proxmox-backup/src/pxar/binary_search_tree.rs

//! Helpers to generate a binary search tree stored in an array from a
//! sorted array.
//!
//! Specifically, for any given sorted array 'input' permute the
//! array so that the following rule holds:
//!
//! For each array item with index i, the item at 2i+1 is smaller and
//! the item 2i+2 is larger.
//!
//! This structure permits efficient (meaning: O(log(n)) binary
//! searches: start with item i=0 (i.e. the root of the BST), compare
//! the value with the searched item, if smaller proceed at item
//! 2i+1, if larger proceed at item 2i+2, and repeat, until either
//! the item is found, or the indexes grow beyond the array size,
//! which means the entry does not exist.
//!
//! Effectively this implements bisection, but instead of jumping
//! around wildly in the array during a single search we only search
//! with strictly monotonically increasing indexes.
//!
//! Algorithm is from casync (camakebst.c), simplified and optimized
//! for rust. Permutation function originally by L. Bressel, 2017. We
//! pass permutation info to user provided callback, which actually
//! implements the data copy.
//!
//! The Wikipedia Artikel for [Binary
//! Heap](https://en.wikipedia.org/wiki/Binary_heap) gives a short
//! intro howto store binary trees using an array.

use std::cmp::Ordering;

#[allow(clippy::many_single_char_names)]
fn copy_binary_search_tree_inner<F:  FnMut(usize, usize)>(
    copy_func: &mut F,
    // we work on input array input[o..o+n]
    n: usize,
    o: usize,
    e: usize,
    i: usize,
) {
    let p = 1 << e;

    let t = p + (p>>1) - 1;

    let m = if n > t {
        // |...........p.............t....n........(2p)|
        p - 1
    } else {
        // |...........p.....n.......t.............(2p)|
        p - 1 - (t-n)
    };

    (copy_func)(o+m, i);

    if m > 0 {
        copy_binary_search_tree_inner(copy_func, m, o, e-1, i*2+1);
    }

    if (m + 1) < n {
        copy_binary_search_tree_inner(copy_func, n-m-1, o+m+1, e-1, i*2+2);
    }
}

/// This function calls the provided `copy_func()` with the permutaion
/// info.
///
/// ```
/// # use proxmox_backup::pxar::copy_binary_search_tree;
/// copy_binary_search_tree(5, |src, dest| {
///    println!("Copy {} to {}", src, dest);
/// });
/// ```
///
/// This will produce the folowing output:
///
/// ```no-compile
/// Copy 3 to 0
/// Copy 1 to 1
/// Copy 0 to 3
/// Copy 2 to 4
/// Copy 4 to 2
/// ```
///
/// So this generates the following permuation: `[3,1,4,0,2]`.

pub fn copy_binary_search_tree<F:  FnMut(usize, usize)>(
    n: usize,
    mut copy_func: F,
) {
    if n == 0 { return };
    let e = (64 - n.leading_zeros() - 1) as usize; // fast log2(n)

