Analysis of Algorithms
Minimum Spanning Trees
Uri Zwick
February 2014
1
Find a minimum spanning tree
11
16
22
5
17
8
1
13
18
12
3
30
25
9
2
15
2
Kruskal’s algorithm
11
16
22
5
17
8
1
13
18
12
3
30
25
9
2
15
3
Prim’s algorithm
11
16
22
5
17
8
1
13
18
12
3
30
25
9
2
15
4
Boruvka’s algorithm
11
16
22
5
17
8
1
13
18
12
3
30
25
9
2
15
5
MST verification
11
16
22
5
17
8
1
13
18
12
3
30
25
9
2
15
6
Comparison-based MST algorithms
Running time
Kruskal (1956)
Prim (1957)
Boruvka (1926)

Gabow-GalilSpencer-Tarjan (1986)
Chazelle (2000)
Karger-Klein-Tarjan
(1995)
O(m logn)
O(m + n logn)
O(m logn)
O(m log (m,n))
Deterministic
Algorithm
O(m (m,n))
Rand.
O(m + n)
Assume for simplicity that
all edge weights are distinct
The MST is then unique
S
Cut rule
VS
The lightest edge in a cut is
contained in the MST
Cycle rule
C
The heaviest edge on a cycle is
not contained in the MST
Cuts and cycles
The intersection between
a cut and a cycle is of even size
Fundamental cycles
Tree + non-tree edge  unique cycle
The removal of any tree edge on the cycle
generates a new tree
Cut rule - proof
S
VS
w'
w
w < w'
The lightest edge in a cut is
contained in the MST