1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541
//! Functions to perform merge-sorts.
//!
//! The goal of merge-sort is to merge two sorted arrays, `[a0, a1]`, `merge_sort(a0, a1)`,
//! so that the resulting array is sorted, i.e. the following invariant upholds:
//! `sort(merge_sort(a0, a1)) == merge_sort(a0, a1)` for any two sorted arrays `a0` and `a1`.
//!
//! Given that two sorted arrays are more likely to be partially sorted within each other,
//! and that the resulting array is built by taking elements from each array, it is
//! advantageous to `take` slices of items, not items, from each array.
//! As such, this module's main data representation is `(i: usize, start: usize, len: usize)`,
//! which represents a slice of array `i`.
//!
//! In this context, `merge_sort` is composed by two main operations:
//!
//! 1. compute the array of slices `v` that construct a new sorted array from `a0` and `a1`.
//! 2. `take_arrays` from `a0` and `a1`, creating the sorted array.
//!
//! In the extreme case where the two arrays are already sorted between then (e.g. `[0, 2]`, `[3, 4]`),
//! we need two slices, `v = vec![(0, 0, a0.len()), (1, 0, a1.len())]`. The higher the
//! inter-leave between the two arrays, the more slices will be needed, and
//! generally the more expensive the `take` operation will be.
//!
//! ## Merge-sort multiple arrays
//!
//! The main advantage of merge-sort over `sort` is that it can be parallelized.
//! For example, given a set of arrays `[a0, a1, a2, a3]` representing the same field,
//! e.g. over 4 batches of arrays, they can be sorted in parallel as follows (pseudo-code):
//!
//! ```rust,ignore
//! // in parallel
//! let a0 = sort(a0);
//! let a1 = sort(a1);
//! let a2 = sort(a2);
//! let a3 = sort(a3);
//!
//! // in parallel and recursively
//! let slices1 = merge_sort_slices(a0, a1);
//! let slices2 = merge_sort_slices(a2, a3);
//! let slices = merge_sort_slices(slices1, slices2);
//!
//! let array = take_arrays(&[a0, a1, a2, a3], slices, None);
//! ```
//!
//! A common operation in query engines is to merge multiple fields based on the
//! same sorting field (e.g. merge-sort multiple batches of arrays).
//! To perform this, use the same idea as above, but use `take_arrays` over
//! each independent field (which can again be parallelized):
//!
//! ```rust,ignore
//! // `slices` computed before-hand
//! // in parallel
//! let array1 = take_arrays(&[a0, a1, a2, a3], slices, None);
//! let array2 = take_arrays(&[b0, b1, b2, b3], slices, None);
//! ```
//!
//! To serialize slices, e.g. for checkpointing or transfer via Arrow's IPC, you can store
//! them as 3 non-null primitive arrays (e.g. `PrimitiveArray<i64>`).
use ahash::AHashMap;
use std::cmp::Ordering;
use std::iter::once;
use itertools::Itertools;
use crate::array::{
growable::make_growable,
ord::{build_compare, DynComparator},
Array,
};
pub use crate::compute::sort::SortOptions;
use crate::error::Result;
/// A slice denoting `(array_index, start, len)` representing a slice from one of N arrays.
/// This is used to keep track of contiguous blocks of slots.
/// An array of MergeSlice, `[MergeSlice]`, represents inter-leaved array slices.
/// For example, `[(0, 0, 2), (1, 0, 1), (0, 2, 3)]` represents 2 arrays (a0 and a1) arranged as follows:
/// `[a0[0..2], a1[0..1], a0[2..3]]`
/// This representation is useful when building arrays in memory as it allows to memcopy slices of arrays.
/// This is particularly useful in merge-sort because sorted arrays (passed to the merge-sort) are more likely
/// to have contiguous blocks of sorted elements (than by random).
pub type MergeSlice = (usize, usize, usize);
/// Takes N arrays together through `slices` under the assumption that the slices have
/// a total coverage of the arrays.
/// I.e. they are such that all elements on all arrays are picked (which is the case in sorting).
/// # Panic
/// This function panics if:
/// * `max(slices[i].0) >= arrays.len()`, as it indicates that the slices point to an array out of bounds from `arrays`.
