1. O(N 1.5 ) divide-and-conquer technique for Minimum Spanning Tree problem Step 1: Divide the graph into  N sub-graph by clustering. Step 2: Solve each sub-problem separately using Prim's algorithm (quadratic complexity) Step 3: Merge the sub-solutions: Construct a meta graph where each node corresponds to one cluster Solve MST for the meta graph Add links from the meta graph to the original graph to complete the solution

2. Example 1: Algorithm finds sub-optimal solution 5 8 1 3 2 6 5 5 8 10 2 5 3 4 7 5 3 1 6

3. Step 1: Divide the graph into  N sub-graph by clustering 5 8 1 3 2 6 5 5 8 10 2 5 3 4 7 5 3 1 6

4. 5 8 1 3 2 6 5 5 8 2 5 3 4 7 5 3 1 6 Step 2: Solve each sub-problem by Prim’s algorithm

5. 5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.1 (a): Select center point for each cluster

6. 5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.1 (b): Connect the nodes of this meta graph

7. 5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.1 (c): Set the weights based on shortest distances 1 3 4

8. 5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.2: Solve MST for the meta graph 1 3 4

9. 5 8 3 2 6 5 8 10 2 5 4 7 3 1 6 Step 3.3: Select the corresponding links 1 3 3 1

10. 5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.3: Add links from the MST of the meta graph Total weight = 21

11. 5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Better solution: Total weight = 20 Optimality of the solution? Add Remove

12. 5 8 1 3 2 6 6 8 10 2 5 3 4 7 3 1 5 Total weight = 20 Example 2: Algorithm finds optimal solution