ストゥージソートで配列を並び替える
ストゥージソートを使用する
ストゥージソート (stooge sort) は、区間の両端を比較して必要なら入れ替えたあと、先頭 2/3・末尾 2/3・再び先頭 2/3 の順に同じ処理を再帰的に繰り返して昇順に整える。
アメリカのコメディトリオ「スリースタージーズ」になぞらえて名付けられた、意図的に非効率な整列アルゴリズムとして知られる。
- 両端の比較: 部分配列
A[lo .. hi]についてA[lo] > A[hi]なら入れ替える。 - 基底: 要素数が 2 以下ならここで終了する。
- 三分割: \(t = \lfloor (hi - lo + 1) / 3 \rfloor\) とし、先頭 2/3 (lo から hi - t)、末尾 2/3 (lo + t から hi)、再び先頭 2/3 (lo から hi - t) と 3 回の再帰を行う。
- 全体: 配列全体
[0 .. n-1]に対して上記を適用する。
procedure stooge_sort(A, lo, hi)
if A[lo] > A[hi] then
swap(A[lo], A[hi])
if hi - lo + 1 <= 2 then
return
t = floor((hi - lo + 1) / 3)
stooge_sort(A, lo, hi - t)
stooge_sort(A, lo + t, hi)
stooge_sort(A, lo, hi - t)
最悪計算量は \(O(n^{\log 3 / \log 1.5}) \approx O(n^{2.709})\) で、\(O(n^2)\) より遅く \(O(n^3)\) よりは速い。 再帰の深さは \(O(\log n)\) だが、同じ区間を何度も整え直すため実時間は非常に大きい。不安定である。
類似アルゴリズムとの相違点
ノームソートやバブルソートは局所的な交換を繰り返すが、 ストゥージソートは同じ部分区間に対して 3 回の再帰を必ず行う。 ロムート分割型クイックソートも再帰的だが、ピボット分割で区間が実質的に縮むのに対し、 本アルゴリズムは末尾 2/3 の整列のあと先頭 2/3 をもう一度整え直す点が特徴的である。
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 | 0.003979 | 0.006489 | 0 | 0 |
| 512 | 0.033008 | 0.055551 | 0 | 0 |
| 1024 | 0.094145 | 0.137719 | 0 | 0 |
| 2048 | 0.751892 | 1.579949 | 0 | 0 |
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 = 11
let RUNS: Int = 8192
func stooge_sort_range(_ a: UnsafeMutableBufferPointer<Int>, _ lo: Int, _ hi: Int) {
if a[lo] > a[hi] {
a.swapAt(lo, hi)
}
if hi - lo + 1 <= 2 {
return
}
let t = (hi - lo + 1) / 3
stooge_sort_range(a, lo, hi - t)
stooge_sort_range(a, lo + t, hi)
stooge_sort_range(a, lo, hi - t)
}
func stooge_sort(_ a: inout [Int]) {
a.withUnsafeMutableBufferPointer { stooge_sort($0) }
}
func stooge_sort(_ a: UnsafeMutableBufferPointer<Int>) {
if a.count <= 1 {
return
}
stooge_sort_range(a, 0, a.count - 1)
}
func benchmark_sort(_ array: inout [Int]) {
stooge_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)
}