59 lines
2.4 KiB
Markdown
59 lines
2.4 KiB
Markdown
---
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id: merging
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title: "Small-To-Large Merging"
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author: Michael Cao
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prerequisites:
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- Silver - Depth First Search
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description: ?
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---
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## Additional Reading
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- CPH 18.4 - Merging Data Structures
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- CF Blogs
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- [Arpa](https://codeforces.com/blog/entry/44351)
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- [tuwuna](https://codeforces.com/blog/entry/67696)
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# Merging Sets
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Let's consider a tree, rooted at node $1$, where each node has a color (see [CSES Distinct Colors](https://cses.fi/problemset/task/1139)).
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For each node, let's store a set containing only that node, and we want to merge the sets in the nodes subtree together such that each node has a set consisting of all colors in the nodes subtree. Doing this allows us to solve a variety of problems, such as query the number of distinct colors in each subtree. Doing this naively, however, yields a runtime complexity of $O(N^2)$.
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However, with just a few lines of code, we can significantly speed this up.
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```cpp
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if(a.size() < b.size()){ //for two sets a and b
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swap(a,b);
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}
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```
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In other words, by merging the smaller set into the larger one, the runtime complexity becomes $O(N\log N).$
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<details>
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<summary> Proof </summary>
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When merging two sets, you move from the smaller set to the larger set. If the size of the smaller set is $X$, then the size of the resulting set is at least $2X$. Thus, an element that has been moved $Y$ times will be in a set of size $2^Y$, and since the maximum size of a set is $N$ (the root), each element will be moved at most $O(\log N$) times leading to a total complexity of $O(N\log N)$.
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</details>
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Additionally, a set doesn't have to be an `std::set`. Many data structures can be merged, such as `std::map` or even adjacency lists.
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<info-block title="Challenge">
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Prove that if you instead merge sets that have size equal to the depths of the subtrees, then small to large merging does $O(N)$ insert calls.
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(be specific about what this means?)
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</info-block>
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## Problems
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- USACO
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- [Gold - Favorite Colors](http://www.usaco.org/index.php?page=viewproblem2&cpid=1042)
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- merge adjacency lists
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- not required to get AC
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- [Plat - Disruption](http://www.usaco.org/index.php?page=viewproblem2&cpid=842)
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- [Plat - Promotion Counting](http://www.usaco.org/index.php?page=viewproblem2&cpid=696)
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- merge indexed sets
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- Other
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- [CSES Distinct Colors](https://cses.fi/problemset/task/1139)
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- [CF Lomsat gelral](https://codeforces.com/contest/600/problem/E)
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