lctree

Rust implementation of Link-cut tree: self-balancing data structure to maintain a dynamic forest of (un)rooted trees under the following operations that take O(logn) amortized time:

• link(v, w): creates an edge between nodes v and w.
• cut(v, w): removes the edge between nodes v and w.
• connected(v, w): returns true if nodes v and w are in the same tree.
• path(v, w): performs calculations on a path between nodes v and w.

Usage

This example shows how to link and cut edges:

use lctree::LinkCutTree;

fn main() {
    // We form a link-cut tree for the following forest:
    // (the numbers in parentheses are the weights of the nodes):
    //            a(9)
    //           /    \
    //         b(1)    e(2)
    //        /   \      \
    //      c(8)  d(10)   f(4)
    let mut lctree = LinkCutTree::default();
    let a = lctree.make_tree(9.);
    let b = lctree.make_tree(1.);
    let c = lctree.make_tree(8.);
    let d = lctree.make_tree(10.);
    let e = lctree.make_tree(2.);
    let f = lctree.make_tree(4.);

    lctree.link(b, a);
    lctree.link(c, b);
    lctree.link(d, b);
    lctree.link(e, a);
    lctree.link(f, e);

    // Checking connectivity:
    assert!(lctree.connected(c, f)); // connected

    // Path aggregation:
    // We find the node with max weight on the path between c to f,
    // where a has the maximum weight of 9.0:
    let heaviest_node = lctree.path(c, f);
    assert_eq!(heaviest_node.idx, a);
    assert_eq!(heaviest_node.weight, 9.0);

    // We cut node e from its parent a:
    lctree.cut(e, a);

    // The forest should now look like this:
    //            a(9)
    //           /    
    //         b(1)      e(2)
    //        /   \        \
    //      c(8)  d(10)    f(4)

    // We check connectivity again:
    assert!(!lctree.connected(c, f)); // not connected anymore
}

Advanced usage include operations on paths:

Common path aggregates

Various kinds of calculations can be performed on a path between two nodes, such as findmax, findmin, or findsum:

use lctree::{LinkCutTree, FindMax, FindMin, FindSum};

fn main() {
    // We form a link-cut tree from the following rooted tree
    // (the numbers in parentheses are the weights of the nodes):
    //           a(9)
    //           /  \
    //         b(1)  e(2)
    //        /   \    \
    //      c(8)  d(10)  f(4)

    // Use FindMax, FindMin or FindSum, depending on your usage:
    let mut lctree: LinkCutTree<FindSum> = lctree::LinkCutTree::new();
    let a = lctree.make_tree(9.);
    let b = lctree.make_tree(1.);
    let c = lctree.make_tree(8.);
    let d = lctree.make_tree(10.);
    let e = lctree.make_tree(2.);
    let f = lctree.make_tree(4.);

    lctree.link(b, a);
    lctree.link(c, b);
    lctree.link(d, b);
    lctree.link(e, a);
    lctree.link(f, e);

    // We find the sum of the weights on the path between c to f,
    let result = lctree.path(c, f);
    assert_eq!(result.sum, 8. + 1. + 9. + 2. + 4.);
}

Custom path aggregate function

A custom path aggregate function can be defined by using the Path trait:

use lctree::{LinkCutTree, Path};

#[derive(Copy, Clone)]
pub struct FindXor {
    pub xor: u64,
}

impl Path for FindXor {
    fn default(weight: f64, _: usize) -> Self {
        FindXor {
            xor: weight as u64,
        }
    }

    fn aggregate(&mut self, other: Self) {
        self.xor ^= other.xor;
    }
}

fn main() {
    // We form a link-cut tree from the following rooted tree
    // (the numbers in parentheses are the weights of the nodes):
    //           a(9)
    //           /  \
    //         b(1)  e(2)
    //        /   \    \
    //      c(8)  d(10)  f(4)
    let mut lctree: LinkCutTree<FindXor> = LinkCutTree::new();
    let a = lctree.make_tree(9.);
    let b = lctree.make_tree(1.);
    let c = lctree.make_tree(8.);
    let d = lctree.make_tree(10.);
    let e = lctree.make_tree(2.);
    let f = lctree.make_tree(4.);

    lctree.link(b, a);
    lctree.link(c, b);
    lctree.link(d, b);
    lctree.link(e, a);
    lctree.link(f, e);

    // We find the xor of the weights on the path between c to f,
    let result = lctree.path(c, f);
    assert_eq!(result.xor, 8 ^ 1 ^ 9 ^ 2 ^ 4);
}

Benchmark

The overall running time for performing a number of random operations (link(v, w), cut(v, w), connected(v, w) or findmax(v, w)) on forests of varying sizes (check benchmark details here).

# Nodes	# Operations	lctree	brute-force
100	10K	4.8161 ms	18.013 ms
200	20K	11.091 ms	69.855 ms
500	50K	31.623 ms	429.53 ms
1000	100K	68.649 ms	1.8746 s
5000	500K	445.83 ms	46.854 s
10K	1M	964.64 ms	183.24 s

Credits

This crate applies the core concepts and ideas presented in the following sources:

• "A data structure for dynamic trees" by D. Sleator and R. E. TarJan (published in STOC '81).
• Link-cut tree source code by the author D. Sleator.
• MIT's lecture on dynamic graphs: lecture, notes, and source code.
• Helpful blog posts on the concepts of rooted trees, rerooting and splay operations.

License

This project is licensed under the Apache License, Version 2.0 - See the LICENSE.md file for details.

Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be licensed as above, without any additional terms or conditions.