Minimum Spanning Tree (Data Structure and Algorithm)

Graphs
19CSE212 : Data Structures and Algorithms
Dr. Chandan Kumar
Department of Computer Science and Engineering
Jan - Jun 2024
Dr Chandan Kumar Jan - Jun 2024

TREE
B
A
F
E
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 Cornwel
l
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 3
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 (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
ED 2
AB 3
CD 4
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
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)

Prim’s Algorithm
Step 1: Select a starting vertex
Step 2: Repeat Steps 3 and 4 until there are fringe
vertices
Step 3: Select an edge e connecting the tree vertex
and fringe vertex that has minimum weight
Step 4: Add the selected edge and the vertex to the
minimum spanning tree T
[END OF LOOP]
Step 5: EXIT

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 5
A
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
Complexity:
Outer loop: n-1 times
Inner loop: n times
O(n2)

a
b
h
c d
e
f
g
i
4
8 7
9
10
14
4
2
2
6
1
7
11
8
Example

a
b
h
c d
e
f
g
i
4
8 7
9
10
14
4
2
2
6
1
7
11
8
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
edge connected to that
vertex
edge connected to any
vertex already
connected
4. Repeat step 3 until all
vertices have been
connected

List of References
Dr Chandan Kumar Jan - Jun 2024
https://www.javatpoint.com/
https://www.tutorialspoint.com
https://www.geeksforgeeks.org/
https://www.programiz.com/dsa/
https://chat.openai.com/

Minimum Spanning Tree (Data Structure and Algorithm)

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