Minimum Spanning Tree Problem - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

Download Minimum Spanning Tree Problem - Discrete Mathematics - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity!

Graphs and Trees

This handout:

• Trees
• Minimum Spanning Tree Problem

Terminology of Graphs:

Cycles, Connectivity and Trees

• A path that begins and ends at the same node is called a cycle.

Example:

• Two nodes are connected if there is a path between them.
• A graph is connected if every pair of its nodes is connected.
• A graph is acyclic if it doesn’t have any cycle.
• A graph is called a tree if it is connected and acyclic.

Example :

Properties of Trees

 Definition: Let T be a tree.

If T has at least 3 nodes, then

• A node of degree 1 in T is called a leaf (or a terminal node).
• A node of degree greater than 1 is called an internal node.

 Lemma: Every tree with more than one node

has at least one leaf.

 Theorem: For any positive integer n,

any tree with n nodes has n-1 edges. Proof by induction ( blackboard ).

Properties of Trees

 Lemma : If G is any connected graph,

C is a cycle in G, and one of the edges of C is removed from G, then the graph that remains is still connected.

 Theorem : For any positive integer n,

if G is a connected graph with n vertices and n-1 edges, then G is a tree.

 Theorem : For any positive integer n,

if G is an acyclic graph with n vertices and n-1 edges, then G is a tree.

Proofs on blackboard.

Minimum Spanning Tree Problem

• Given : Graph G=(V, E), |V|=n

Cost function c: E → R.

• Goal : Find a minimum-cost spanning tree for V

i.e., find a subset of arcs E* ⊆ E which

connects any two nodes of V

with minimum possible cost.

• Example:

e

b

c

d

a

e

b

c

d

a

G=(V,E) Min. span. tree: G=(V,E)

Red bold arcs are in E*Docsity.com

Algorithm for solving the Minimum Spanning Tree Problem

• Initialization: Select any node arbitrarily,

connect to its nearest node.

• Repeat
  • Identify the unconnected node which is closest to a connected node
  • Connect these two nodes

Until all nodes are connected

Note : Ties for the closest node are broken arbitrarily.

The algorithm applied to our example

• Iteration 2 : The unconnected node closest to a connected

node is node d (3 far from node b). Connect nodes b and d.

• Iteration 3: The only unconnected node left is node e. Its closest connected node is node c (distance between c and e is 4). Connect node e to node c.
• All nodes are connected. The bold

arcs give a min. spanning tree.

e

b

c

d

a

e

b

c

d

a