Minimum spanning tree

Minimum Spanning Tree
Presented by:
Hinal Lunagariya

TREE
B
A
FE
C
D
Connected acyclic graph
Tree with n nodes contains
exactly n-1 edges.
GRAPH
Graph with n nodes contains
less than or equal to n(n-1)/2
edges.

A connected,
undirected
graph
Four of the spanning trees of the graph
SPANNING TREE...
Suppose you have a connected undirected graph
Connected: every node is reachable from every
other node
Undirected: edges do not have an associated
direction
...then a spanning tree of the graph is a connected
subgraph in which there are no cycles

Minimizing costs
Suppose you want to supply a set of houses (say, in a new
subdivision) with:
electric power
water
sewage lines
telephone lines
To keep costs down, you could connect these houses with a
spanning tree (of, for example, power lines)
However, the houses are not all equal distances apart
To reduce costs even further, you could connect the houses
with a minimum-cost spanning tree

A cable company want to connect five villages to their network
which currently extends to the market town of Avonford.
What is the minimum length of cable needed?
Avonford Fingley
Brinleigh Cornwell
Donster
Edan
2
7
4
5
8 6
4
5
3
8
Example

MINIMUM SPANNING TREE
Let G = (N, A) be a connected, undirected graph where
N is the set of nodes and A is the set of edges. Each
edge has a given nonnegative length. The problem is to
find a subset T of the edges of G such that all the
nodes remain connected when only the edges in T are
used, and the sum of the lengths of the edges in T is as
small as possible possible. Since G is connected, at
least one solution must exist.

Finding Spanning Trees
• There are two basic algorithms for finding minimum-cost
spanning trees, and both are greedy algorithms
• Kruskal’s algorithm:
Created in 1957 by Joseph Kruskal
• Prim’s algorithm
Created by Robert C. Prim

We model the situation as a network, then the problem is
to find the minimum connector for the network
A F
B C
D
E
2
7
4
5
8 6
4
5
3
8

A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
List the edges in
order of size:
ED 2
AB 3
AE 4
CD 4
BC 5
EF 5
CF 6
AF 7
BF 8
CF 8
Kruskal’s Algorithm

Select the shortest
edge in the network
ED 2
A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8

Select the next
shortest
edge which does not
create a cycle
ED 2
AB 3A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8

Select the next
shortest
edge which does not
create a cycle
ED 2
AB 3
CD 4 (or AE 4)
A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8

Select the next
shortest edge which
does not create a cycle
E
A
C
A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8

Select the next shortest
edge which does not
create a cycle
ED 2
AB 3
CD 4
AE 4
BC 5 – forms a cycle
EF 5
A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8

All vertices have been
connected.
The solution is
ED 2
AB 3
CD 4
AE 4
EF 5
Total weight of tree:
18
A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8

Algorithm
function Kruskal (G=(N,A): graph ; length : AR+
):set of edges
{initialisation}
sort A by increasing length
N  the number of nodes in N
T  Ø {will contain the edges of the minimum spanning tree}
initialise n sets, each containing the different element of N
{greedy loop}
repeat
e  {u , v}  shortest edge not yet considered
ucomp  find(u)
vcomp  find(v)
if ucomp ≠ vcomp then
merge(ucomp , vcomp)
T  T Ú {e}
until T contains n-1 edges
return T

Kruskal’s Algorithm: complexity
Sorting loop: O(a log n)
Initialization of components: O(n)
Finding and merging: O(a log n)
O(a log n)

A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
Select any vertex
A
Select the shortest
edge connected to
that vertex
AB 3
Prim’s Algorithm

A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
Select the shortest
edge connected to
any vertex already
connected.
AE 4
Prim’s Algorithm

Select the shortest
edge connected to
any vertex already
connected.
ED 2
A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
Prim’s Algorithm

Select the shortest
edge connected to
any vertex already
connected.
DC 4
A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
Prim’s Algorithm

Select the shortest
edge connected to
any vertex already
connected.
EF 5A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
Prim’s Algorithm

A
F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
All vertices have been
connected.
The solution is
AB 3
AE 4
ED 2
DC 4
EF 5
Total weight of tree:
18
Prim’s Algorithm

Prim’s Algorithm
function Prim(G = <N,A>: graph ; length : A R+
) : set of
edges
{initialisation}
T  Ø
B  {an arbitrary member of N}
While B ≠ N do
find e = {u , v} of minimum length such
u € B and v € N B
T T υ {e}
B  B υ {v}
Return T
Complexity:
Outer loop: n-1 times
Inner loop: n times
O(n2
)

a
b
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c d
e
fg
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8 7
9
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14
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7
11
8
Example

a
b
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c d
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fg
i
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Solution

Minimum Connector Algorithms
Kruskal’s algorithm
1. Select the shortest edge in
a network
2. Select the next shortest
edge which does not create a
cycle
3. Repeat step 2 until all
vertices have been
connected
Prim’s algorithm
1. Select any vertex
2. Select the shortest edge
connected to that vertex
3. Select the shortest edge
connected to any vertex
already connected
4. Repeat step 3 until all
vertices have been
connected

Minimum spanning tree

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