サンプルソートで配列を並び替える
サンプルソートを使用する
サンプルソート (sample sort) は、入力から少数の標本を取り出して整列し、その値を分割点(スプリッター)として全体をバケットに分け、各バケットを整列して連結する。
並列計算向けに設計された手法として知られ、プロセッサ数 p に対して p−1 個のスプリッターを選ぶ構成が典型である。
クイックソートが 1 つのピボットで左右に分けるのに対し、サンプルソートは複数のスプリッターで p 個(デモでは要素数に応じた複数個)のバケットに一度に仕分ける。分割の偏りを抑えやすく、各バケットを別プロセッサで同時に整列できる点が並列版の利点である。
- サンプリング: 配列から
s個の要素を(等間隔に)選び、サンプル集合を得る。文献では \(s \approx \sqrt{n}\) やs = p−1などが用いられる。 - サンプルの整列: サンプルだけを整列し、昇順のスプリッター列 \(t_0 \le t_1 \le \ldots\) を得る。
- 仕分け: 各要素
xを、\(t_0, t_1, \ldots\) と比較して属するバケット番号を決める(例: \(x \le t_0\) ならバケット 0、\(t_0 < x \le t_1\) ならバケット 1、…)。 - バケット整列: 各バケットを独立に整列する(再帰的にサンプルソート、または十分小さければ挿入ソートなど)。
- 連結: バケット
0, 1, …の順に並べれば全体が昇順になる。
procedure sample_sort(A, s)
if length(A) <= SMALL then
insertion_sort(A)
return
S = choose_s_samples(A, s)
sort(S)
splitters = S
buckets = empty list of s + 1 arrays
for each x in A
b = bucket_index(x, splitters)
append x to buckets[b]
for each bucket B in buckets
sample_sort(B, s)
A = concatenate(buckets)
逐次では標本の整列とバケットへの仕分けを合わせておおむね \(O(n \log n)\)、プロセッサ p 台では \(O(n/p)\) 程度の並列時間を狙える。バケット境界での分割は交換を伴うため不安定である。
比例拡張ソートのように「小さな整列済み標本から分割点を得る」発想はサンプルソートと共通する。バブルソートのように隣接交換だけで進む単純ソートや、クイックソートの単一ピボット再帰と比べると、標本に基づく多分割と並列化しやすい区切りが特徴的である。
以下のデモでは 15 要素から 4 個のサンプルを取り、スプリッター 4 個で 5 バケットに仕分けたあと、各バケットを挿入ソートで仕上げる。
外部ソートや分散整列の文脈では I/O 効率のよい分割としても使われる。サンプル数や再帰の打ち切り、等値要素の扱いが性能を左右する。
類似アルゴリズムとの相違点
ロムート分割型クイックソートが 1 つのピボットで 2 分割するのに対し、標本から複数スプリッターを求めて同時に多分割する。並列・外部整列向きである。バケットソートは値域を等分する点が異なる。
時間計算量および空間計算量を計測する
| Size | Average time (s) | Maximum time (s) | Average memory (KiB) | Maximum memory (KiB) |
|---|---|---|---|---|
| 256 | 0.000010 | 0.000058 | 4 | 6 |
| 512 | 0.000021 | 0.000146 | 7 | 10 |
| 1024 | 0.000040 | 0.000242 | 14 | 19 |
| 2048 | 0.000079 | 0.000140 | 27 | 32 |
| 4096 | 0.000152 | 0.000427 | 52 | 62 |
| 8192 | 0.000331 | 0.002167 | 101 | 120 |
| 16384 | 0.000557 | 0.000948 | 198 | 217 |
| 32768 | 0.001094 | 0.001764 | 389 | 429 |
| 65536 | 0.002252 | 0.004152 | 766 | 819 |
| 131072 | 0.004685 | 0.008009 | 1518 | 1612 |
| 262144 | 0.009153 | 0.015627 | 3012 | 3175 |
計測に使用したコードを表示する
#!/usr/bin/env swift
import Foundation
// This standalone Swift driver creates the same temporary Docker build
// context as the former shell wrapper. The benchmark program itself remains
// embedded below so readers can copy one complete, reproducible file.
