"""Determine new partition divisions using approximate percentiles. We use a custom algorithm to calculate approximate, evenly-distributed percentiles of arbitrarily-ordered data for any dtype in a distributed fashion with one pass over the data. This is used to determine new partition divisions when changing the index of a dask.dataframe. We claim no statistical guarantees, but we use a variety of heuristics to try to provide reliable, robust results that are "good enough" and can scale to large number of partitions. Our approach is similar to standard approaches such as t- and q-digest, GK, and sampling-based algorithms, which consist of three parts: 1. **Summarize:** create summaries of subsets of data 2. **Merge:** combine summaries to make a new summary 3. **Compress:** periodically compress a summary into a smaller summary We summarize the data in each partition by calculating several percentiles. The value at each percentile is given a weight proportional to the length of the partition and the differences between the current percentile and the adjacent percentiles. Merging summaries is simply a ``merge_sorted`` of the values and their weights, which we do with a reduction tree. Percentiles is a good choice for our case, because we are given a numpy array of the partition's data, and percentiles is a relatively cheap operation. Moreover, percentiles are, by definition, much less susceptible to the underlying distribution of the data, so the weights given to each value--even across partitions--should be comparable. Let us describe this to a child of five. We are given many small cubes (of equal size) with numbers on them. Split these into many piles. This is like the original data. Let's sort and stack the cubes from one of the piles. Next, we are given a bunch of unlabeled blocks of different sizes, and most are much larger than the the original cubes. Stack these blocks until they're the same height as our first stack. Let's write a number on each block of the new stack. To do this, choose the number of the cube in the first stack that is located in the middle of an unlabeled block. We are finished with this stack once all blocks have a number written on them. Repeat this for all the piles of cubes. Finished already? Great! Now take all the stacks of the larger blocks you wrote on and throw them into a single pile. We'll be sorting these blocks next, which may be easier if you carefully move the blocks over and organize... ah, nevermind--too late. Okay, sort and stack all the blocks from that amazing, disorganized pile you just made. This will be very tall, so we had better stack it sideways on the floor like so. This will also make it easier for us to split the stack into groups of approximately equal size, which is our final task... This, in a nutshell, is the algorithm we deploy. The main difference is that we don't always assign a block the number at its median (ours fluctuates around the median). The numbers at the edges of the final groups is what we use as divisions for repartitioning. We also need the overall min and max, so we take the 0th and 100th percentile of each partition, and another sample near each edge so we don't give disproportionate weights to extreme values. Choosing appropriate percentiles to take in each partition is where things get interesting. The data is arbitrarily ordered, which means it may be sorted, random, or follow some pathological distribution--who knows. We hope all partitions are of similar length, but we ought to expect some variation in lengths. The number of partitions may also be changing significantly, which could affect the optimal choice of percentiles. For improved robustness, we use both evenly-distributed and random percentiles. If the number of partitions isn't changing, then the total number of percentiles across all partitions scales as ``npartitions**1.5``. Although we only have a simple compression operation (step 3 above) that combines weights of equal values, a more sophisticated one could be added if needed, such as for extremely large ``npartitions`` or if we find we need to increase the sample size for each partition. """ from __future__ import annotations import math import numpy as np import pandas as pd from pandas.api.types import is_datetime64_dtype, is_integer_dtype from tlz import merge, merge_sorted, take from dask.base import tokenize from dask.dataframe.core import Series from dask.dataframe.dispatch import tolist_dispatch from dask.dataframe.utils import is_series_like from dask.utils import is_cupy_type, random_state_data def