ビンゴソートを使用する

ビンゴソート (bingo sort) は、選択ソートの変形で、未整列範囲の最小値(ビンゴ)を一度見つけたら、その値と等しい要素をすべて確定位置へまとめて移し、次のビンゴを同じ走査の中で探す。

選択ソートが「要素ごと」に未整列範囲を走査するのに対し、ビンゴソートは「相異なる値ごと」に 1 回の走査で足りる。重複が多い入力では走査回数が減り、同値だらけのデータでは選択ソートより速くなりやすい。名前は、目的の値が現れたときに「ビンゴ!」と叫ぶイメージに由来する。

  1. 初回のビンゴ: 配列全体の最小値を bingo、最大値を largest とする。nextBingo をいったん largest に置き、書き込み位置 nextPos0 から始める。
  2. 同値の配置と次値の探索: nextPos から末尾まで走査する。A[i] = bingo なら A[i]A[nextPos] を交換して nextPos を進める。それ以外で A[i] < nextBingo なら nextBingo を更新する(次パスで使う次の最小値候補)。
  3. 次のビンゴへ: 1 パス終了後に bingo = nextBingo とし、nextBingo を再び largest に戻す。
  4. 終了条件: bingo < nextBingo のあいだ手順 2〜3 を繰り返す。最後に残る最大値の塊は、より小さい値がすべて左側へ寄った時点で既に正しい位置にあるため、明示的な最終パスは不要である。
procedure bingo_sort(A)
  n = length(A)
  if n <= 1 then return
  bingo = minimum(A)
  largest = maximum(A)
  nextBingo = largest
  nextPos = 0
  while bingo < nextBingo
    startPos = nextPos
    for i from startPos to n - 1
      if A[i] = bingo then
        swap(A[i], A[nextPos])
        nextPos = nextPos + 1
      else if A[i] < nextBingo then
        nextBingo = A[i]
    bingo = nextBingo
    nextBingo = largest

最悪・平均は相異なる値の個数を m として \(\Theta(n m)\)(すべて相異なれば選択ソートと同じ \(\Theta(n^2)\))、最良は \(\Theta(n + m^2)\) とされる。追加配列は不要(インプレース)で、不安定である。

類似アルゴリズムとの相違点

選択ソートは最小を 1 要素ずつ確定するため、重複があってもほぼ毎回フル走査が要る。ビンゴソートは同値をまとめて確定し、走査回数を相異なる値の個数に近づける。カウンティングソートも重複に強いが、値域幅ぶんの計数配列を使う非比較ソートであり、ビンゴソートは比較ベースのインプレース変形である。

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 0.000030 0.000263 0 0
512 0.000127 0.001205 0 0
1024 0.000401 0.000777 0 0
2048 0.001360 0.002486 0 0
4096 0.004998 0.007768 0 0
8192 0.019377 0.035094 0 0
16384 0.085506 0.153013 0 0
32768 0.292541 0.641881 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 = 15
let RUNS: Int = 8192


func bingo_sort(_ a: inout [Int]) {
    a.withUnsafeMutableBufferPointer { bingo_sort($0) }
}

func bingo_sort(_ a: UnsafeMutableBufferPointer<Int>) {
    let n = a.count
    if n <= 1 {
        return
    }

    var bingo = a.min()!
    let largest = a.max()!
    var next_bingo = largest
    var next_pos = 0

    while bingo < next_bingo {
        let start_pos = next_pos
        for i in start_pos..<n {
            if a[i] == bingo {
                a.swapAt(i, next_pos)
                next_pos += 1
            } else if a[i] < next_bingo {
                next_bingo = a[i]
            }
        }
        bingo = next_bingo
        next_bingo = largest
    }
}


func benchmark_sort(_ array: inout [Int]) {

    bingo_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)
}