archetypes/default.md

@@ -5,18 +5,15 @@ draft: false # I don't care for draft mode, git has branches for that
 description: ""
 tags:
   - accounting
-  - algorithms
   - c++
   - ci/cd
   - cli
-  - data structures
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-  - python
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content/posts/2024-06-24-union-find/index.md

@@ -1,154 +0,0 @@
----
-title: "Union Find"
-date: 2024-06-24T21:07:49+01:00
-draft: false # I don't care for draft mode, git has branches for that
-description: "My favorite data structure"
-tags:
-  - algorithms
-  - data structures
-  - python
-categories:
-  - programming
-series:
-  - Cool algorithms
-favorite: false
-disable_feed: false
----
-
-To kickoff the [series]({{< ref "/series/cool-algorithms/">}}) of posts about
-algorithms and data structures I find interesting, I will be talking about my
-favorite one: the [_Disjoint Set_][wiki]. Also known as the _Union-Find_ data
-structure, so named because of its two main operations: `ds.union(lhs, rhs)` and
-`ds.find(elem)`.
-
-[wiki]: https://en.wikipedia.org/wiki/Disjoint-set_data_structure
-
-<!--more-->
-
-## What does it do?
-
-The _Union-Find_ data structure allows one to store a collection of sets of
-elements, with operations for adding new sets, merging two sets into one, and
-finding the representative member of a set. Not only does it do all that, but it
-does it in almost constant (amortized) time!
-
-Here is a small motivating example for using the _Disjoint Set_ data structure:
-
-```python
-def connected_components(graph: Graph) -> list[set[Node]]:
-    # Initialize the disjoint set so that each node is in its own set
-    ds: DisjointSet[Node] = DisjointSet(graph.nodes)
-    # Each edge is a connection, merge both sides into the same set
-    for (start, dest) in graph.edges:
-        ds.union(start, dest)
-    # Connected components share the same (arbitrary) root
-    components: dict[Node, set[Node]] = defaultdict(set)
-    for n in graph.nodes:
-        components[ds.find(n)].add(n)
-    # Return a list of disjoint sets corresponding to each connected component
-    return list(components.values())
-```
-
-## Implementation
-
-I will show how to implement `UnionFind` for integers, though it can easily be
-extended to be used with arbitrary types (e.g: by mapping each element
-one-to-one to a distinct integer, or using a different set representation).
-
-### Representation
-
-Creating a new disjoint set is easy enough:
-
-```python
-class UnionFind:
-    _parent: list[int]
-    _rank: list[int]
-
-    def __init__(self, size: int):
-        # Each node is in its own set, making it its own parent...
-        self._parents = list(range(size))
-        # ... And its rank 0
-        self._rank = [0] * size
-```
-
-We represent each set through the `_parent` field: each element of the set is
-linked to its parent, until the root node which is its own parent. When first
-initializing the structure, each element is in its own set, so we initialize
-each element to be a root and make it its own parent (`_parent[i] == i` for all
-`i`).
-
-The `_rank` field is an optimization which we will touch on in a later section.
-
-### Find
-
-A naive Implementation of `find(...)` is simple enough to write:
-
-```python
-def find(self, elem: int) -> int:
-    # If `elem` is its own parent, then it is the root of the tree
-    if (parent: = self._parent[elem]) == elem:
-        return elem
-    # Otherwise, recurse on the parent
-    return self.find(parent)
-```
-
-However, going back up the chain of parents each time we want to find the root
-node (an `O(n)` operation) would make for disastrous performance. Instead we can
-do a small optimization called _path splitting.
-
-```python
-def find(self, elem: int) -> int:
-    while (parent: = self._parent[elem]) != elem:
-        # Replace each parent link by a link to the grand-parent
-        elem, self._parent[elem] = parent, self._parent[parent]
-    return elem
-```
-
-This flattens the links so that each node links directly to the root, making
-each subsequent `find(...)` constant time.
-
-Other compression schemes exist, along the spectrum between faster shortening
-the chain faster earlier, or updating `_parent` fewer times per `find(...)`.
-
-### Union
-
-A naive implementation of `union(...)` is simple enough to write:
-
-```python
-def union(self, lhs: int, rhs: int) -> int:
-    # Replace both element by their root parent
-    lhs = self.find(lhs)
-    rhs = self.find(rhs)
-    # arbitrarily merge one into the other
-    self._parent[rhs] = lhs
-    # Return the new root
-    return lhs
-```
-
-Once again, improvements can be made. Depending on the order in which we call
-`union(...)`, we might end up creating a long chain from the leaf of the tree to
-the root node, leading to slower `find(...)` operations. If at all possible, we
-would like to keep the trees as shallow as possible.
-
-To do so, we want to avoid merging taller trees into smaller ones, so as to keep
-them as balanced as possible. Since a higher tree will result in a slower
-`find(...)`, keeping the trees balanced will lead to increased performance.
-
-This is where the `_rank` field we mentioned earlier comes in: the _rank_ of an
-element is an upper bound on its height in the tree. By keeping track of this
-_approximate_ height, we can keep the trees balanced when merging them.
-
-```python
-def union(self, lhs: int, rhs: int) -> int:
-    lhs = self.find(lhs)
-    rhs = self.find(rhs)
-    # Always keep `lhs` as the taller tree
-    if (self._rank[lhs] < self._rank[rhs])
-        lhs, rhs = rhs, lhs
-    # Merge the smaller tree into the taller one
-    self._parent[rhs] = lhs
-    # Update the rank when merging trees of approximately the same size
-    if self._rank[lhs] == self._rank[rhs]:
-        self._rank[lhs] += 1
-    return lhs
-```