posts: union-find: add 'find'
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@ -78,3 +78,34 @@ 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 links so that each node links directly to the root, making
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each subsequent `find(...)` constant time.
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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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