Lecture 8

Cut Property §

Definition:

• For $G=(V,E)$, consider $X\subset E$ where: $X\subset T$ for a MST T
• Take any subset S of vertices where No edge of X crosses $S\leftrightarrow \overline{S}$
• Let $e^{\ast }$ be any minimum weight edge of E crossing $S\leftrightarrow \overline{S}$
• Then: $XU{e}^{\ast }\subset T^{\prime }$ for a MST T’. (Thus $XU{e}^{\ast }$ is a partial MST)

Cut Property Proof §

We know that $X\subset T$ for MST T. There are two cases $e^{\ast }\in T$ or $e^{\ast }\notin T$.

Case 1: $e^{\ast }\in T$
Since $X\subset T$ and $e^{\ast }\in T$
Then $X\cup e^{\ast }\subset T$ and $T$ is a MST

Case 2: $e^{\ast }\notin T$
Say $e^{\ast }=(y,z)$, add $e^{\ast }$ to T.
T is a tree so there is exactly 1 path P between y & z Then, $T\cup U{e}^{\ast }$ has a cycle C = $P\cup e^{\ast }$

Union-Find Data Structure §

The Union-Find data structure is a collection of sets where each set has a unique name.

Operations:

• Makeset(x): Creates set
• Find(x): Returns root
• Union(x,y): Points root of smaller rank root to larger rank root

Properties:

• Each set is a directed tree:
  • Edges point up to the root
  • The name of the set is the root element
• Each node has the following:
  • rank(x) = height of subtree below x
  • $\prod$(x) = parent of x, note that x is the root the parent of x is x
• Rank only increases when merging two trees of same rank

Union Find Claims §

Claim 1 §

The max depth of a subtree is ${\log }_{2}n$

Proof:
Let l = # of nodes of rank = k
Then, $(l)(2^k)\le n$ so $l\le \frac{n}{2k}$
Let $k={\log }_{2}n+1$ then $l\le \frac{1}{2}\le 1$ so no nodes of rank ${\log }_{2}n+1$

Claim 2 §

Root of rank $k$ has $\ge$ 2${}^k$ nodes in its subtree (including itself)

Proof:
Induct on $k$.
Base case: $k=0$, since count root then $2^0=1$
Assume true for trees of rank $k\le k-1$
To get node $v$ of rank $k$ we merge two trees of rank $k-1$, thus $v$’s subtree has $\ge 2^{k-1}+2^{k-1}=2^k$