python's sorting methods.
reverse: (optional) whether to order things smallest to largest, rather
than largest to smallest.
weighted: (optional) if set to True, the transform returns weighted
quantiles. The input PCollection is then expected to contain tuples of
input values with the corresponding weight.
"""
def __init__(self, num_quantiles, key=None, reverse=False, weighted=False):
self._num_quantiles = num_quantiles
self._key = key
self._reverse = reverse
self._weighted = weighted
[docs] def expand(self, pcoll):
return pcoll | CombinePerKey(
ApproximateQuantilesCombineFn.create(
num_quantiles=self._num_quantiles,
key=self._key,
reverse=self._reverse,
weighted=self._weighted))
[docs] def display_data(self):
return ApproximateQuantiles._display_data(
num_quantiles=self._num_quantiles,
key=self._key,
reverse=self._reverse,
weighted=self._weighted)
class _QuantileBuffer(object):
"""A single buffer in the sense of the referenced algorithm.
(see http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.6.6513&rep=rep1
&type=pdf and ApproximateQuantilesCombineFn for further information)"""
def __init__(self, elements, weighted, level=0, weight=1):
# In case of weighted quantiles, elements are tuples of values and weights.
self.elements = elements
self.weighted = weighted
self.level = level
self.weight = weight
def __lt__(self, other):
if self.weighted:
return [element[0] for element in self.elements
] < [element[0] for element in other.elements]
else:
return self.elements < other.elements
def sized_iterator(self):
class QuantileBufferIterator(object):
def __init__(self, elem, weighted, weight):
self._iter = iter(elem)
self.weighted = weighted
self.weight = weight
def __iter__(self):
return self
def __next__(self):
if self.weighted:
return next(self._iter)
else:
value = next(self._iter)
return (value, self.weight)
next = __next__ # For Python 2
return QuantileBufferIterator(self.elements, self.weighted, self.weight)
class _QuantileState(object):
"""
Compact summarization of a collection on which quantiles can be estimated.
"""
min_val = None # Holds smallest item in the list
max_val = None # Holds largest item in the list
def __init__(self, buffer_size, num_buffers, unbuffered_elements, buffers):
self.buffer_size = buffer_size
self.num_buffers = num_buffers
self.buffers = buffers
# The algorithm requires that the manipulated buffers always be filled to
# capacity to perform the collapse operation. This operation can be extended
# to buffers of varying sizes by introducing the notion of fractional
# weights, but it's easier to simply combine the remainders from all shards
# into new, full buffers and then take them into account when computing the
# final output.
self.unbuffered_elements = unbuffered_elements
def is_empty(self):
"""Check if the buffered & unbuffered elements are empty or not."""
return not self.unbuffered_elements and not self.buffers
class ApproximateQuantilesCombineFn(CombineFn):
"""
This combiner gives an idea of the distribution of a collection of values
using approximate N-tiles. The output of this combiner is the list of size of
the number of quantiles (num_quantiles), containing the input values of the
minimum value item of the list, the intermediate values (n-tiles) and the
maximum value item of the list, in the sort order provided via key (similar
to the key argument of Python's sorting methods).
If there are fewer values to combine than the number of quantile
(num_quantiles), then the resulting list will contain all the values being
combined, in sorted order.
If no `key` is provided, then the results are sorted in the natural order.
To evaluate the quantiles, we use the "New Algorithm" described here:
[MRL98] Manku, Rajagopalan & Lindsay, "Approximate Medians and other
Quantiles in One Pass and with Limited Memory", Proc. 1998 ACM SIGMOD,
Vol 27, No 2, p 426-435, June 1998.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.6.6513&rep=rep1
&type=pdf
The default error bound is (1 / N) for uniformly distributed data and
min(1e-2, 1 / N) for weighted case, though in practice the accuracy tends to
be much better.
Args:
num_quantiles: Number of quantiles to produce. It is the size of the final
output list, including the mininum and maximum value items.
buffer_size: The size of the buffers, corresponding to k in the referenced
paper.
num_buffers: The number of buffers, corresponding to b in the referenced
paper.
key: (optional) Key is a mapping of elements to a comparable key, similar
to the key argument of Python's sorting methods.
reverse: (optional) whether to order things smallest to largest, rather
than largest to smallest.
weighted: (optional) if set to True, the combiner produces weighted
quantiles. The input elements are then expected to be tuples of input
values with the corresponding weight.
