# Union Find

May 12, 2021
Tech

#### Introduction

Imagine you’re back in elementary school and its recess and you and your friends have chosen captains to form teams. No single player can belong to more than one team and at some point after you have played some games some players can switch teams. What if someone came while you guys are on break and wanted to know to which team a player belonged to?

The Union Find data structure was invented by Bernard Galler and Michael Fischer to efficiently determine which non-overlapping group and item belongs to (for ex. which team a player belongs to) and to quickly join two items into a single group (i.e. the captain selects you into his team). It has the following two operations:

• join(item A, item B) -> joins two items together
• find(item) -> group it belongs to

#### How It Works

Like many other data structures, the union find data structure is modeled as a tree, but like the common heap implementation the tree operations are performed on an array. The key to the union find data structure is assigning a representative value as an id to each disjoint (non-overlapping) group.

We allocate an array to store the representative id of the immediate parent of each of our N items. Initially each item belongs to its own group the parent[i] = i. We will also allocate an array to store the rank (which is the height of a tree beginning at the node itself) for each item.

Then if we want to join two items into the same group, we first have to check to see if they’re already in the same group. If they are not, then we will assign the root of the item with the smaller rank to be a direct child of the root of the item with the larger rank so as not to increase the maximum rank between the two items. If the two items have exactly the same rank, we can arbitrarily choose one item and assign it as the parent of the other.

For the find operation we will walk back parents until we land on the root node. And as we walk back we can reduce the distance for our next search by assigning to each descendent of the root node, the root node as its parent, so next time we call find we can immediately find it. This is called path compression. The implementation of the data structure follows below:

The key thing to observe is that in the recursive find function above we'll continue to call find on the parent until the parent becomes itself while setting the parent of any descendants of the root to the root. The path compression allows us to achieve a worst case O(LogN) time for the find and join operations (In fact they are slightly faster than O(LogN). You can read more here). This is how we compress the path. It's usually the case that if a problem can be solved using the Union Find data structure, then it can alternatively be solved using BFS or DFS as well. It's good to practice all three approaches.

MEDIUM

HARD