バイトニックソートを使用する

バイトニックソート (bitonic sort) は、部分列を昇順・降順の2つの単調列(バイトニック列)に組み立て、距離 k の要素同士を比較・交換するバイトニックマージで整列させる。

  1. 分割: 長さ n(2の冪)の区間を半分に分け、前半を昇順・後半を降順に整えるよう再帰する。
  2. バイトニック列の形成: 再帰の底で長さ2の区間は1回の比較で昇順または降順になる。
  3. バイトニックマージ: 区間の前半と後半を距離 n/2 でペアにし、方向に応じて比較交換する。その後、半分の長さで同じ処理を再帰する。
  4. 全体: 最上位の呼び出しで昇順方向を指定すれば、配列全体が昇順になる。
procedure compare_exchange(A, i, j, dir_up)
  if dir_up and A[i] > A[j] then
    swap(A[i], A[j])
  if not dir_up and A[i] < A[j] then
    swap(A[i], A[j])

procedure bitonic_merge(A, lo, cnt, dir_up)
  if cnt <= 1 then
    return
  k = cnt / 2
  for i from lo to lo + k - 1
    compare_exchange(A, i, i + k, dir_up)
  bitonic_merge(A, lo, k, dir_up)
  bitonic_merge(A, lo + k, k, dir_up)

procedure bitonic_sort(A, lo, cnt, dir_up)
  if cnt <= 1 then
    return
  k = cnt / 2
  bitonic_sort(A, lo, k, true)
  bitonic_sort(A, lo + k, k, false)
  bitonic_merge(A, lo, cnt, dir_up)

並列比較ネットワーク向けで、逐次実行では O(n log² n) となり、要素数は 2 の冪を前提とする実装が多い。

類似アルゴリズムとの相違点

奇偶マージソートも固定距離の比較ネットワークだが、バイトニックは列を昇順・降順の 2 列に組み立ててからマージする。マージソートは可変長の部分列を併合する点が異なる。

計算時間量および空間計算量を計測する

Size Average time Maximum time Average memory Maximum memory
256 0.000016 0.000192 62 68
512 0.000035 0.000534 58 64
1024 0.000077 0.000287 58 64
2048 0.000172 0.002404 62 68
4096 0.000378 0.001847 74 80
8192 0.000801 0.002242 62 68
16384 0.001766 0.016892 57 64
32768 0.003937 0.022379 74 80
65536 0.008867 0.020473 62 68
131072 0.023818 0.049083 58 64
262144 0.054617 0.127214 74 80
計測に使用したコードを表示する

set -euo pipefail

WORKDIR="$(mktemp -d)"
trap 'rm -rf "$WORKDIR"' EXIT

cat > "$WORKDIR/Dockerfile" <<'EOF'
FROM rust:1.95.0

WORKDIR /app

RUN mkdir -p src

RUN cat > Cargo.toml <<'CARGO'
[package]
name = "rust-benchmark"
version = "0.1.0"
edition = "2021"

[profile.release]
lto = true
codegen-units = 1
panic = "abort"
CARGO

RUN cat > src/main.rs <<'RUST'
use std::{
    env,
    process::Command,
    time::{Duration, Instant},
};
const MIN_POWER: u32 = 8;
const MAX_POWER: u32 = 18;
const RUNS: usize = 8192;


fn compare_exchange(a: &mut [usize], i: usize, j: usize, dir_up: bool) {
    let swap = if dir_up {
        a[i] > a[j]
    } else {
        a[i] < a[j]
    };
    if swap {
        a.swap(i, j);
    }
}

fn bitonic_merge(a: &mut [usize], lo: usize, cnt: usize, dir_up: bool) {
    if cnt <= 1 {
        return;
    }
    let k = cnt / 2;
    for i in lo..lo + k {
        compare_exchange(a, i, i + k, dir_up);
    }
    bitonic_merge(a, lo, k, dir_up);
    bitonic_merge(a, lo + k, k, dir_up);
}

