158 lines
5.4 KiB
Markdown
158 lines
5.4 KiB
Markdown
---
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title: "Union Find"
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date: 2024-06-24T21:07:49+01:00
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draft: false # I don't care for draft mode, git has branches for that
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description: "My favorite data structure"
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tags:
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- algorithms
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- data structures
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- python
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categories:
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- programming
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series:
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- Cool algorithms
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favorite: false
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disable_feed: false
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---
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To kickoff the [series]({{< ref "/series/cool-algorithms/" >}}) of posts about
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algorithms and data structures I find interesting, I will be talking about my
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favorite one: the [_Disjoint Set_][wiki]. Also known as the _Union-Find_ data
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structure, so named because of its two main operations: `ds.union(lhs, rhs)` and
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`ds.find(elem)`.
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[wiki]: https://en.wikipedia.org/wiki/Disjoint-set_data_structure
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<!--more-->
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## What does it do?
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The _Union-Find_ data structure allows one to store a collection of sets of
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elements, with operations for adding new sets, merging two sets into one, and
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finding the representative member of a set. Not only does it do all that, but it
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does it in almost constant (amortized) time!
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Here is a small motivating example for using the _Disjoint Set_ data structure:
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```python
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def connected_components(graph: Graph) -> list[set[Node]]:
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# Initialize the disjoint set so that each node is in its own set
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ds: DisjointSet[Node] = DisjointSet(graph.nodes)
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# Each edge is a connection, merge both sides into the same set
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for (start, dest) in graph.edges:
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ds.union(start, dest)
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# Connected components share the same (arbitrary) root
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components: dict[Node, set[Node]] = defaultdict(set)
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for n in graph.nodes:
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components[ds.find(n)].add(n)
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# Return a list of disjoint sets corresponding to each connected component
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return list(components.values())
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```
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## Implementation
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I will show how to implement `UnionFind` for integers, though it can easily be
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extended to be used with arbitrary types (e.g: by mapping each element
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one-to-one to a distinct integer, or using a different set representation).
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### Representation
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Creating a new disjoint set is easy enough:
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```python
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class UnionFind:
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_parent: list[int]
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_rank: list[int]
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def __init__(self, size: int):
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# Each node is in its own set, making it its own parent...
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self._parents = list(range(size))
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# ... And its rank 0
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self._rank = [0] * size
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```
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We represent each set through the `_parent` field: each element of the set is
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linked to its parent, until the root node which is its own parent. When first
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initializing the structure, each element is in its own set, so we initialize
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each element to be a root and make it its own parent (`_parent[i] == i` for all
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`i`).
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The `_rank` field is an optimization which we will touch on in a later section.
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### Find
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A naive Implementation of `find(...)` is simple enough to write:
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```python
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def find(self, elem: int) -> int:
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# If `elem` is its own parent, then it is the root of the tree
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if (parent := self._parent[elem]) == elem:
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return elem
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# Otherwise, recurse on the parent
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return self.find(parent)
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```
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However, going back up the chain of parents each time we want to find the root
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node (an `O(n)` operation) would make for disastrous performance. Instead we can
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do a small optimization called _path splitting_.
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```python
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def find(self, elem: int) -> int:
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while (parent := self._parent[elem]) != elem:
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# Replace each parent link by a link to the grand-parent
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elem, self._parent[elem] = parent, self._parent[parent]
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return elem
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```
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This flattens the chain so that each node links more directly to the root (the
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length is reduced by half), making each subsequent `find(...)` faster.
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Other compression schemes exist, along the spectrum between faster shortening
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the chain faster earlier, or updating `_parent` fewer times per `find(...)`.
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### Union
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A naive implementation of `union(...)` is simple enough to write:
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```python
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def union(self, lhs: int, rhs: int) -> int:
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# Replace both element by their root parent
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lhs = self.find(lhs)
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rhs = self.find(rhs)
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# arbitrarily merge one into the other
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self._parent[rhs] = lhs
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# Return the new root
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return lhs
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```
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Once again, improvements can be made. Depending on the order in which we call
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`union(...)`, we might end up creating a long chain from the leaf of the tree to
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the root node, leading to slower `find(...)` operations. If at all possible, we
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would like to keep the trees as shallow as possible.
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To do so, we want to avoid merging taller trees into smaller ones, so as to keep
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them as balanced as possible. Since a higher tree will result in a slower
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`find(...)`, keeping the trees balanced will lead to increased performance.
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This is where the `_rank` field we mentioned earlier comes in: the _rank_ of an
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element is an upper bound on its height in the tree. By keeping track of this
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_approximate_ height, we can keep the trees balanced when merging them.
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```python
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def union(self, lhs: int, rhs: int) -> int:
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lhs = self.find(lhs)
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rhs = self.find(rhs)
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# Bail out early if they already belong to the same set
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if lhs == rhs:
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return lhs
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# Always keep `lhs` as the taller tree
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if (self._rank[lhs] < self._rank[rhs])
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lhs, rhs = rhs, lhs
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# Merge the smaller tree into the taller one
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self._parent[rhs] = lhs
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# Update the rank when merging trees of approximately the same size
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if self._rank[lhs] == self._rank[rhs]:
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self._rank[lhs] += 1
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return lhs
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```
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