---
id: dsu
title: "Disjoint Set Union"
author: Benjamin Qi, Michael Cao 
prerequisites:
 - Silver - Depth First Search
description: The Disjoint Set Union (DSU) data structure allows you to add edges to an initially empty graph and test whether two vertices of the graph are connected.
frequency: 3
---

import { Problem } from "../models";

export const metadata = {
  problems: {
    sample: [
      new Problem("YS", "Union Find", "unionfind", "Intro|Easy", false, []),
    ],
    general: [
      new Problem("Gold", "Closing the Farm", "646", "Easy", false, [], "Simulate process in reverse and maintain the # of connected components. Similar to [CSES Network Breakdown](https://cses.fi/problemset/task/1677)"),
      new Problem("Gold", "Mootube", "789", "Normal", false, [], "Answer queries in decreasing order of $k$. Maintain size of each connected component. Same as [CSES Road Construction](https://cses.fi/problemset/task/1676)"),
    ],
  }
};

<problems-list problems={metadata.problems.sample} />

## Resources

<resources>
  <resource source="CSA" title="Disjoint Data Sets" url="disjoint_data_sets" starred>both optimizations, diagrams</resource>
  <resource source="CPH" title="15.2 - Union-Find">small to large, diagrams</resource>
  <resource source="IUSACO" title="10.6 - Disjoint-Set Data Structure">path compression, diagrams</resource>
  <resource source="PAPS" title="11.1 - Disjoint Sets">both optimizations, no diagrams</resource>
  <resource source="TC" title="Disjoint Set Data Structures" url="disjoint-set-data-structures">diagrams</resource>
  <resource source="CPC" title="3 - Data Structures" url="03_data_structures"></resource>
</resources>

Note: You may prefer to use this in place of DFS for computing connected components.

<optional-content title="DSU Complexity Proofs">

   - [$\log^* n$](https://en.wikipedia.org/wiki/Proof_of_O(log*n)\_time_complexity\_of_union%E2%80%93find)
   - [$\alpha (m,n)$](https://dl.acm.org/doi/pdf/10.1145/321879.321884)

</optional-content>


## Problems

<problems-list problems={metadata.problems.general} />