Tree Chain Partitioning

Tree Chain Partitioning

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Background

Tree chain partitioning means that cut a tree into several lines

Some definitions

Heavy son: the son with the most Subnodes.

Light son: other sons except heavy son.

Heavy edge: the edge between root and heavy son

Light edge: the edge between root and light son.

Heavy chain： the chain that is only made of heavy edges.

Several theories

Noumenon

The most important theory: every node going to root will go through at most logN light edges.

And several easy theory:

1. Every node is at most in a heavy chain

2. There is no intersection between heavy chains.

Provement

I don't want to spend ink proving the second and third theory because we can prove it just by the definition of heavy edges.

I will just prove the first theory.

If a node go through a light edge, we can know that there are at least one subtree is bigger than it.

So, the number of subnodes will increase at least one time. 

So, it is true that very node going to root will go through at most logN light sons or .heavy chains.

Algorithm core