1 / 12

O( N 1.5 ) divide-and-conquer technique for Minimum Spanning Tree problem

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:

misty
Download Presentation

O( N 1.5 ) divide-and-conquer technique for Minimum Spanning Tree problem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. O(N1.5) 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 8 5 5 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3

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

  4. Step 2: Solve each sub-problem by Prim’s algorithm 8 5 5 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3

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

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

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

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

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

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

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

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

More Related