/// * the arrays do not have the same [`crate::datatypes::DataType`] (as it makes no sense to take together from them)
pub fn take_arrays<I: IntoIterator<Item = MergeSlice>>(
arrays: &[&dyn Array],
slices: I,
limit: Option<usize>,
) -> Box<dyn Array> {
let slices = slices.into_iter();
let len = arrays.iter().map(|array| array.len()).sum();
let limit = limit.unwrap_or(len);
let limit = limit.min(len);
let mut growable = make_growable(arrays, false, limit);
if limit != len {
let mut current_len = 0;
for (index, start, len) in slices {
if len + current_len >= limit {
growable.extend(index, start, limit - current_len);
break;
} else {
growable.extend(index, start, len);
current_len += len;
}
}
} else {
for (index, start, len) in slices {
growable.extend(index, start, len);
}
}
growable.as_box()
}
/// Combines two sorted [Array]s of the same [`crate::datatypes::DataType`] into a single sorted array.
/// If the arrays are not sorted (which this function does not check), the result is wrong.
/// # Error
/// This function errors when:
/// * the arrays have a different [`crate::datatypes::DataType`]
/// * the arrays have a [`crate::datatypes::DataType`] that has no order relationship
/// # Example
/// ```rust
/// use arrow2::array::Int32Array;
/// use arrow2::compute::merge_sort::{merge_sort, SortOptions};
/// # use arrow2::error::Result;
/// # fn main() -> Result<()> {
/// let a = Int32Array::from_slice(&[2, 4, 6]);
/// let b = Int32Array::from_slice(&[0, 1, 3]);
/// let sorted = merge_sort(&a, &b, &SortOptions::default(), None)?;
/// let expected = Int32Array::from_slice(&[0, 1, 2, 3, 4, 6]);
/// assert_eq!(expected, sorted.as_ref());
/// # Ok(())
/// # }
/// ```
pub fn merge_sort(
lhs: &dyn Array,
rhs: &dyn Array,
options: &SortOptions,
limit: Option<usize>,
) -> Result<Box<dyn Array>> {
let arrays = &[lhs, rhs];
let pairs: &[(&[&dyn Array], &SortOptions)] = &[(arrays, options)];
let comparator = build_comparator(pairs)?;
let lhs = (0, 0, lhs.len());
let rhs = (1, 0, rhs.len());
let slices = merge_sort_slices(once(&lhs), once(&rhs), &comparator);
Ok(take_arrays(arrays, slices, limit))
}
/// Returns a vector of slices from different sorted arrays that can be used to create sorted arrays.
/// `pairs` is an array representing multiple sorted array sets. The expected format is
///
/// pairs: [([a00, a01], o1), ([a10, a11], o2), ...]
/// where aj0.len() == aj0.len()
/// aj1.len() == aj1.len()
/// ...
/// In other words, `pairs.i.0[j]` must be an array coming from a batch of equal len arrays.
/// # Example
/// ```rust
/// use arrow2::array::Int32Array;
/// use arrow2::compute::merge_sort::{slices, SortOptions};
/// # use arrow2::error::Result;
/// # fn main() -> Result<()> {
/// let a = Int32Array::from_slice(&[2, 4, 6]);
/// let b = Int32Array::from_slice(&[0, 1, 3]);
/// let slices = slices(&[(&[&a, &b], &SortOptions::default())])?;
/// assert_eq!(slices, vec![(1, 0, 2), (0, 0, 1), (1, 2, 1), (0, 1, 2)]);
///
/// # Ok(())
/// # }
/// ```
/// # Error
/// This function errors if the arrays `a0i` are not pairwise sortable. This happens when either
/// they have not the same [`crate::datatypes::DataType`] or when their [`crate::datatypes::DataType`]
/// does not correspond to a sortable type.
/// # Panic
/// This function panics if:
/// * `pairs` has no elements
/// * the length condition above is not fulfilled
pub fn slices(pairs: &[(&[&dyn Array], &SortOptions)]) -> Result<Vec<MergeSlice>> {
assert!(!pairs.is_empty());
let comparator = build_comparator(pairs)?;
// pairs: [([a00, a01], o1), ([a10, a11], o2), ...]
// slices: [(0, 0, len), (1, 0, len)]
let slices = pairs[0]
.0
.iter()
.enumerate()
.map(|(index, array)| vec![(index, 0, array.len())])
.collect::<Vec<_>>();
let slices = slices
.iter()
.map(|slice| slice.as_ref())
.collect::<Vec<_>>();
Ok(recursive_merge_sort(&slices, &comparator))
}
/// recursively sort-merges multiple `slices` representing slices of sorted arrays according
/// to a comparison function between those arrays.
/// Note that `slices` is an array of arrays, `slices[i][j]`. The index `i` represents
/// the set of arrays `i`, while the index `j` represents
/// the array `j` within that set.
