Tree: Definition and equvalent notions

A tree is a connected, acyclic graph. (By default, a generic tree is undirected.) Equivalently,

• Unique paths: A tree is an undirected graph \(T\) such that, for any two vertices \(u\) and \(v\) in \(T\), there is a unique path from \(u\) to \(v\).
• \(V-1\) edges: A tree is a connected, undirected graph with \(V-1\) edges.
  • Can’t add an edge: Adding an edge to a tree makes a cycle.

Minimum Spanning Tree of a Weighted Graph

• A graph is called weighted if there is an assignment of real numbers (weights) to the edges.
  
• Given a weighted graph \(G\), a minimum spanning tree \(T\) is a subgraph of \(G\) such that:
  • \(T\) contains all the vertices of \(G\), and
  • The sum of weights of the edges of \(T\) is as small as possible.
• Applications: network optimization, image recognition, data analysis, etc.

(See Jamboard from last time.)

Table Groups

Smallest edge

1. Let \(G\) be a weighted graph with an edge \(e\) that is lighter (i.e., smaller) than all the other edges. Prove that any minimum spanning tree must contain \(e\).

Hints:

• Suppose to the contrary that \(T\) is a minimum spanning tree of \(G\) and that ___________________.
• What happens if you add an edge to \(T\)?
• Show you can make a lighter spanning tree (contradiction).

Simplifying assumption: distinct weights

Without losing generality, we can assume our edge weights are distinct.

Why is this OK?

• In most applications, they are distinct.
• We could break ties by adding an insignificant \(\varepsilon\) to a weight.
• We could break ties some other systematic way.
• Two floats are “never” equal (at least you shouldn’t expect them to be).

sqrt(3)^2 == 3

[1] FALSE

Two minimum spanning trees?

Suppose a connected, weighted, undirected graph \(G\) has distinct edge weights and two minimum spanning trees \(T\) and \(T'\).

• Suppose to the contrary that \(T \neq T'\).
  • Then there are edges in \(T\) that are not in \(T'\), and edges in \(T'\) that are not in \(T\). Let \(e\) be the smallest of these edges.
  • In other words, \(e\) is the smallest edge in \((T\setminus T')\cup(T'\setminus T)\).
  • WLOG, \(e\in T\setminus T'\).
• Adding this edge to \(T'\) creates a cycle in \(T'\).
  • One of the edges in this cycle is not in \(T\).
  • Exchange this edge for \(e\), and you reduce the weight of \(T'\), a contradiction.

The minimum spanning tree

We have proved the following.

Lemma. If all the edge weights of a graph are distinct, then its minimum spanning tree is unique.

So we will henceforth refer to “the” minimum spanning tree of a graph.

Intermediate spanning forests

• We can build a spanning tree by selecting edges.
• At each stage, we avoid cycles, so we have a forest (maybe not a tree).

An intermediate spanning forest of a weighted graph \(G\) is a subgraph of the minimum spanning tree of \(G\).

Safe and Useless edges

For any intermediate spanning forest \(F\) of a graph \(G\),

• An edge of \(G\) is useless if it is not an edge of \(F\), but both its endpoints are in the same component of \(F\).
• An edge of \(G\) is safe if it is the minimum-weight edge with exactly one endpoint in some component of \(F\).

Example

In the following graph \(G\), the bold edges form an intermediate spanning forest \(F\).

Safe and Useless

2. Identify all the safe edges and all the useless edges.

3. What happens if you add a useless edge to \(F\)?

4. Add all the safe edges to \(F\). Is the resulting forest \(F'\) still an intermediate spanning forest?

5. Repeat: Add all of the safe edges of \(F'\) to \(F'\), and continue. How will this process terminate?