Union–find
one step at a time.
Each group is a tree named by its root; union links two roots, and compression keeps the trees flat.
Free to use. Step forward and back at your pace, and share a link to the exact step you are stuck on.
What you actually watch.
The part a dry run on paper loses: the state, at every step.
Each group is a tree named by its root; union links two roots, and compression keeps the trees flat. That sentence is easy to agree with and hard to hold in your head on step nine, which is where most people lose the thread.
So the visualiser keeps it in front of you: the frontier spreading, and which nodes are settled for good. The line of code being executed is highlighted as the picture changes, so the drawing and the code never drift apart. Step forward, step back, or let it play.
Put your own input in. Union–find on a clean example teaches you the shape; union–find on an empty input, a single element, or one with duplicates teaches you the thing an interviewer asks about.
How it runs, step by step.
The same order the visualiser moves in, so the list and the picture match.
- 1Start from a source and mark it seen.
- 2Take the next node from the frontier: a queue spreads in layers, a stack goes deep.
- 3Look at its neighbours, and add the ones never seen before.
- 4Never revisit a node, so every edge is examined once.
Where people get this one wrong.
These are not made up. Each one is a switch in the tool: turn it on, run the same input, and watch the state go wrong.
- Union by size + path compressionHangs the smaller tree under the larger and points every node on a find path straight at the root. Trees stay flat, so finds take almost no steps.
Inputs worth trying.
The cases that teach something. Each one is a preset in the tool, with the reason it is there.
- Random unionsChange the seed for a different set of unions.
What it costs.
The two numbers an interviewer will ask for, and what they mean here.
- Time
- O(α(n)) with compressioncounted as real operations while it runs, not quoted from a table
- Space
- O(n)what the algorithm holds besides the input itself
- Pattern
- GraphsExplore in layers, or go deep, and never visit twice.
Once you can see it, go and use it.
Watching is the first ten minutes. The pattern sticks when you solve with it.
Everything in the app, in one place.
Practice, interviews and the tools, all free to start. Swipe to see the rest.
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About union–find.
Open it in the app and the answers are easier to see than to read.
Open the visualiserWhat does the union–find visualiser show?
Every step of the algorithm on input you choose: the frontier spreading, and which nodes are settled for good, with the line of code being executed highlighted beside it. You can step forward, step back, or let it play. Spelled visualizer or visualiser, it is the same tool.
What is the time complexity of union–find?
O(α(n)) with compression time and O(n) space. The visualiser counts the real operations as it runs, so you can watch the number grow instead of taking the formula on trust.
Can I use my own input?
Yes. Type your own array, string or graph and the algorithm runs on that, including the edge cases that break a solution in an interview.
Is it free?
Yes. The app is free to start, with 100 AI credits when you sign up and 100 every month, and watching an algorithm costs no credits.