/// Note that this does not split to the smallest element as arrays: the smallest unit is a `slice`
fn recursive_merge_sort(slices: &[&[MergeSlice]], comparator: &Comparator) -> Vec<MergeSlice> {
let n = slices.len();
let m = n / 2;
if n == 1 {
// slices are assumed sort arrays
return slices[0].to_vec();
}
if n == 2 {
return merge_sort_slices(slices[0].iter(), slices[1].iter(), comparator)
.collect::<Vec<_>>();
}
// split in 2 and sort
let lhs = recursive_merge_sort(&slices[0..m], comparator);
let rhs = recursive_merge_sort(&slices[m..n], comparator);
// merge-sort the splits
merge_sort_slices(lhs.iter(), rhs.iter(), comparator).collect::<Vec<_>>()
}
/// An iterator adapter that merge-sorts two iterators of `MergeSlice` into a single `MergeSlice`
/// such that the resulting `MergeSlice`s are ordered according to `comparator`.
pub struct MergeSortSlices<'a, L, R>
where
L: Iterator<Item = &'a MergeSlice>,
R: Iterator<Item = &'a MergeSlice>,
{
lhs: L,
rhs: R,
comparator: &'a Comparator<'a>,
left: Option<(MergeSlice, usize)>, // current left pile and index
right: Option<(MergeSlice, usize)>, // current right pile and index
// track the current slice being constructed (from left or right)
has_started: bool,
current_start: usize,
current_len: usize,
current_is_left: bool,
}
impl<'a, L, R> MergeSortSlices<'a, L, R>
where
L: Iterator<Item = &'a MergeSlice>,
R: Iterator<Item = &'a MergeSlice>,
{
fn new(lhs: L, rhs: R, comparator: &'a Comparator<'a>) -> Self {
Self {
lhs,
rhs,
comparator,
left: None,
right: None,
has_started: false,
current_start: 0,
current_len: 0,
current_is_left: true,
}
}
fn next_left(&mut self) {
match self.lhs.next() {
Some(slice) => {
self.left = Some((*slice, slice.1));
self.current_start = slice.1;
}
None => self.left = None,
}
}
fn next_right(&mut self) {
match self.rhs.next() {
Some(slice) => {
self.right = Some((*slice, slice.1));
self.current_start = slice.1;
}
None => self.right = None,
}
}
/// Collect the MergeSortSlices to be a vec for reusing
#[warn(dead_code)]
pub fn to_vec(self, limit: Option<usize>) -> Vec<MergeSlice> {
match limit {
Some(limit) => {
let mut v = Vec::with_capacity(limit);
let mut current_len = 0;
for (index, start, len) in self {
if len + current_len >= limit {
v.push((index, start, limit - current_len));
break;
} else {
v.push((index, start, len));
}
current_len += len;
}
v
}
None => self.into_iter().collect(),
}
}
}
impl<'a, L, R> Iterator for MergeSortSlices<'a, L, R>
where
L: Iterator<Item = &'a MergeSlice>,
R: Iterator<Item = &'a MergeSlice>,
{
type Item = MergeSlice;
fn next(&mut self) -> Option<Self::Item> {
if !self.has_started {
// first call of `next`
self.next_left();
self.next_right();
}
match (self.left, self.right) {
(None, None) => {
// both ended
None
}
(Some((left_slice, left_index)), None) => {
// right ended => push left
self.next_left();
// pushing from left
if left_index != left_slice.1 {
// we are in the middle of some slice: push the
// remaining of that slice
Some((
left_slice.0,
left_index,
left_slice.2 - (left_index - left_slice.1),
))
} else {
Some(left_slice)
}
}
(None, Some((right_slice, right_index))) => {
// left ended => push right
self.next_right();
if right_index != right_slice.1 {
// we are in the middle of some slice: push the
// remaining of that slice
Some((
right_slice.0,
right_index,
right_slice.2 - (right_index - right_slice.1),
))
} else {
Some(right_slice)
}
}
// both sides have elements
(Some((left_slice, mut left_index)), Some((right_slice, mut right_index))) => {
if !self.has_started {
let ordering =
(self.comparator)(left_slice.0, left_index, right_slice.0, right_index);
if ordering == Ordering::Greater {
self.current_is_left = false;
self.current_start = right_index;
} else {
self.current_is_left = true;
self.current_start = left_index;
}
self.has_started = true;
}
// advance left_index or right_index until the next split
while (left_index < left_slice.1 + left_slice.2)
&& (right_index < right_slice.1 + right_slice.2)
{
match (
(self.comparator)(left_slice.0, left_index, right_slice.0, right_index),
self.current_is_left,
) {
(Ordering::Less, true) | (Ordering::Equal, true) => {
// on the left and take from the left
self.current_len += 1;
left_index += 1;
}
(Ordering::Greater, false) | (Ordering::Equal, false) => {
