1. Network Optimization Problems: Models and Algorithms This handout: Minimum Spanning Tree Problem

2. Terminology of Graphs A graph (or network) consists of – a set of points – a set of lines connecting certain pairs of the points. The points are called nodes (or vertices). The lines are called arcs (or edges or links). Example:

3. Terminology of Graphs: Paths A path between two nodes is a sequence of distinct nodes and edges connecting these nodes. Example: a b

4. 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:

5. 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: 2 3 3 4 4 5 7 8 e b c d a 2 3 3 4 4 5 7 8 e b c d a G=(V,E) Min. span. tree: G*=(V,E*) Red bold arcs are in E*

6. 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.

7. The algorithm applied to our example Initialization: Select node a to start. Its closest node is node b. Connect nodes a and b. Iteration 1: There are two unconnected node closest to a connected node: nodes c and d (both are 3 units far from node b). Break the tie arbitrarily by connecting node c to node b. 2 3 3 4 4 5 7 8 e b c d a 2 3 3 4 4 5 7 8 e b c d a Red bold arcs are in E*; thin arcs represent potential links.

8. 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. 2 3 3 4 4 5 7 8 e b c d a 2 3 3 4 4 5 7 8 e b c d a