    copy_binary_search_tree_inner(&mut copy_func, n, 0, e, 0);
}


/// This function searches for the index where the comparison by the provided
/// `compare()` function returns `Ordering::Equal`.
/// The order of the comparison matters (noncommutative) and should be search
/// value compared to value at given index as shown in the examples.
/// The parameter `skip_multiples` defines the number of matches to ignore while
/// searching before returning the index in order to lookup duplicate entries in
/// the tree.
///
/// ```
/// # use proxmox_backup::pxar::{copy_binary_search_tree, search_binary_tree_by};
/// let mut vals = vec![0,1,2,2,2,3,4,5,6,6,7,8,8,8];
///
/// let clone = vals.clone();
/// copy_binary_search_tree(vals.len(), |s, d| {
///     vals[d] = clone[s];
/// });
/// let should_be = vec![5,2,8,1,3,6,8,0,2,2,4,6,7,8];
/// assert_eq!(vals, should_be);
///
/// let find = 8;
/// let skip_multiples = 0;
/// let idx = search_binary_tree_by(0, vals.len(), skip_multiples, |idx| find.cmp(&vals[idx]));
/// assert_eq!(idx, Some(2));
///
/// let find = 8;
/// let skip_multiples = 1;
/// let idx = search_binary_tree_by(2, vals.len(), skip_multiples, |idx| find.cmp(&vals[idx]));
/// assert_eq!(idx, Some(6));
///
/// let find = 8;
/// let skip_multiples = 1;
/// let idx = search_binary_tree_by(6, vals.len(), skip_multiples, |idx| find.cmp(&vals[idx]));
/// assert_eq!(idx, Some(13));
///
/// let find = 5;
/// let skip_multiples = 1;
/// let idx = search_binary_tree_by(0, vals.len(), skip_multiples, |idx| find.cmp(&vals[idx]));
/// assert!(idx.is_none());
///
/// let find = 5;
/// let skip_multiples = 0;
/// // if start index is equal to the array length, `None` is returned.
/// let idx = search_binary_tree_by(vals.len(), vals.len(), skip_multiples, |idx| find.cmp(&vals[idx]));
/// assert!(idx.is_none());
///
/// let find = 5;
/// let skip_multiples = 0;
/// // if start index is larger than length, `None` is returned.
/// let idx = search_binary_tree_by(vals.len() + 1, vals.len(), skip_multiples, |idx| find.cmp(&vals[idx]));
/// assert!(idx.is_none());
/// ```

pub fn search_binary_tree_by<F: Copy + Fn(usize) -> Ordering>(
    start: usize,
    size: usize,
    skip_multiples: usize,
    compare: F
) -> Option<usize> {
    if start >= size {
        return None;
    }

    let mut skip = skip_multiples;
    let cmp = compare(start);
    if cmp == Ordering::Equal {
        if skip == 0 {
            // Found matching hash and want this one
            return Some(start);
        }
        // Found matching hash, but we should skip the first `skip_multiple`,
        // so continue search with reduced skip count.
        skip -= 1;
    }

    if cmp == Ordering::Less || cmp == Ordering::Equal {
        let res = search_binary_tree_by(2 * start + 1, size, skip, compare);
        if res.is_some() {
            return res;
        }
    }

    if cmp == Ordering::Greater || cmp == Ordering::Equal {
        let res = search_binary_tree_by(2 * start + 2, size, skip, compare);
        if res.is_some() {
            return res;
        }
    }

    None
}

#[test]
fn test_binary_search_tree() {

    fn run_test(len: usize) -> Vec<usize> {

        const MARKER: usize = 0xfffffff;
        let mut output = vec![];
        for _i in 0..len { output.push(MARKER); }
        copy_binary_search_tree(len, |s, d| {
            assert!(output[d] == MARKER);
            output[d] = s;
        });
        if len < 32 { println!("GOT:{}:{:?}", len, output); }
        for i in 0..len {
            assert!(output[i] != MARKER);
        }
        output
    }

    assert!(run_test(0).len() == 0);
    assert!(run_test(1) == [0]);
    assert!(run_test(2) == [1,0]);
    assert!(run_test(3) == [1,0,2]);
    assert!(run_test(4) == [2,1,3,0]);
    assert!(run_test(5) == [3,1,4,0,2]);
    assert!(run_test(6) == [3,1,5,0,2,4]);
    assert!(run_test(7) == [3,1,5,0,2,4,6]);
    assert!(run_test(8) == [4,2,6,1,3,5,7,0]);
    assert!(run_test(9) == [5,3,7,1,4,6,8,0,2]);
    assert!(run_test(10) == [6,3,8,1,5,7,9,0,2,4]);
    assert!(run_test(11) == [7,3,9,1,5,8,10,0,2,4,6]);
    assert!(run_test(12) == [7,3,10,1,5,9,11,0,2,4,6,8]);
    assert!(run_test(13) == [7,3,11,1,5,9,12,0,2,4,6,8,10]);
    assert!(run_test(14) == [7,3,11,1,5,9,13,0,2,4,6,8,10,12]);
    assert!(run_test(15) == [7,3,11,1,5,9,13,0,2,4,6,8,10,12,14]);
    assert!(run_test(16) == [8,4,12,2,6,10,14,1,3,5,7,9,11,13,15,0]);
    assert!(run_test(17) == [9,5,13,3,7,11,15,1,4,6,8,10,12,14,16,0,2]);

    for len in 18..1000 {
        run_test(len);
    }
}