// on the right and take from the right
self.current_len += 1;
right_index += 1;
}
(Ordering::Less, false) => {
// switch from right side to left side => push new slice from the right
let start = self.current_start;
let len = self.current_len;
self.current_is_left = true;
self.current_len = 0;
self.current_start = left_index;
if len > 0 {
self.left = Some((left_slice, left_index));
self.right = Some((right_slice, right_index));
return Some((right_slice.0, start, len));
}
}
(Ordering::Greater, true) => {
// switch from left side to right side => push slice from the left
let start = self.current_start;
let len = self.current_len;
self.current_is_left = false;
self.current_len = 0;
self.current_start = right_index;
if len > 0 {
self.left = Some((left_slice, left_index));
self.right = Some((right_slice, right_index));
return Some((left_slice.0, start, len));
}
}
}
}
let start = self.current_start;
let len = self.current_len;
if left_index == left_slice.1 + left_slice.2 {
// reached end of left slice => push it
self.current_len = 0;
self.next_left();
Some((left_slice.0, start, len))
} else {
debug_assert_eq!(right_index, right_slice.1 + right_slice.2);
// reached end of right slice => push it
self.current_len = 0;
self.next_right();
Some((right_slice.0, start, len))
}
}
}
}
}
/// Given two iterators of slices representing two sets of sorted [`Array`]s, and a `comparator` bound to those [`Array`]s,
/// returns a new iterator of slices denoting how to `take` slices from each of the arrays such that the resulting
/// array is sorted according to `comparator`
pub fn merge_sort_slices<
'a,
L: Iterator<Item = &'a MergeSlice>,
R: Iterator<Item = &'a MergeSlice>,
>(
lhs: L,
rhs: R,
comparator: &'a Comparator,
) -> MergeSortSlices<'a, L, R> {
MergeSortSlices::new(lhs, rhs, comparator)
}
// (left index, left row), (right index, right row)
type Comparator<'a> = Box<dyn Fn(usize, usize, usize, usize) -> Ordering + 'a>;
type IsValid<'a> = Box<dyn Fn(usize) -> bool + 'a>;
/// returns a comparison function between any two arrays of each pair of arrays, according to `SortOptions`.
pub fn build_comparator<'a>(
pairs: &'a [(&'a [&'a dyn Array], &SortOptions)],
) -> Result<Comparator<'a>> {
build_comparator_impl(pairs, &build_compare)
}
/// returns a comparison function between any two arrays of each pair of arrays, according to `SortOptions`.
/// Implementing custom `build_compare_fn` for unsupportd data types.
pub fn build_comparator_impl<'a>(
pairs: &'a [(&'a [&'a dyn Array], &SortOptions)],
build_compare_fn: &dyn Fn(&dyn Array, &dyn Array) -> Result<DynComparator>,
) -> Result<Comparator<'a>> {
// prepare the comparison function of _values_ between all pairs of arrays
let indices_pairs = (0..pairs[0].0.len())
.combinations(2)
.map(|indices| (indices[0], indices[1]));
let data = indices_pairs
.map(|(lhs_index, rhs_index)| {
let multi_column_comparator = pairs
.iter()
.map(move |(arrays, _)| {
Ok((
Box::new(move |row| arrays[lhs_index].is_valid(row)) as IsValid<'a>,
Box::new(move |row| arrays[rhs_index].is_valid(row)) as IsValid<'a>,
build_compare_fn(arrays[lhs_index], arrays[rhs_index])?,
))
})
.collect::<Result<Vec<_>>>()?;
Ok(((lhs_index, rhs_index), multi_column_comparator))
})
.collect::<Result<AHashMap<(usize, usize), Vec<(IsValid, IsValid, DynComparator)>>>>()?;
// prepare a comparison function taking into account _nulls_ and sort options
let cmp = move |left_index, left_row, right_index, right_row| {
let data = data.get(&(left_index, right_index)).unwrap();
//data.iter().zip(pairs.iter()).for_each()
for c in 0..pairs.len() {
let descending = pairs[c].1.descending;
let null_first = pairs[c].1.nulls_first;
let (l_is_valid, r_is_valid, value_comparator) = &data[c];
let mut result = match ((l_is_valid)(left_row), (r_is_valid)(right_row)) {
(true, true) => (value_comparator)(left_row, right_row),
(false, true) => {
if null_first {
Ordering::Less
} else {
Ordering::Greater
}
}
(true, false) => {
if null_first {
Ordering::Greater
} else {
Ordering::Less
}
}
(false, false) => Ordering::Equal,
};
if descending {
result = result.reverse();
};
if result != Ordering::Equal {
// we found a relevant comparison => short-circuit and return it
return result;
}
}
Ordering::Equal
};
Ok(Box::new(cmp))
}