To kickoff the series of posts about algorithms and data structures I find interesting, I will be talking about my favorite one: the Disjoint Set. Also known as the Union-Find data structure, so named because of its two main operations: ds.union(lhs, rhs) and ds.find(elem).

What does it do?

The Union-Find data structure allows one to store a collection of sets of elements, with operations for adding new sets, merging two sets into one, and finding the representative member of a set. Not only does it do all that, but it does it in almost constant (amortized) time!

Here is a small motivating example for using the Disjoint Set data structure:

def connected_components(graph: Graph) -> list[set[Node]]:
    # Initialize the disjoint set so that each node is in its own set
    ds: DisjointSet[Node] = DisjointSet(graph.nodes)
    # Each edge is a connection, merge both sides into the same set
    for (start, dest) in graph.edges:
        ds.union(start, dest)
    # Connected components share the same (arbitrary) root
    components: dict[Node, set[Node]] = defaultdict(set)
    for n in graph.nodes:
        components[ds.find(n)].add(n)
    # Return a list of disjoint sets corresponding to each connected component
    return list(components.values())

Implementation

I will show how to implement UnionFind for integers, though it can easily be extended to be used with arbitrary types (e.g: by mapping each element one-to-one to a distinct integer, or using a different set representation).

Representation

Creating a new disjoint set is easy enough:

class UnionFind:
    _parent: list[int]
    _rank: list[int]

    def __init__(self, size: int):
        # Each node is in its own set, making it its own parent...
        self._parents = list(range(size))
        # ... And its rank 0
        self._rank = [0] * size

We represent each set through the _parent field: each element of the set is linked to its parent, until the root node which is its own parent. When first initializing the structure, each element is in its own set, so we initialize each element to be a root and make it its own parent (_parent[i] == i for all i).

The _rank field is an optimization which we will touch on in a later section.

Find

A naive Implementation of find(...) is simple enough to write:

def find(self, elem: int) -> int:
    # If `elem` is its own parent, then it is the root of the tree
    if (parent := self._parent[elem]) == elem:
        return elem
    # Otherwise, recurse on the parent
    return self.find(parent)

However, going back up the chain of parents each time we want to find the root node (an O(n) operation) would make for disastrous performance. Instead we can do a small optimization called path splitting.

def find(self, elem: int) -> int:
    while (parent := self._parent[elem]) != elem:
        # Replace each parent link by a link to the grand-parent
        elem, self._parent[elem] = parent, self._parent[parent]
    return elem

This flattens the chain so that each node links more directly to the root (the length is reduced by half), making each subsequent find(...) faster.

Other compression schemes exist, along the spectrum between faster shortening the chain faster earlier, or updating _parent fewer times per find(...).

Union

A naive implementation of union(...) is simple enough to write:

def union(self, lhs: int, rhs: int) -> int:
    # Replace both element by their root parent
    lhs = self.find(lhs)
    rhs = self.find(rhs)
    # arbitrarily merge one into the other
    self._parent[rhs] = lhs
    # Return the new root
    return lhs

Once again, improvements can be made. Depending on the order in which we call union(...), we might end up creating a long chain from the leaf of the tree to the root node, leading to slower find(...) operations. If at all possible, we would like to keep the trees as shallow as possible.

To do so, we want to avoid merging taller trees into smaller ones, so as to keep them as balanced as possible. Since a higher tree will result in a slower find(...), keeping the trees balanced will lead to increased performance.

This is where the _rank field we mentioned earlier comes in: the rank of an element is an upper bound on its height in the tree. By keeping track of this approximate height, we can keep the trees balanced when merging them.

def union(self, lhs: int, rhs: int) -> int:
    lhs = self.find(lhs)
    rhs = self.find(rhs)
    # Bail out early if they already belong to the same set
    if lhs == rhs:
      return lhs
    # Always keep `lhs` as the taller tree
    if (self._rank[lhs] < self._rank[rhs])
        lhs, rhs = rhs, lhs
    # Merge the smaller tree into the taller one
    self._parent[rhs] = lhs
    # Update the rank when merging trees of approximately the same size
    if self._rank[lhs] == self._rank[rhs]:
        self._rank[lhs] += 1
    return lhs