struct BenchmarkError: Error, CustomStringConvertible {
let message: String
var description: String { message }
init(_ message: String) {
self.message = message
}
}
func runCommand(_ executable: String, _ arguments: [String]) throws {
let process = Process()
process.executableURL = URL(fileURLWithPath: "/usr/bin/env")
process.arguments = [executable] + arguments
process.standardInput = FileHandle.standardInput
process.standardOutput = FileHandle.standardOutput
process.standardError = FileHandle.standardError
do {
try process.run()
} catch {
throw BenchmarkError("Could not start \(executable): \(error)")
}
process.waitUntilExit()
guard process.terminationStatus == 0 else {
throw BenchmarkError(
"Command failed (\(process.terminationStatus)): " +
"\(executable) \(arguments.joined(separator: " "))"
)
}
}
do {
// The UUID avoids collisions when two benchmark copies are run at once.
let workdir = FileManager.default.temporaryDirectory
.appendingPathComponent("swift-sort-benchmark-\(UUID().uuidString)")
try FileManager.default.createDirectory(at: workdir, withIntermediateDirectories: true)
defer { try? FileManager.default.removeItem(at: workdir) }
// A raw Swift string is used so the nested main.swift keeps its own
// interpolation expressions such as \(seed) until Docker compiles it.
let dockerfile = #"""
FROM swift:6.0
WORKDIR /app
RUN cat > alloc_track.c <<'ALLOC'
#define _GNU_SOURCE
#include <dlfcn.h>
#include <malloc.h>
#include <stdatomic.h>
#include <stddef.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
static atomic_size_t live_bytes = 0;
static atomic_size_t peak_bytes = 0;
static void *(*real_malloc)(size_t) = NULL;
static void *(*real_calloc)(size_t, size_t) = NULL;
static void *(*real_realloc)(void *, size_t) = NULL;
static void (*real_free)(void *) = NULL;
static void init_reals(void) {
if (real_malloc) {
return;
}
real_malloc = (void *(*)(size_t))dlsym(RTLD_NEXT, "malloc");
real_calloc = (void *(*)(size_t, size_t))dlsym(RTLD_NEXT, "calloc");
real_realloc = (void *(*)(void *, size_t))dlsym(RTLD_NEXT, "realloc");
real_free = (void (*)(void *))dlsym(RTLD_NEXT, "free");
}
static void record_alloc(size_t size) {
size_t live = atomic_fetch_add(&live_bytes, size) + size;
size_t peak = atomic_load(&peak_bytes);
while (live > peak) {
if (atomic_compare_exchange_weak(&peak_bytes, &peak, live)) {
break;
}
}
}
void alloc_track_reset_peak(void) {
atomic_store(&peak_bytes, atomic_load(&live_bytes));
}
size_t alloc_track_live(void) { return atomic_load(&live_bytes); }
size_t alloc_track_peak(void) { return atomic_load(&peak_bytes); }
void *malloc(size_t size) {
init_reals();
void *p = real_malloc(size);
if (p) {
record_alloc(malloc_usable_size(p));
}
return p;
}
void *calloc(size_t nmemb, size_t size) {
init_reals();
void *p = real_calloc(nmemb, size);
if (p) {
record_alloc(malloc_usable_size(p));
}
return p;
}
void *realloc(void *ptr, size_t size) {
init_reals();
size_t old_size = 0;
if (ptr) {
old_size = malloc_usable_size(ptr);
}
void *p = real_realloc(ptr, size);
if (p) {
atomic_fetch_sub(&live_bytes, old_size);
record_alloc(malloc_usable_size(p));
} else if (size == 0) {
atomic_fetch_sub(&live_bytes, old_size);
}
return p;
}
void free(void *ptr) {
init_reals();
if (ptr) {
atomic_fetch_sub(&live_bytes, malloc_usable_size(ptr));
real_free(ptr);
}
}
ALLOC
RUN cat > main.swift <<'SWIFT'
import Foundation
#if canImport(Glibc)
import Glibc
#elseif canImport(Darwin)
import Darwin
#endif
@_silgen_name("alloc_track_live") func alloc_track_live() -> Int
@_silgen_name("alloc_track_peak") func alloc_track_peak() -> Int