"""
# For alternating between biasing up and down in the above even weight
# collapse operation.
_offset_jitter = 0
# The cost (in time and space) to compute quantiles to a given accuracy is a
# function of the total number of elements in the data set. If an estimate is
# not known or specified, we use this as an upper bound. If this is too low,
# errors may exceed the requested tolerance; if too high, efficiency may be
# non-optimal. The impact is logarithmic with respect to this value, so this
# default should be fine for most uses.
_MAX_NUM_ELEMENTS = 1e9
_qs = None # Refers to the _QuantileState
def __init__(
self,
num_quantiles,
buffer_size,
num_buffers,
key=None,
reverse=False,
weighted=False):
def _comparator(a, b):
if key:
a, b = key(a), key(b)
retval = int(a > b) - int(a < b)
if reverse:
return -retval
return retval
self._comparator = _comparator
self._num_quantiles = num_quantiles
self._buffer_size = buffer_size
self._num_buffers = num_buffers
if weighted:
self._key = (lambda x: x[0]) if key is None else (lambda x: key(x[0]))
else:
self._key = key
self._reverse = reverse
self._weighted = weighted
@classmethod
def create(
cls,
num_quantiles,
epsilon=None,
max_num_elements=None,
key=None,
reverse=False,
weighted=False):
"""
Creates an approximate quantiles combiner with the given key and desired
number of quantiles.
Args:
num_quantiles: Number of quantiles to produce. It is the size of the
final output list, including the mininum and maximum value items.
epsilon: (optional) The default error bound is `epsilon`, which holds as
long as the number of elements is less than `_MAX_NUM_ELEMENTS`.
Specifically, if one considers the input as a sorted list x_1, ...,
x_N, then the distance between each exact quantile x_c and its
approximation x_c' is bounded by `|c - c'| < epsilon * N`. Note that
these errors are worst-case scenarios. In practice the accuracy tends
to be much better.
max_num_elements: (optional) The cost (in time and space) to compute
quantiles to a given accuracy is a function of the total number of
elements in the data set.
key: (optional) Key is a mapping of elements to a comparable key, similar
to the key argument of Python's sorting methods.
reverse: (optional) whether to order things smallest to largest, rather
than largest to smallest.
weighted: (optional) if set to True, the combiner produces weighted
quantiles. The input elements are then expected to be tuples of values
with the corresponding weight.
"""
max_num_elements = max_num_elements or cls._MAX_NUM_ELEMENTS
if not epsilon:
epsilon = min(1e-2, 1.0 / num_quantiles) \
if weighted else (1.0 / num_quantiles)
b = 2
while (b - 2) * (1 << (b - 2)) < epsilon * max_num_elements:
b = b + 1
b = b - 1
k = max(2, int(math.ceil(max_num_elements / float(1 << (b - 1)))))
return cls(
num_quantiles=num_quantiles,
buffer_size=k,
num_buffers=b,
key=key,
reverse=reverse,
weighted=weighted)
def _add_unbuffered(self, qs, elements):
"""
Add a new buffer to the unbuffered list, creating a new buffer and
collapsing if needed.
"""
qs.unbuffered_elements.extend(elements)
if len(qs.unbuffered_elements) >= qs.buffer_size:
qs.unbuffered_elements.sort(key=self._key, reverse=self._reverse)
while len(qs.unbuffered_elements) >= qs.buffer_size:
to_buffer = qs.unbuffered_elements[:qs.buffer_size]
heapq.heappush(
qs.buffers,
_QuantileBuffer(
elements=to_buffer,
weighted=self._weighted,
weight=sum([element[1] for element in to_buffer])
if self._weighted else 1))
qs.unbuffered_elements = qs.unbuffered_elements[qs.buffer_size:]
self._collapse_if_needed(qs)
def _offset(self, new_weight):
"""
If the weight is even, we must round up or down. Alternate between these
two options to avoid a bias.
"""
if new_weight % 2 == 1:
return (new_weight + 1) / 2
else:
self._offset_jitter = 2 - self._offset_jitter
return (new_weight + self._offset_jitter) / 2
def _collapse(self, buffers):
new_level = 0
new_weight = 0
for buffer_elem in buffers:
# As presented in the paper, there should always be at least two
# buffers of the same (minimal) level to collapse, but it is possible
# to violate this condition when combining buffers from independently
# computed shards. If they differ we take the max.
new_level = max([new_level, buffer_elem.level + 1])
new_weight = new_weight + buffer_elem.weight
if self._weighted:
step = new_weight / (self._buffer_size - 1)
offset = new_weight / (2 * self._buffer_size)
else:
step = new_weight
offset = self._offset(new_weight)
new_elements = self._interpolate(buffers, self._buffer_size, step, offset)
return _QuantileBuffer(new_elements, self._weighted, new_level, new_weight)
def _collapse_if_needed(self, qs):
while len(qs.buffers) > self._num_buffers:
to_collapse = []
to_collapse.append(heapq.heappop(qs.buffers))
to_collapse.append(heapq.heappop(qs.buffers))
min_level = to_collapse[1].level
while len(qs.buffers) > 0 and qs.buffers[0].level == min_level:
to_collapse.append(heapq.heappop(qs.buffers))
heapq.heappush(qs.buffers, self._collapse(to_collapse))
def _interpolate(self, i_buffers, count, step, offset):
"""
Emulates taking the ordered union of all elements in buffers, repeated
according to their weight, and picking out the (k * step + offset)-th
elements of this list for `0 <= k < count`.