sample_percentiles(num_old, num_new, chunk_length, upsample=1.0, random_state=None): """Construct percentiles for a chunk for repartitioning. Adapt the number of total percentiles calculated based on the number of current and new partitions. Returned percentiles include equally spaced percentiles between [0, 100], and random percentiles. See detailed discussion below. Parameters ---------- num_old: int Number of partitions of the current object num_new: int Number of partitions of the new object chunk_length: int Number of rows of the partition upsample : float Multiplicative factor to increase the number of samples Returns ------- qs : numpy.ndarray of sorted percentiles between 0, 100 Constructing ordered (i.e., not hashed) partitions is hard. Calculating approximate percentiles for generic objects in an out-of-core fashion is also hard. Fortunately, partition boundaries don't need to be perfect in order for partitioning to be effective, so we strive for a "good enough" method that can scale to many partitions and is reasonably well-behaved for a wide variety of scenarios. Two similar approaches come to mind: (1) take a subsample of every partition, then find the best new partitions for the combined subsamples; and (2) calculate equally-spaced percentiles on every partition (a relatively cheap operation), then merge the results. We do both, but instead of random samples, we use random percentiles. If the number of partitions isn't changing, then the ratio of fixed percentiles to random percentiles is 2 to 1. If repartitioning goes from a very high number of partitions to a very low number of partitions, then we use more random percentiles, because a stochastic approach will be more stable to potential correlations in the data that may cause a few equally- spaced partitions to under-sample the data. The more partitions there are, then the more total percentiles will get calculated across all partitions. Squaring the number of partitions approximately doubles the number of total percentiles calculated, so num_total_percentiles ~ sqrt(num_partitions). We assume each partition is approximately the same length. This should provide adequate resolution and allow the number of partitions to scale. For numeric data, one could instead use T-Digest for floats and Q-Digest for ints to calculate approximate percentiles. Our current method works for any dtype. """ # *waves hands* random_percentage = 1 / (1 + (4 * num_new / num_old) ** 0.5) num_percentiles = upsample * num_new * (num_old + 22) ** 0.55 / num_old num_fixed = int(num_percentiles * (1 - random_percentage)) + 2 num_random = int(num_percentiles * random_percentage) + 2 if num_fixed + num_random + 5 >= chunk_length: return np.linspace(0, 100, chunk_length + 1) if not isinstance(random_state, np.random.RandomState): random_state = np.random.RandomState(random_state) q_fixed = np.linspace(0, 100, num_fixed) q_random = random_state.rand(num_random) * 100 q_edges = [60 / (num_fixed - 1), 100 - 60 / (num_fixed - 1)] qs = np.concatenate([q_fixed, q_random, q_edges, [0, 100]]) qs.sort() # Make the divisions between percentiles a little more even qs = 0.5 * (qs[:-1] + qs[1:]) return qs def tree_width(N, to_binary=False): """Generate tree width suitable for ``merge_sorted`` given N inputs The larger N is, the more tasks are reduced in a single task. In theory, this is designed so all tasks are of comparable effort. """ if N < 32: group_size = 2 else: group_size = int(math.log(N)) num_groups = N // group_size if to_binary or num_groups < 16: return 2 ** int(math.log(N / group_size, 2)) else: return num_groups def tree_groups(N, num_groups): """Split an integer N into evenly sized and spaced groups. >>> tree_groups(16, 6) [3, 2, 3, 3, 2, 3] """ # Bresenham, you so smooth! group_size = N // num_groups dx = num_groups dy = N - group_size * num_groups D = 2 * dy - dx rv = [] for _ in range(num_groups): if D < 0: rv.append(group_size) else: rv.append(group_size + 1) D -= 2 * dx D += 2 * dy return rv def create_merge_tree(func, keys, token, level=0): """Create a task tree that merges all the keys with a reduction function. Parameters ---------- func: callable Reduction function that accepts a single list of values to reduce. keys: iterable Keys to reduce from the source dask graph. token: object Included in each key of the returned dict. level: int, default 0 The token-level to begin with. This creates a k-ary tree where k depends on the current level and is greater the further away a node is from the root node. This reduces the total number of nodes (thereby reducing scheduler