@_silgen_name("alloc_track_reset_peak") func alloc_track_reset_peak()
extension UnsafeMutableBufferPointer where Element == Int {
func swapAt(_ i: Int, _ j: Int) {
let t = self[i]; self[i] = self[j]; self[j] = t
}
}
let MIN_POWER: Int = 8
let MAX_POWER: Int = 18
let RUNS: Int = 8192
func insertion_sort(_ a: inout [Int]) {
a.withUnsafeMutableBufferPointer { insertion_sort($0) }
}
func insertion_sort(_ a: UnsafeMutableBufferPointer<Int>) {
if a.count < 2 {
return
}
for i in 1..<a.count {
var j = i
while j > 0 && a[j - 1] > a[j] {
a.swapAt(j - 1, j)
j -= 1
}
}
}
func partition_at(_ a: UnsafeMutableBufferPointer<Int>, _ lo: Int, _ hi: Int, _ pivot_idx: Int) -> Int {
a.swapAt(pivot_idx, hi)
let pivot = a[hi]
var i = lo
for j in lo..<hi {
if a[j] < pivot {
a.swapAt(i, j)
i += 1
}
}
a.swapAt(i, hi)
return i
}
func partition(_ a: UnsafeMutableBufferPointer<Int>, _ lo: Int, _ hi: Int) -> Int {
partition_at(a, lo, hi, lo + (hi - lo) / 2)
}
func quick_sort_range(_ a: UnsafeMutableBufferPointer<Int>, _ lo: Int, _ hi: Int) {
if hi <= lo {
return
}
if hi - lo < 16 {
insertion_sort(UnsafeMutableBufferPointer(rebasing: a[lo..<(hi + 1)]))
return
}
let p = partition(a, lo, hi)
if p > 0 {
quick_sort_range(a, lo, p - 1)
}
quick_sort_range(a, p + 1, hi)
}
func quick_sort(_ a: inout [Int]) {
a.withUnsafeMutableBufferPointer { quick_sort($0) }
}
func quick_sort(_ a: UnsafeMutableBufferPointer<Int>) {
if a.count > 0 {
let hi = a.count - 1
quick_sort_range(a, 0, hi)
}
}
func sample_sort(_ a: inout [Int]) {
a.withUnsafeMutableBufferPointer { sample_sort($0) }
}
func sample_sort(_ a: UnsafeMutableBufferPointer<Int>) {
if a.count <= 32 {
insertion_sort(a)
return
}
let sample_count = Int((Double(a.count)).squareRoot())
let step = max(a.count / max(sample_count, 1), 1)
var splitters = [Int]()
var si = step - 1
while si < a.count && splitters.count < sample_count {
splitters.append(a[si])
si += step
}
quick_sort(&splitters)
// A single distinct splitter sends every element to bucket 0, so recursing would
// never shrink the input (33+ equal keys); finish such a run with quick sort.
if splitters.first == splitters.last {
quick_sort(a)
return
}
var buckets = [[Int]](repeating: [], count: splitters.count + 1)
for i in 0..<a.count {
let value = a[i]
// partition_point where predicate value > splitter is false
var lo = 0
var hi = splitters.count
while lo < hi {
let mid = lo + (hi - lo) / 2
if value > splitters[mid] {
lo = mid + 1
} else {
hi = mid
}
}
buckets[lo].append(value)
}
var pos = 0
for b in 0..<buckets.count {
sample_sort(&buckets[b])
for value in buckets[b] {
a[pos] = value
pos += 1
}
}
}
func benchmark_sort(_ array: inout [Int]) {
sample_sort(&array)
}
func is_non_decreasing(_ a: [Int]) -> Bool {
guard a.count >= 2 else { return true }
for i in 1..<a.count {
if a[i - 1] > a[i] { return false }
}
return true
}
func same_multiset(_ a: [Int], _ b: [Int]) -> Bool {
if a.count != b.count {
return false
}
var left = a
var right = b
left.sort()
right.sort()
return left == right
}
func check_correctness_case(_ label: String, _ input: [Int]) {
var input = input
let original = input
benchmark_sort(&input)
if !is_non_decreasing(input) {
fatalError("correctness case \(label): output is not sorted")
}
if !same_multiset(input, original) {
fatalError("correctness case \(label): elements were lost or added")
}
}
// Skip cases larger than the algorithm's measured size cap (MAX_POWER). That
// cap exists because larger inputs are impractically slow; forcing them here
// would stall the published measurement script before any table rows print.
