ヒープソートを使用する

ヒープソート (heap sort) は、配列を二分ヒープに整え、根(最大値)と末尾を入れ替えて確定させながらヒープを縮めていく。

  1. 準備: 配列をインデックス付きの完全二分木とみなす。親 i (i >= 0) に対し左の子は 2i+1、右の子は 2i+2 になる。
  2. 構築: 親の値が子の値以上という条件を満たすよう、葉に近い内部ノードから順に沈降して最大ヒープにする。
  3. 抽出: 根(位置 0)が全体の最大なので、それとヒープの末尾を交換する。末尾側はもうソート済み領域として固定し、残り [0 .. heapEnd-1] だけをヒープとして見なす。
  4. 反復: 新しい根がヒープ条件を満たすまで子と比較・交換して下へ沈める。ヒープ長が 1 になるまで手順 3〜4 を繰り返す。
procedure heap_sort(A)
  n = length(A)
  for i from floor(n / 2) - 1 down to 0
    sift_down(A, i, n)
  for end from n - 1 down to 1
    swap(A[0], A[end])
    sift_down(A, 0, end)

procedure sift_down(A, start, heap_end)
  i = start
  while true
    l = 2 * i + 1
    r = 2 * i + 2
    largest = i
    if l < heap_end and A[l] > A[largest] then
      largest = l
    if r < heap_end and A[r] > A[largest] then
      largest = r
    if largest = i then
      break
    swap(A[i], A[largest])
    i = largest

バブルソートの O(n²) と比べて大きな入力では優位になりやすく、クイックソートと異なり最悪計算量も O(n log n) に抑えられる反面、定数倍やキャッシュ効率では実装やデータに依存してクイックソートに劣ることもある。

優先度付きキューの内部にもヒープが使われるため、アルゴリズムの理解はヒープソートだけでなく広い実装の読み取りにもつながる。

類似アルゴリズムとの相違点

クイックソートより最悪計算量の保証は強いが、キャッシュ効率では負けることがある。スムースソートもインプレースで 最悪計算量 O(n log n) だが、整列済み入力ではさらに軽い。

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 0.000008 0.000386 1662 1668
512 0.000017 0.000390 1666 1672
1024 0.000039 0.000383 1674 1680
2048 0.000083 0.000380 1690 1696
4096 0.000185 0.000454 1722 1728
8192 0.000410 0.001943 1786 1792
16384 0.000846 0.001191 1918 1924
32768 0.001867 0.013515 2177 2184
65536 0.004028 0.022833 2690 2696
131072 0.008586 0.021879 3714 3720
262144 0.019282 0.044340 5762 5768
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 sift_down(a: &mut [usize], mut root: usize, end: usize) {
    loop {
        let child = root * 2 + 1;
        if child > end {
            break;
        }
        let mut swap_idx = child;
        if child < end && a[child] < a[child + 1] {
            swap_idx = child + 1;
        }
        if a[root] >= a[swap_idx] {
            break;
        }
        a.swap(root, swap_idx);
        root = swap_idx;
    }
}

fn heap_sort(a: &mut [usize]) {
    if a.len() <= 1 {
        return;
    }
    for start in (0..a.len() / 2).rev() {
        sift_down(a, start, a.len() - 1);
    }
    for end in (1..a.len()).rev() {
        a.swap(0, end);
        sift_down(a, 0, end - 1);
    }
}





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

    heap_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],
    );
    for seed in 0..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