Kruskal’s Algorithm: Finding the Minimum Spanning Tree

KRUSKAL’S
ALGORITHM
DETAILED ANALYSIS ON:

JOSEPH KRUSKAL
• JOSEPH KRUSKAL WAS AN AMERICAN MATHEMATICIAN, STATISTICIAN AND
COMPUTER SCIENTIST.
• HE WAS BORN IN JANUARY 29, 1928.
• HE WAS A STUDENT AT THE UNIVERSITY OF CHICAGO EARNING A BACHELOR OF
SCIENCE IN MATHEMATICS IN THE YEAR OF 1948 AND A MASTER OF SCIENCE IN
MATHEMATICS IN THE FOLLOWING YEAR 1949.
• AFTER HIS TIME AT THE UNIVERSITY OF CHICAGO, KRUSKAL ATTENDED PRINCETON
UNIVERSITY, WHERE HE COMPLETED HIS PH.D. IN MATHEMATICS IN THE YEAR OF 1954.
• IN STATISTICS, KRUSKAL'S MOST INFLUENTIAL WORK IS HIS SEMINAL CONTRIBUTION TO THE
FORMULATION OF MULTIDIMENSIONAL SCALING.
• IN COMPUTER SCIENCE, HIS BEST KNOWN WORK IS KRUSKAL'S ALGORITHM FOR COMPUTING
THE MINIMAL SPANNING TREE (MST) OF A WEIGHTED GRAPH.
• HE DIED IN SEPTEMBER 19, 2010.

WHAT IS AN ALGORITHM?
AN ALGORITHM IS A STEP-BY-STEP
PROCEDURE TO SOLVE A GIVEN
PROBLEM.

KRUSKAL'S ALGORITHM IS AN ALGORITHM
FOR
COMPUTING THE MINIMAL SPANNING
TREE (MST)
OF A WEIGHTED GRAPH.
KRUSKAL’S ALGORITHM
A Minimal Spanning Tree is a Spanning Tree of
a connected, undirected graph which connects
all the vertices together with the Minimal total
weightage for its edges and it always have V-1
Edges.
A
B
C
D
1 3
2

KRUSKAL'S ALGORITHM IS AN ALGORITHM
FOR
COMPUTING THE MINIMAL SPANNING
TREE (MST)
OF A WEIGHTED GRAPH.
A weighted graph is a graph
whose vertices or edges
have been assigned weights.
A
B
C
D
1 3
2
2
5

LETS SEE A GRAPH…!
A
B
C
D
1 3
2
2
5
Vertices
Edge
Weight of an
Edge

HOW TO FIND MINIMAL SPANNING TREE (MST) BY USING
KRUSKAL’S ALGORITHM?
We will always start from the
smallest weight edge and keep
selecting edges that does not
form any circuit with the
previously selected edges.
REMEMBER…
.

A
B
C
D
1 3
2
2
5
We will always start from the smallest weight edge and keep selecting edges
that does not form any circuit with the previously selected edges.
We have 5 edges.
So our edge table will have 5 columns.
Create the edge table.
(An edge table will have name
of all the edges along with their
weight in ascending order)
Edge
Weight
STEPS TO FIND MINIMAL SPANNING TREE (MST) BY USING

A
B
C
D
1 3
2
2
5
Edge
Weight
Now look at the graph and fill the
1st column with the edge of minimum weight.
In this case edge BC is of least weight
so we will select it.
BC
1
Now look at the graph again and find the
edge of next minimum weight.
In this case edge AB and AC are having
the minimum weight so we will select
them both.
Note!
In case we have two or more
edges with same weight then
we can write them in any order
in the edge table.
AB
2
AC
2

A
B
C
D
1 3
2
2
5
Edge
Weight
BC
1
Now look at the graph again and find the
edge of next minimum weight.
In this case edge AB and AC are having
the minimum weight so we will select
them both.
Note!
In case we have two or more
edges with same weight then
we can write them in any order
in the edge table.
AB
2
AC
2

A
B
C
D
1 3
2
2
5
Edge
Weight
BC
1
AB
2
AC
2
Now look at the graph again and
find the edge of next minimum edge.
In this case edge DC has the
minimum weight so we will select it.
DC
2
Now look at the graph again
and find the edge of next
minimum edge.
In this case edge BD has the
minimum weight so we will
select it.
BD
5

A
B
C
D
1 3
2
2
5
TIME TO FIND MINIMAL SPANNING TREE (MST) BY USING
Edge
Weight
BC
1
AB
2
AC
2
DC
2
BD
5
Note! Our graph has 4 vertices.
So, our MST will have 3 edges.

Edge
Weight
BC
1
AB
2
AC
2
DC
2
BD
5
1 is the smallest weight.
So we will select edge BC.
B
C
1
A
B
C
D
1 3
2
2
5

A
2
A
B
C
D
1 3
2
2
5
Edge
Weight
BC
1
AB
2
AC
2
DC
2
BD
5
2 is the next smallest
weight and edge AB does
not form a circuit with the
previously selected edges.
So we will select it.
B
C
1

A
2
A
B
C
D
1 3
2
2
5
Edge
Weight
BC
1
AB
2
AC
2
DC
2
BD
5
2 is the smallest weight.
But if we select edge AC
then it will form a circuit. So
we are going to reject edge AC.
B
C
1

D
3
A
2
A
B
C
D
1 3
2
2
5
Edge
Weight
BC
1
AB
2
AC
2
DC
2
BD
5
3 is the next smallest weight and
edge DC does not form a circuit
with the previously selected edges.
So we will select it.
B
C
1

D
3
A
2
A
B
C
D
1 3
2
2
5
Edge
Weight
BC
1
AB
2
AC
2
DC
2
BD
5
B
C
1
Since we have got the 3 edges.
So we will stop here.

THEREFORE, OUR REQUIRED MINIMUM SPANNING TREE IS
D
3
A
2
B
C
1

Below are the steps for finding MST using Kruskal’s algorithm:
#Sort all the edges in non-decreasing order of their weight.
#Pick the smallest edge.
#Check if it forms a cycle with the spanning tree formed so far.
#If the cycle is not formed, include this edge.
Else, discard it.
#Repeat step#2 until there are (V-1) edges in the spanning tree.
KRUSKAL’S ALGORITHM IMPLEMENTATION IN CODE

 Make_Set(V) will try to think each vertices a individual tree
 Find_Set is the function that checks the other vertices that are reachable
 Find_Set is a trick to check whether it will create a cycle or not
 UNION merges two trees
KRUSKAL’S ALGORITHM IMPLEMENTATION IN CODE

TIME COMPLEXITY FOR GRAPH
 For Graph G=(V,E)
 For Most Efficient Time Complexity here, we use either Merge or Heap Sort
 Where time complexity is O(n log n)
 Thus, For this Graph
Time Complexity using Kruskal’s Algorithm is O(E log E) or O(E log V)
MST-KRUSKAL(G, w)
1 A = ∅
2 for each vertex v G.V
∈
3 MAKE-SET(v)
4 sort the edges of G.E into nondecreasing order by weight w
5 for each edge (u, v) G.E, taken in nondecreasing order by weight
∈
6 if FIND-SET(u) ≠ FIND-SET(v)
7 A = A {(u, v)}
∪
8 UNION(u, v)
9 return A

Kruskal’s Algorithm: Finding the Minimum Spanning Tree

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