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Intermediate Topic 31 of 52 Data Structures

Path Compression

Flattening find paths to speed up disjoint sets.

How it works

Path compression is an optimization in union-find: during a find operation, every node visited on the way to the root is re-pointed directly at the root, flattening the tree. Future finds on those nodes take one step. Combined with union by rank, it yields amortized time per operation of the inverse Ackermann function, effectively constant.

Step-by-step walkthrough

  1. A long chain d to c to b to root makes find slow.
  2. find(d) walks the whole chain up to the root.
  3. It reaches the root.
  4. Every node on the path is re-pointed directly at the root.
  5. Later finds on those nodes take one step.

Where it is used

Disjoint set union implementations in Kruskal's algorithm, connectivity queries and online equivalence classes use path compression. It can be written recursively or with an iterative two-pass or halving variant.

Common pitfall

The recursive version may overflow the stack on very deep trees before compression has occurred. Path compression alone does not preserve rank values, so rank is only an upper bound. Be careful to return the root, not only to assign parents, and test on chain-shaped inputs.

Where it fits in the course

Path Compression is topic 31 of 52 in Data Structures, rated Intermediate. It follows Disjoint Set Union. The next topic is Skip Lists.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Data Structures quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Path Compression?
Flattening find paths to speed up disjoint sets. It belongs to Data Structures and is an intermediate-level topic.
Where do people use Path Compression?
Disjoint set union implementations in Kruskal's algorithm, connectivity queries and online equivalence classes use path compression.
What is the most common mistake with Path Compression?
The recursive version may overflow the stack on very deep trees before compression has occurred.
What should I learn before this topic?
Start with Disjoint Set Union, the topic just before it in Data Structures.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Data Structures quiz.
Is Path Compression hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Path Compression?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Path Compression?
Disjoint Set Union, Skip Lists, Adjacency Matrix are closely connected. The full list is in the related topics section of this page.
What should I study after Path Compression?
Continue with Skip Lists, the next topic in Data Structures.

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