An in-memory topological sort algorithm for trees based on Group Theory
Documentation

# Tree-Memory-Sort

An in-memory topological sort algorithm for trees based on Group Theory

### Design

This algorithm uses in-memory swapping directly on the array which nodes are stored. Since swap operations satisfy the axioms of Group Theory, the topological sort can be made more efficient by using a group generator.

A group generator is an array which stores an index for every node index. When swap operations are performed on the array instead of the data, it is possible to predict where nodes will be stored in the solution without changing the meaning of the current node indices.

Once a solution has been found, the group generator can be used to retrace the swapping operations required to order the tree.

The order which swaps are retraced might be different than the solving phase:

`a, b, c` => `a, (c, b)` => `(c, a), b` => `c, a, b` (solving phase)
`c, a, b` => `(b), a, (c)` => `(a, b), c` => `a, b, c` (retrace phase)

### Primes example

This example shows how the algorithm works using some simple numbers.

Assume that you have two equations:

12 = 2 * 6
6 = 3 * 2

If you arrange these equations as a tree, you will naturally start at the top 12 and list children of each node in the same order as in the equations.

When using an automated theorem prover that re-writes a tree like this, it can get messy. Some algorithms relies on a well-ordered tree to perform efficiently.

By performing topological sort on the tree, it can be restored to a well-ordered form:

Tree            i       i'
--------------------------
12              3   =>  0
|- 2            2   =>  1
|- 6            1   =>  2
|- 3         4   =>  3
|- 2         0   =>  4

The algorithm does not change the relative connections inside the tree, just how nodes are stored in memory. However, it needs to change the indices such they point to the correct nodes.

Source code:

extern crate tree_mem_sort;

use tree_mem_sort::sort;

#[derive(Debug)]
pub struct Number {
/// The value of the number.
pub value: u32,
/// Which number this was factored from.
pub parent: Option<usize>,
/// Prime factors.
pub children: Vec<usize>,
}

fn main() {
let mut nodes = vec![
Number {
value: 2,
parent: Some(1),
children: vec![],
}, // 0
Number {
value: 6,
parent: Some(0),
children: vec![4, 0],
}, // 1
Number {
value: 2,
parent: Some(2),
children: vec![],
}, // 2
Number {
value: 12,
parent: None,
children: vec![2, 1],
}, // 3
Number {
value: 3,
parent: Some(2),
children: vec![],
}, // 4
];
for i in 0..nodes.len() {
println!("{}: {:?}", i, nodes[i]);
}
// Prints `[2, 6, 2, 12, 3]`
println!("{:?}", nodes.iter().map(|n| n.value).collect::<Vec<u32>>());

sort(&mut nodes, |n| &mut n.parent, |n| &mut n.children);
println!("");
for i in 0..nodes.len() {
println!("{}: {:?}", i, nodes[i]);
}
// Prints `[12, 2, 6, 3, 2]`
println!("{:?}", nodes.iter().map(|n| n.value).collect::<Vec<u32>>());
}

### Limitations

The sort algorithm assumes that each node is referenced by maximum one parent. If you share nodes between parent nodes, the algorithm might enter an infinite loop.

One can use sort_dag to sort a tree where nodes can have multiple parents. In order for the algorithm to work with shared nodes, the tree must be a Directed Acyclic Graph (DAG). If the tree is not a DAG, the algorithm will run in an infinite loop.

### Why topological sort on trees? Why not use DAG representation?

The idea is to preserve the following properties, and otherwise minimize work:

• Each child is greater than their parent
• Each sibling is greater than previous siblings

Topological sorts is often associated with Directed Acyclic Graphs (DAG) and not trees. This algorithm works on DAGs, but not on all trees with shared nodes.

• If every node is referenced by maximum one parent, then it is automatically a DAG
• Trees with ordered children encode arrows among siblings, which affects whether it is a DAG

For example, the following tree with shared nodes is not a DAG:

A
|- B
|- D
|- C
|- C
|- D

Notice that B orders D before C, so D is less than C. However, since D is a child of C it must be greater than C. This leads to a contradiction.

If you try to sort the tree above using sort_dag, it will run in an infinite loop.

Trees are easy to reason about and has a more efficient encoding for this library's common usage. For N children, the arrows of an equivalent DAG requires at least N arrows. In addition, these arrows must be arranged in a such way that the children becomes a total order. This is necessary to determine the order for every pair of children. By using a tree with ordered children, the memory required for arrows is zero, because the children are stored in an array anyway.

A left-child first traversal of a tree without shared nodes can be used to produce a topological order. However, building indices from traversal of a tree makes all indices of a right-child larger than the indices of a left-child. This moves nodes around more than desirable.

For forests, a tree traversal also requires an extra pass through all nodes. Here, the algorithm that sorts trees also works for forests without modification.

A topological sort of a tree has the property that if you push a new node at the end of the array storing the nodes, which node's parent is any existing node, then the new tree is topologically sorted. The same is not true for indices built from tree traversal.