14.5 Weighted Quick Union with Path Compression

Weighted Quick Union is pretty good, but we can do even better!
Professor Hug's explanation on Weighted Quick Union with Path Compression
The clever insight is realizing that whenever we call find(x) we have to traverse the path from x to root. So, along the way we can connect all the items we visit to their root at no extra asymptotic cost.
Connecting all the items along the way to the root will help make our tree shorter with each call to find.
Recall that both connect(x, y) and isConnected(x, y) always call find(x) and find(y). Thus, after calling connect or isConnected enough, essentially all elements will point directly to their root.
By extension, the average runtime of connect and isConnected becomes almost constant in the long term! This is called the amortized runtime.
More specifically, for M operations on N elements, WQU with Path Compression is in
O(N+M(lgN))O(N + M (lg* N))
. lg* is the iterated logarithm which is less than 5 for any real-world input.

Summary

N: number of elements in Disjoint Set
Implementation
isConnected
connect
Quick Find
Θ(1)
Θ(N)
Quick Union
O(N)
O(N)
Weighted Quick Union (WQU)
O(log N)
O(log N)
WQU with Path Compression
O(α(N))*
O(α(N))*
*behaves as constant in long term.