1. Divide-and-Conquer MST
Pasi Fränti

2. Minimum spanning tree Prim’s algorithm: N nodes and M links.
Creates tree by adding links one-by-one
Always shortest link that connects one new node
Even best solutions lower limited byO(M)
Time complexity for full graph O(N2) because M=N2. 5 8 1 3 2 6 10 4 7

3. Prim’s algorithm example5 8 1 3 2 6 10 4 7

4. O(N 1.5) divide-and-conquer algorithm for Minimum Spanning Tree problem
C. Zhong, M.I. Malinen, D. Miao and P. Fränti, "A fast minimum spanning tree algorithm based on K-means", Information Sciences 295, 1-17, February 2015.
Step 1: Divide the graph into N clusters
Step 2: Solve each cluster by Prim's algorithm
Step 3: Merge the results of clusters:
Create meta graph where one node per cluster
Solve MST for the meta graph
Combine MSTs of the meta graph and clusters

5. Example 8 5 5 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3
Example by: Michiyo Ashida  (IMPIT’2010)

6. Step 1: Divide the graph into N sub-graph by clustering8 5 5 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3

7. Step 2: Solve each sub-problem by Prim’s algorithm8 5 5 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3

8. Step 3.1 (a): Select center point for each cluster8 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3

9. Step 3.1 (b): Connect the nodes of this meta graph8 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3

10. Step 3.1 (c): Set the weights based on shortest distances8 5 1 3 6 1 2 6 8 2 3 4 4 5 5 10 1 7 3 3

11. Step 3.2: Solve MST for the meta graph8 5 1 3 6 1 2 6 8 2 3 4 4 5 5 10 1 7 3 3

12. Step 3.3: Select the corresponding links8 5 1 3 6 1 2 6 8 2 3 4 5 5 10 1 7 3 3

13. Step 3.3: Add links from the MST of the meta graph8 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3
Total weight = 21

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

15. Bigger example Clustering
Dataset
Clustering

16. Bigger example Creation of MST
Cluster-level MST
Connecting clusters

17. Time complexity Step 1 Step 2
Cluster to N sub-graphs: O(k∙N) = N∙N = N 1.5
Step 2
k=N sub-problems of size N /k = N /N 1/2 = N 1/2 = N
Prim's for one: (N)2 = N
For All sub-problems: N∙N = N 1.5
Step 3
Construct meta graph: (N)∙(N)2 = N 1.5
Solve meta graph: (N)2 = N
Add links: N

18. Refinement stage
Step 1: Calculate midpoints of links in the meta graph
Step 2: Cluster based on midpoints
Step 3: Solve each cluster by Prim's algorithm
Step 4: Merge the 1st MST with the 2nd MST

19. Mid-points

20. Secondary clustering

21. Secondary MST within clusters8 5 3 1 2 6 6 8 2 4 5 5 10 1 7 3 3

22. Secondary MST between clusters8 5 3 1 2 6 6 8 2 4 5 5 10 1 7 3 3

23. Merge of 1st and 2nd MST 8 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3

24. Final MST 8 5 3 6 1 2 6 8 2 4 5 5 10 1 7 3 3
Total weight = 20

25. Illustration with collinear partition

26. Experiments t4.8k
Processing time
Approximation error

27. Experiments ConfLongDemo
Processing time
Approximation error

28. Effect of cluster size n

29. Effect of cluster size n

30. Blank space for notes