Optimal disjoint-set based implementation

We can do two things to improve the simple and sub-optimal disjoint-set subalgorithms:

1. Path compression heuristic: findSet does not need to ever handle a tree with height bigger than 2. If it ends up iterating such a tree, it can link the lower nodes directly to the root, optimizing future traversals;

subalgo findSet(v: a node):
    if v.parent != v
        v.parent = findSet(v.parent)
    return v.parent

1. Height-based merging heuristic: for each node, store the height of its subtree. When merging, make the taller tree the parent of the smaller one, thus not increasing anyone’s height.

subalgo unionSet(u, v: nodes):
    vRoot = findSet(v)
    uRoot = findSet(u)

if vRoot == uRoot:
    return

if vRoot.height < uRoot.height:
    vRoot.parent = uRoot
else if vRoot.height > uRoot.height:
    uRoot.parent = vRoot
else:
    uRoot.parent = vRoot
    uRoot.height =  uRoot.height + 1

This leads to O(alpha(n)) time for each operation, where alpha is the inverse of the fast-growing Ackermann function, thus it is very slow growing, and can be considered O(1) for practical purposes.

This makes the entire Kruskal’s algorithm O(m log m + m) = O(m log m), because of the initial sorting.

Note

Path compression may reduce the height of the tree, hence comparing heights of the trees during union operation might not be a trivial task. Hence to avoid the complexity of storing and calculating the height of the trees the resulting parent can be picked randomly:

subalgo unionSet(u, v: nodes):
    vRoot = findSet(v)
    uRoot = findSet(u)

if vRoot == uRoot:
    return
if random() % 2 == 0:
    vRoot.parent = uRoot
else:
    uRoot.parent = vRoot

In practice this randomised algorithm together with path compression for findSet operation will result in comparable performance, yet much simpler to implement.