> For the complete documentation index, see [llms.txt](https://cs61b-2.gitbook.io/cs61b-textbook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://cs61b-2.gitbook.io/cs61b-textbook/14.-disjoint-sets/14.5-weighted-quick-union-with-path-compression.md).

# 14.5 Weighted Quick Union with Path Compression

{% embed url="<https://www.youtube.com/watch?v=DZKzDebT4gU>" %}
Professor Hug's explanation on Weighted Quick Union with Path Compression
{% endembed %}

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 $$O(N + M (lg\* N))$$. lg\* is the [iterated logarithm](https://en.wikipedia.org/wiki/Iterated_logarithm) which is less than 5 for any real-world input.

### Summary <a href="#summary" id="summary"></a>

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.


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter:

```
GET https://cs61b-2.gitbook.io/cs61b-textbook/14.-disjoint-sets/14.5-weighted-quick-union-with-path-compression.md?ask=<question>&goal=<endgoal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.