fn bitonic_sort_range(a: &mut [usize], lo: usize, cnt: usize, dir_up: bool) {
    if cnt <= 1 {
        return;
    }
    let k = cnt / 2;
    bitonic_sort_range(a, lo, k, true);
    bitonic_sort_range(a, lo + k, k, false);
    bitonic_merge(a, lo, cnt, dir_up);
}

fn bitonic_sort(a: &mut [usize]) {
    let n = a.len();
    if n <= 1 {
        return;
    }
    if n.is_power_of_two() {
        bitonic_sort_range(a, 0, n, true);
        return;
    }
    // Classic bitonic sort assumes a power-of-two length; pad for other sizes.
    let k = n.next_power_of_two();
    let mut buf = vec![usize::MAX; k];
    buf[..n].copy_from_slice(a);
    bitonic_sort_range(&mut buf, 0, k, true);
    a.copy_from_slice(&buf[..n]);
}


fn benchmark_sort(array: &mut [usize]) {

    bitonic_sort(array);

}

fn is_non_decreasing(a: &[usize]) -> bool {
    a.windows(2).all(|w| w[0] <= w[1])
}

fn same_multiset(a: &[usize], b: &[usize]) -> bool {
    if a.len() != b.len() {
        return false;
    }

    let mut left = a.to_vec();
    let mut right = b.to_vec();
    left.sort_unstable();
    right.sort_unstable();
    left == right
}

fn check_correctness_case(label: &str, mut input: Vec<usize>) {
    let original = input.clone();

    benchmark_sort(&mut input);

    if !is_non_decreasing(&input) {
        panic!("correctness case {}: output is not sorted", label);
    }

    if !same_multiset(&input, &original) {
        panic!("correctness case {}: elements were lost or added", label);
    }
}

fn few_unique_values(size: usize, unique: usize, seed: u64) -> Vec<usize> {
    let mut state = seed;

    (0..size)
        .map(|_| {
            state ^= state << 13;
            state ^= state >> 7;
            state ^= state << 17;
            (state as usize % unique) + 1
        })
        .collect()
}

fn run_correctness_checks() {
    check_correctness_case("empty", vec![]);
    check_correctness_case("single", vec![42]);
    check_correctness_case("duplicates", vec![3, 1, 3, 2, 1, 2]);
    check_correctness_case("sorted", vec![1, 2, 3, 4, 5]);
    check_correctness_case("reverse", vec![5, 4, 3, 2, 1]);
    check_correctness_case("all_equal", vec![7, 7, 7, 7]);
    check_correctness_case("skewed_range", vec![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",
        vec![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(
            &format!("few_keys_len32_seed_{seed}"),
            few_unique_values(32, 4, seed),
        );
    }
}


fn shuffled(size: usize, seed: u64) -> Vec<usize> {
    let mut v: Vec<usize> = (1..=size).collect();

    let mut state = seed;

    for i in (1..size).rev() {
        state ^= state << 13;
        state ^= state >> 7;
        state ^= state << 17;

        let j = (state as usize) % (i + 1);

        v.swap(i, j);
    }

    v
}

fn memory_usage_kb() -> usize {
    // VmHWM (peak RSS, KiB). Reported memory subtracts a per-size baseline that only
    // holds the input array, so the table reflects auxiliary space during sorting.
    let contents = std::fs::read_to_string("/proc/self/status")
        .unwrap_or_default();

    for line in contents.lines() {
        if let Some(rest) = line.strip_prefix("VmHWM:") {
            let kb = rest
                .split_whitespace()
                .next()
                .unwrap_or("0")
                .parse::<usize>()
                .unwrap_or(0);

            return kb;
        }
    }

    0
}

fn micros(d: Duration) -> u128 {
    d.as_micros()
}

fn input_array(size: usize, seed: u64) -> Vec<usize> {
    shuffled(size, seed)
}

fn run_baseline(size: usize) -> usize {
    let _hold = input_array(size, 1);
    memory_usage_kb()
}

fn run_once(size: usize, seed: usize) -> (u128, usize) {
    let mut array = input_array(size, seed as u64);

    let start = Instant::now();

    benchmark_sort(&mut array);

    let elapsed = start.elapsed();
    let mem = memory_usage_kb();

    let expected: Vec<usize> = (1..=size).collect();
    if array != expected {
        panic!(
            "sort failed with seed {} for size {}",
            seed,
            size
        );
    }