overhead), but still has beneficial properties of trees. For reasonable numbers of keys, N < 1e5, the total number of nodes in the tree is roughly ``N**0.78``. For 1e5 < N < 2e5, is it roughly ``N**0.8``. """ prev_width = len(keys) prev_keys = iter(keys) rv = {} while prev_width > 1: width = tree_width(prev_width) groups = tree_groups(prev_width, width) keys = [(token, level, i) for i in range(width)] for num, key in zip(groups, keys): rv[key] = (func, list(take(num, prev_keys))) prev_width = width prev_keys = iter(keys) level += 1 return rv def percentiles_to_weights(qs, vals, length): """Weigh percentile values by length and the difference between percentiles >>> percentiles = np.array([0., 25., 50., 90., 100.]) >>> values = np.array([2, 3, 5, 8, 13]) >>> length = 10 >>> percentiles_to_weights(percentiles, values, length) ([2, 3, 5, 8, 13], [125.0, 250.0, 325.0, 250.0, 50.0]) The weight of the first element, ``2``, is determined by the difference between the first and second percentiles, and then scaled by length: >>> 0.5 * length * (percentiles[1] - percentiles[0]) 125.0 The second weight uses the difference of percentiles on both sides, so it will be twice the first weight if the percentiles are equally spaced: >>> 0.5 * length * (percentiles[2] - percentiles[0]) 250.0 """ if length == 0: return () diff = np.ediff1d(qs, 0.0, 0.0) weights = 0.5 * length * (diff[1:] + diff[:-1]) try: # Try using tolist_dispatch first return tolist_dispatch(vals), weights.tolist() except TypeError: return vals.tolist(), weights.tolist() def merge_and_compress_summaries(vals_and_weights): """Merge and sort percentile summaries that are already sorted. Each item is a tuple like ``(vals, weights)`` where vals and weights are lists. We sort both by vals. Equal values will be combined, their weights summed together. """ vals_and_weights = [x for x in vals_and_weights if x] if not vals_and_weights: return () it = merge_sorted(*[zip(x, y) for x, y in vals_and_weights]) vals = [] weights = [] vals_append = vals.append weights_append = weights.append val, weight = prev_val, prev_weight = next(it) for val, weight in it: if val == prev_val: prev_weight += weight else: vals_append(prev_val) weights_append(prev_weight) prev_val, prev_weight = val, weight if val == prev_val: vals_append(prev_val) weights_append(prev_weight) return vals, weights def process_val_weights(vals_and_weights, npartitions, dtype_info): """Calculate final approximate percentiles given weighted vals ``vals_and_weights`` is assumed to be sorted. We take a cumulative sum of the weights, which makes them percentile-like (their scale is [0, N] instead of [0, 100]). Next we find the divisions to create partitions of approximately equal size. It is possible for adjacent values of the result to be the same. Since these determine the divisions of the new partitions, some partitions may be empty. This can happen if we under-sample the data, or if there aren't enough unique values in the column. Increasing ``upsample`` keyword argument in ``df.set_index`` may help. """ dtype, info = dtype_info if not vals_and_weights: try: return np.array(None, dtype=dtype) except Exception: # dtype does not support None value so allow it to change return np.array(None, dtype=np.float64) vals, weights = vals_and_weights vals = np.array(vals) weights = np.array(weights) # We want to create exactly `npartition` number of groups of `vals` that # are approximately the same weight and non-empty if possible. We use a # simple approach (more accurate algorithms exist): # 1. Remove all the values with weights larger than the relative # percentile width from consideration (these are `jumbo`s) # 2. Calculate percentiles with "interpolation=left" of percentile-like # weights of the remaining values. These are guaranteed to be unique. # 3. Concatenate the values from (1) and (2), sort, and return. # # We assume that all values are unique, which happens in the previous # step `merge_and_compress_summaries`. if len(vals) == npartitions + 1: rv = vals elif len(vals) < npartitions + 1: # The data is under-sampled if np.issubdtype(vals.dtype, np.number) and not isinstance( dtype, pd.CategoricalDtype ): # Interpolate extra divisions q_weights = np.cumsum(weights) q_target = np.linspace(q_weights[0], q_weights[-1], npartitions + 1) rv = np.interp(q_target, q_weights, vals) else: # Distribute the empty partitions duplicated_index = np.linspace( 0, len(vals) - 1, npartitions - len(vals) + 1, dtype=int ) duplicated_vals = vals[duplicated_index] rv = np.concatenate([vals, duplicated_vals]) rv.sort() else: target_weight = weights.sum() / npartitions jumbo_mask = weights >= target_weight jumbo_vals = vals[jumbo_mask] trimmed_vals = vals[~jumbo_mask] trimmed_weights = weights[~jumbo_mask] trimmed_npartitions = npartitions - len(jumbo_vals) # percentile-like, but scaled by weights q_weights = np.cumsum(trimmed_weights) q_target = np.linspace(0, q_weights[-1], trimmed_npartitions + 1) left = np.searchsorted(q_weights, q_target, side="left") right = np.searchsorted(q_weights, q_target, side="right") - 1 # stay inbounds np.maximum(right, 0, right) lower = np.minimum(left, right) trimmed = trimmed_vals[lower] rv = np.concatenate([trimmed, jumbo_vals]) rv.sort() if isinstance(dtype, pd.CategoricalDtype): rv = pd.Categorical.from_codes(rv, info[0], info[1]) elif isinstance(dtype, pd.DatetimeTZDtype): rv = pd.DatetimeIndex(rv).tz_localize(dtype.tz) elif "datetime64" in str(dtype): rv = pd.DatetimeIndex(rv, dtype=dtype) elif rv.dtype != dtype: if is_integer_dtype(dtype) and pd.api.types.is_float_dtype(rv.dtype): # pandas EA raises instead of truncating rv = np.floor(rv) rv = pd.array(rv, dtype=dtype) return rv def percentiles_summary(df, num_old, num_new, upsample, state): """Summarize data using percentiles and derived weights. These summaries can be merged, compressed, and converted back into approximate percentiles. Parameters ---------- df: pandas.Series Data to summarize num_old: int Number of partitions of the current object num_new: int Number of partitions of the new object upsample: float Scale factor to increase the number of percentiles calculated in each partition. Use to improve accuracy. """ length = len(df) if length == 0: return () random_state = np.random.RandomState(state) qs = sample_percentiles(num_old, num_new, length, upsample, random_state) data = df interpolation = "linear" if isinstance(data.dtype, pd.CategoricalDtype): data = data.cat.codes interpolation = "nearest" elif is_datetime64_dtype(data.dtype) or is_integer_dtype(data.dtype): interpolation = "nearest" try: # Plan A: Use `Series.quantile` vals = data.quantile(q=qs / 100, interpolation=interpolation) except (TypeError, NotImplementedError): # Series.quantile doesn't work with some data types (e.g. strings) # Fall back to DataFrame.quantile with "nearest" interpolation interpolation = "nearest" vals = ( data.to_frame() .quantile( q=qs / 100, interpolation=interpolation, numeric_only=False, method="table", ) .iloc[:, 0] ) # Convert to array if necessary (and possible) if is_series_like(vals): try: vals = vals.values except (ValueError, TypeError): # cudf->cupy won't work if nulls are present, # or if this is a string dtype pass if ( is_cupy_type(data) and interpolation == "linear" and np.issubdtype(data.dtype, np.integer) ): vals = np.round(vals).astype(data.dtype) if qs[0] == 0: # Ensure the 0th quantile is the minimum value of the data vals[0] = data.min() vals_and_weights = percentiles_to_weights(qs, vals, length) return vals_and_weights def dtype_info(df): info = None if isinstance(df.dtype, pd.CategoricalDtype): data = df.values info = (data.categories, data.ordered) return df.dtype, info def partition_quantiles(df, npartitions, upsample=1.0, random_state=None): """Approximate quantiles of Series used for repartitioning""" assert isinstance(df, Series) # currently, only Series has quantile method # Index.quantile(list-like) must be pd.Series, not pd.Index return_type = Series qs = np.linspace(0, 1, npartitions + 1) token = tokenize(df, qs, upsample) if random_state is None: random_state = int(token, 16) % np.iinfo(np.int32).max state_data = random_state_data(df.npartitions, random_state) df_keys = df.__dask_keys__() name0 = "re-quantiles-0-" + token dtype_dsk = {(name0, 0): (dtype_info, df_keys[0])} name1 = "re-quantiles-1-" + token val_dsk = { (name1, i): ( percentiles_summary, key, df.npartitions, npartitions, upsample, state, ) for i, (state, key) in enumerate(zip(state_data, df_keys)) } name2 = "re-quantiles-2-" + token merge_dsk = create_merge_tree(merge_and_compress_summaries, sorted(val_dsk), name2) if not merge_dsk: # Compress the data even if we only have one partition merge_dsk = {(name2, 0, 0): (merge_and_compress_summaries, [list(val_dsk)[0]])} merged_key = max(merge_dsk) name3 = "re-quantiles-3-" + token last_dsk = { (name3, 0): ( pd.Series, # TODO: Use `type(df._meta)` when cudf adds `tolist()` (process_val_weights, merged_key, npartitions, (name0, 0)), qs, None, df.name, ) } dsk = merge(df.dask, dtype_dsk, val_dsk, merge_dsk, last_dsk) new_divisions = [0.0, 1.0] return return_type(dsk, name3, df._meta, new_divisions)