# Optimal disjoint-set based implementation

suggest change

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.