---
id: merging
title: "Small-To-Large Merging"
author: Michael Cao
prerequisites:
- Silver - Depth First Search
description: ?
---
# Merging Sets
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)).
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 $O(N^2)$.
However, with just a few lines of code, we can significantly speed this up.
```cpp
if(a.size() < b.size()){ //for two sets a and b
swap(a,b);
}
```
In other words, by merging the smaller set into the larger one, the runtime complexity becomes $O(N\log N).$
Proof
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 $O(N\log N)$.
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.
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.
(be specific about what this means?)
## Further Reading
- "Merging Data Structures" from CPH 18
## Problems
- [Favorite Colors (USACO Gold)](http://www.usaco.org/index.php?page=viewproblem2&cpid=1042)
- Merge Adjacency Lists
- Promotion Counting
- Merge Indexed Sets
## Small to Large (Offline)
Sample codes: [DSU on Tree code](https://codeforces.com/blog/entry/44351), [explanation of code](https://codeforces.com/blog/entry/67696)
- USACO Disrupt again!
- [CSES Distinct Colors](https://cses.fi/problemset/task/1139)
- [CF Lomsat gelral](https://codeforces.com/contest/600/problem/E)