func check_correctness_case_within_limit(_ label: String, _ input: [Int]) {
if input.count > (1 << MAX_POWER) {
return
}
check_correctness_case(label, input)
}
func few_unique_values(_ size: Int, _ unique: Int, _ seed: UInt64) -> [Int] {
var state = seed
var result = [Int]()
result.reserveCapacity(size)
for _ in 0..<size {
state ^= state << 13
state ^= state >> 7
state ^= state << 17
result.append(Int(state % UInt64(unique)) + 1)
}
return result
}
func run_correctness_checks() {
check_correctness_case("empty", [])
check_correctness_case("single", [42])
check_correctness_case("duplicates", [3, 1, 3, 2, 1, 2])
check_correctness_case("sorted", [1, 2, 3, 4, 5])
check_correctness_case("reverse", [5, 4, 3, 2, 1])
check_correctness_case("all_equal", [7, 7, 7, 7])
check_correctness_case("skewed_range", [1_000_000, 2, 1_000_001, 1, 999_999])
// Static-buffer Grail skips the in-buffer build when key collection is sparse
// (ideal_buffer = false). Exercising that path catches regressions in buffer gating.
check_correctness_case(
"few_keys_len16",
[2, 2, 2, 2, 2, 2, 2, 2, 4, 3, 1, 2, 3, 4, 1, 4]
)
// Seed 0 is a fixed point of the xorshift below, so it would degenerate into
// yet another all-equal case instead of a 4-value mix. Start at 1.
for seed in 1...32 {
check_correctness_case(
"few_keys_len32_seed_\(seed)",
few_unique_values(32, 4, UInt64(seed))
)
}
// Small-input cutoffs (insertion sort below 32 elements, etc.) hide duplicate-key
// bugs in the recursive path, so repeat the duplicate cases at the smallest
// benchmark size, which every algorithm must handle within reasonable time.
check_correctness_case("all_equal_len256", [Int](repeating: 7, count: 256))
for seed in 1...4 {
check_correctness_case(
"few_keys_len256_seed_\(seed)",
few_unique_values(256, 4, UInt64(seed))
)
}
// Blit's equal-key second sweep used to copy the whole range into a fixed
// 512-element swap; lengths above that must still sort without panicking.
// Respect MAX_POWER so algorithms with a low measured-size cap (slow,
// sleep) do not hang here for minutes or months.
check_correctness_case_within_limit("all_equal_len600", [Int](repeating: 7, count: 600))
for seed in 1...4 {
check_correctness_case_within_limit(
"few_keys_len2048_seed_\(seed)",
few_unique_values(2048, 4, UInt64(seed))
)
}
}
func shuffled(_ size: Int, seed: UInt64) -> [Int] {
guard size > 0 else { return [] }
var v = Array(1...size)
var state = seed
if size > 1 {
for i in stride(from: size - 1, through: 1, by: -1) {
state ^= state << 13
state ^= state >> 7
state ^= state << 17
let j = Int(state % UInt64(i + 1))
v.swapAt(i, j)
}
}
return v
}
func micros(_ d: Duration) -> UInt64 {
let c = d.components
let fromSeconds = UInt64(c.seconds) * 1_000_000
let fromAttos = UInt64(max(0, c.attoseconds / 1_000_000_000_000))
return fromSeconds + fromAttos
}
func padLeft(_ value: String, _ width: Int) -> String {
if value.count >= width {
return value
}
return String(repeating: " ", count: width - value.count) + value
}
func formatSeconds(_ micros: UInt64) -> String {
let whole = micros / 1_000_000
let frac = micros % 1_000_000
let fracStr = padLeft(String(frac), 6).replacingOccurrences(of: " ", with: "0")
return "\(whole).\(fracStr)"
}
func input_array(_ size: Int, seed: UInt64) -> [Int] {
shuffled(size, seed: seed)
}
/// Peak heap growth during `benchmark_sort`, in bytes (explicit buffers such as swap).
