posts: union-find: add 'union'
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@ -109,3 +109,46 @@ each subsequent `find(...)` constant time.
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Other compression schemes exist, along the spectrum between faster shortening
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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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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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# 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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