1. Analysis of Algorithms
Minimum Spanning Trees
Uri Zwick
February 2014

2. Find a minimum spanning tree11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

3. Kruskal’s algorithm 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

4. Prim’s algorithm 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

5. Boruvka’s algorithm 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

6. MST verification 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

7. Comparison-based MST algorithms
Running time
Algorithm
O(m log n)
Kruskal (1956)
O(m + n log n)
Prim (1957)
Boruvka (1926) 
O(m log (m,n))
Gabow-Galil- Spencer-Tarjan (1986)
O(m (m,n))
Chazelle (2000)
O(m + n)
Karger-Klein-Tarjan (1995)
Deterministic
Rand.

8. Assume for simplicity that all edge weights are distinct
The MST is then unique

9. The lightest edge in a cut is contained in the MST
Cut rule S
VS
The lightest edge in a cut is contained in the MST

10. The heaviest edge on a cycle is not contained in the MST
Cycle rule C
The heaviest edge on a cycle is not contained in the MST

11. The intersection between a cut and a cycle is of even size
Cuts and cycles
The intersection between a cut and a cycle is of even size

12. Fundamental cycles Tree + non-tree edge  unique cycle
The removal of any tree edge on the cycle generates a new tree

13. The lightest edge in a cut is contained in the MST
Cut rule - proof S
VS w' w
w < w'
The lightest edge in a cut is contained in the MST