/// Kept in bytes so the parent can average before rounding; converting to KiB here
/// would truncate sub-KiB buffers to 0 in every run and hide them from the average.
func run_once(size: Int, seed: Int) -> (UInt64, Int) {
var array = input_array(size, seed: UInt64(seed))
let baseBytes = alloc_track_live()
alloc_track_reset_peak()
let start = ContinuousClock.now
benchmark_sort(&array)
let elapsed = ContinuousClock.now - start
let peakBytes = alloc_track_peak()
let auxBytes = max(0, peakBytes - baseBytes)
let expected: [Int] = size > 0 ? Array(1...size) : []
if array != expected {
fatalError("sort failed with seed \(seed) for size \(size)")
}
return (micros(elapsed), auxBytes)
}
func run_child(_ args: [String]) {
let size = Int(args[2])!
let seed = Int(args[3])!
let (elapsedUs, mem) = run_once(size: size, seed: seed)
print("\(elapsedUs) \(mem)")
}
let args = CommandLine.arguments
if args.count > 1 && args[1] == "--run-once" {
run_child(args)
} else {
run_correctness_checks()
let tableHeader =
"| \(padLeft("Size", 10)) | " +
"\(padLeft("Average time (s)", 16)) | " +
"\(padLeft("Maximum time (s)", 16)) | " +
"\(padLeft("Average memory (KiB)", 20)) | " +
"\(padLeft("Maximum memory (KiB)", 20)) |"
print(tableHeader)
print("|-----------:|-----------------:|-----------------:|---------------------:|---------------------:|")
for power in MIN_POWER...MAX_POWER {
let size = 1 << power
var totalTime: UInt64 = 0
var maxTime: UInt64 = 0
var totalMem = 0
var maxMem = 0
for seed in 1...RUNS {
let process = Process()
process.executableURL = URL(fileURLWithPath: args[0])
process.arguments = ["--run-once", "\(size)", "\(seed)"]
let stdout = Pipe()
let stderr = Pipe()
process.standardOutput = stdout
process.standardError = stderr
do {
try process.run()
} catch {
fatalError("failed to run benchmark child process: \(error)")
}
process.waitUntilExit()
if process.terminationStatus != 0 {
let err = String(data: stderr.fileHandleForReading.readDataToEndOfFile(), encoding: .utf8) ?? ""
fatalError("benchmark child process failed: \(err)")
}
let data = stdout.fileHandleForReading.readDataToEndOfFile()
let stdoutText = String(data: data, encoding: .utf8) ?? ""
let fields = stdoutText.split(whereSeparator: \.isWhitespace)
guard fields.count >= 2,
let elapsedUs = UInt64(fields[0]),
let auxMem = Int(fields[1]) else {
fatalError("invalid child process output: \(stdoutText)")
}
totalTime += elapsedUs
if elapsedUs > maxTime {
maxTime = elapsedUs
}
totalMem += auxMem
if auxMem > maxMem {
maxMem = auxMem
}
}
let avgTime = totalTime / UInt64(RUNS)
// Memory is summed in bytes and converted to KiB once, after averaging.
let avgMemKb = totalMem / RUNS / 1024
let maxMemKb = maxMem / 1024
let tableRow =
"| \(padLeft(String(size), 10)) | " +
"\(padLeft(formatSeconds(avgTime), 16)) | " +
"\(padLeft(formatSeconds(maxTime), 16)) | " +
"\(padLeft(String(avgMemKb), 20)) | " +
"\(padLeft(String(maxMemKb), 20)) |"
print(tableRow)
}
}
SWIFT
RUN clang -O2 -fPIC -shared alloc_track.c -o liballoc_track.so -ldl
RUN swiftc -Ounchecked -whole-module-optimization \
main.swift \
-o swift-benchmark \
-L. -lalloc_track \
-Xlinker -rpath -Xlinker /app
ENV LD_PRELOAD=/app/liballoc_track.so
CMD ["./swift-benchmark"]
"""#
try dockerfile.write(
to: workdir.appendingPathComponent("Dockerfile"),
atomically: true,
encoding: .utf8
)
// Keeping build and run as separate child processes preserves Docker's
// normal output and the original image tag used by the benchmark skill.
try runCommand("docker", ["build", "-t", "swift-benchmark", workdir.path])
try runCommand("docker", ["run", "--rm", "--init", "swift-benchmark"])
} catch {
fputs("\(error)\n", stderr)
exit(1)
}