    (micros(elapsed), mem)
}

fn run_baseline_child(args: &[String]) {
    let size = args[2].parse::<usize>().expect("invalid size");
    let mem = run_baseline(size);
    println!("{}", mem);
}

fn run_child(args: &[String]) {
    let size = args[2].parse::<usize>().expect("invalid size");
    let seed = args[3].parse::<usize>().expect("invalid seed");
    let (elapsed_us, mem) = run_once(size, seed);
    println!("{} {}", elapsed_us, mem);
}

fn main() {
    let args: Vec<String> = env::args().collect();
    if args.get(1).is_some_and(|arg| arg == "--baseline-once") {
        run_baseline_child(&args);
        return;
    }
    if args.get(1).is_some_and(|arg| arg == "--run-once") {
        run_child(&args);
        return;
    }

    run_correctness_checks();

    println!(
        "| {:>10} | {:>15} | {:>15} | {:>15} | {:>15} |",
        "Size",
        "Average time",
        "Maximum time",
        "Average memory",
        "Maximum memory"
    );

    println!(
        "|{:-<11}:|{:-<16}:|{:-<16}:|{:-<16}:|{:-<16}:|",
        "",
        "",
        "",
        "",
        ""
    );

    for power in MIN_POWER..=MAX_POWER {
        let size = 1usize << power;

        let baseline_output = Command::new(env::current_exe().expect("failed to find current executable"))
            .arg("--baseline-once")
            .arg(size.to_string())
            .output()
            .expect("failed to run benchmark baseline process");

        if !baseline_output.status.success() {
            panic!(
                "benchmark baseline process failed: {}",
                String::from_utf8_lossy(&baseline_output.stderr)
            );
        }

        let baseline_stdout = String::from_utf8(baseline_output.stdout)
            .expect("baseline process returned non-UTF-8 output");
        let baseline_mem = baseline_stdout
            .split_whitespace()
            .next()
            .expect("missing baseline memory usage")
            .parse::<usize>()
            .expect("invalid baseline memory usage");

        let mut total_time: u128 = 0;
        let mut max_time: u128 = 0;

        let mut total_mem: usize = 0;
        let mut max_mem: usize = 0;

        for seed in 1..=RUNS {
            let output = Command::new(env::current_exe().expect("failed to find current executable"))
                .arg("--run-once")
                .arg(size.to_string())
                .arg(seed.to_string())
                .output()
                .expect("failed to run benchmark child process");

            if !output.status.success() {
                panic!(
                    "benchmark child process failed: {}",
                    String::from_utf8_lossy(&output.stderr)
                );
            }

            let stdout = String::from_utf8(output.stdout)
                .expect("child process returned non-UTF-8 output");
            let mut fields = stdout.split_whitespace();
            let elapsed_us = fields
                .next()
                .expect("missing elapsed time")
                .parse::<u128>()
                .expect("invalid elapsed time");
            let mem = fields
                .next()
                .expect("missing memory usage")
                .parse::<usize>()
                .expect("invalid memory usage");

            total_time += elapsed_us;

            if elapsed_us > max_time {
                max_time = elapsed_us;
            }

            let aux_mem = mem.saturating_sub(baseline_mem);

            total_mem += aux_mem;

            if aux_mem > max_mem {
                max_mem = aux_mem;
            }
        }

        let avg_time = total_time / RUNS as u128;
        let avg_mem = total_mem / RUNS;

        println!(
            "| {:>10} | {:>15} | {:>15} | {:>15} | {:>15} |",
            size,
            format!("{}.{:06}", avg_time / 1_000_000, avg_time % 1_000_000),
            format!("{}.{:06}", max_time / 1_000_000, max_time % 1_000_000),
            avg_mem,
            max_mem
        );
    }
}
RUST

RUN cargo build --release

CMD ["./target/release/rust-benchmark"]
EOF

docker build -t rust-benchmark "$WORKDIR"
docker run --rm --init rust-benchmark