"""
iterators = []
new_elements = []
compare_key = self._key
if self._key and not self._weighted:
compare_key = lambda x: self._key(x[0])
for buffer_elem in i_buffers:
iterators.append(buffer_elem.sized_iterator())
# Python 3 `heapq.merge` support key comparison and returns an iterator and
# does not pull the data into memory all at once. Python 2 does not
# support comparison on its `heapq.merge` api, so we use the itertools
# which takes the `key` function for comparison and creates an iterator
# from it.
if sys.version_info[0] < 3:
sorted_elem = iter(
sorted(
itertools.chain.from_iterable(iterators),
key=compare_key,
reverse=self._reverse))
else:
sorted_elem = heapq.merge(
*iterators, key=compare_key, reverse=self._reverse)
weighted_element = next(sorted_elem)
current = weighted_element[1]
j = 0
previous = 0
while j < count:
target = j * step + offset
j = j + 1
try:
while current <= target:
weighted_element = next(sorted_elem)
current = current + weighted_element[1]
except StopIteration:
pass
if self._weighted:
new_elements.append((weighted_element[0], current - previous))
previous = current
else:
new_elements.append(weighted_element[0])
return new_elements
def create_accumulator(self):
self._qs = _QuantileState(
buffer_size=self._buffer_size,
num_buffers=self._num_buffers,
unbuffered_elements=[],
buffers=[])
return self._qs
def add_input(self, quantile_state, element):
"""
Add a new element to the collection being summarized by quantile state.
"""
value = element[0] if self._weighted else element
if quantile_state.is_empty():
quantile_state.min_val = quantile_state.max_val = value
elif self._comparator(value, quantile_state.min_val) < 0:
quantile_state.min_val = value
elif self._comparator(value, quantile_state.max_val) > 0:
quantile_state.max_val = value
self._add_unbuffered(quantile_state, elements=[element])
return quantile_state
def add_inputs(self, quantile_state, elements):
"""Add new elements to the collection being summarized by quantile state.
"""
if not elements:
return quantile_state
values = [
element[0] for element in elements
] if self._weighted else elements
min_val = min(values)
max_val = max(values)
if quantile_state.is_empty():
quantile_state.min_val = min_val
quantile_state.max_val = max_val
elif self._comparator(min_val, quantile_state.min_val) < 0:
quantile_state.min_val = min_val
elif self._comparator(max_val, quantile_state.max_val) > 0:
quantile_state.max_val = max_val
self._add_unbuffered(quantile_state, elements=elements)
return quantile_state
def merge_accumulators(self, accumulators):
"""Merges all the accumulators (quantile state) as one."""
qs = self.create_accumulator()
for accumulator in accumulators:
if accumulator.is_empty():
continue
if not qs.min_val or self._comparator(accumulator.min_val,
qs.min_val) < 0:
qs.min_val = accumulator.min_val
if not qs.max_val or self._comparator(accumulator.max_val,
qs.max_val) > 0:
qs.max_val = accumulator.max_val
self._add_unbuffered(qs, accumulator.unbuffered_elements)
qs.buffers.extend(accumulator.buffers)
self._collapse_if_needed(qs)
return qs
def extract_output(self, accumulator):
"""
Outputs num_quantiles elements consisting of the minimum, maximum and
num_quantiles - 2 evenly spaced intermediate elements. Returns the empty
list if no elements have been added.
"""
if accumulator.is_empty():
return []
all_elems = accumulator.buffers
if self._weighted:
unbuffered_weight = sum(
[element[1] for element in accumulator.unbuffered_elements])
total_weight = unbuffered_weight
for buffer_elem in all_elems:
total_weight += sum([element[1] for element in buffer_elem.elements])
if accumulator.unbuffered_elements:
accumulator.unbuffered_elements.sort(
key=self._key, reverse=self._reverse)
all_elems.append(
_QuantileBuffer(
accumulator.unbuffered_elements,
weighted=True,
weight=unbuffered_weight))
step = 1.0 * total_weight / (self._num_quantiles - 1)
offset = (1.0 * total_weight) / (self._num_quantiles - 1)
mid_quantiles = [
element[0] for element in self._interpolate(
all_elems, self._num_quantiles - 2, step, offset)
]
else:
total_weight = len(accumulator.unbuffered_elements)
for buffer_elem in all_elems:
total_weight += accumulator.buffer_size * buffer_elem.weight
if accumulator.unbuffered_elements:
accumulator.unbuffered_elements.sort(
key=self._key, reverse=self._reverse)
all_elems.append(
_QuantileBuffer(accumulator.unbuffered_elements, weighted=False))
step = 1.0 * total_weight / (self._num_quantiles - 1)
offset = (1.0 * total_weight - 1) / (self._num_quantiles - 1)
mid_quantiles = self._interpolate(
all_elems, self._num_quantiles - 2, step, offset)
quantiles = [accumulator.min_val]
quantiles.extend(mid_quantiles)
quantiles.append(accumulator.max_val)
return quantiles