Prim's Minimum Spanning Tree Algorithm

PRIM’S MINIMUM SPANNING TREE

Informatics Department

Parahyangan Catholic University

PRIM’S MST



Used to solve minimum spanning tree problem



Has the property that edges in the set A always forms a single tree



The tree starts from an arbitrary vertex r,

and grows until the tree spans all the vertices

in V

PRIM’S MST

At each step, a light edge is added to the tree A that connects A to an isolated vertex

A

b

d a

c 14

8 5

22

Light edge ≡ edge with smallest weight that connects A

to an isolated vertex

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

A

EXAMPLE :: INITIALIZATION

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

a

⁰

b

^

c

^

d

^

e

^

f

^

g

^

h

^

i

^

Priority queue Q :

EXAMPLE :: ITERATION #1

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

a

⁰

b

⁴

h

⁸

c

^

d

^

e

^

f

^

g

^

i

^

Priority queue Q :

w(a,b) = 4 w(a,h) = 8

EXAMPLE :: ITERATION #2

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

b

⁴

h

⁸

c

⁸

d

^

e

^

f

^

g

^

i

^

Priority queue Q :

w(b,a) = 4 w(b,h) = 11

w(b,c) = 8

a not in Q 11 > key[h]

EXAMPLE :: ITERATION #3

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

h

⁸

g

¹

c

⁸

i

⁷

d

^

e

^

f

^

Priority queue Q :

w(h,a) = 8 w(h,b) = 11

w(h,i) = 7 w(h,g) = 1

a not in Q b not in Q

EXAMPLE :: ITERATION #4

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

g

¹

f

²

i

⁶

c

⁸

d

^

e

^

Priority queue Q :

w(g,h) = 1 w(g,i) = 6 w(g,f) = 2

h not in Q

EXAMPLE :: ITERATION #5

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

f

²

c

⁴

i

⁶

e

¹⁰

d

¹⁴

Priority queue Q :

w(f,g) = 2 w(f,c) = 4 w(f,d) = 14 w(f,e) = 10

g not in Q

EXAMPLE :: ITERATION #6

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

c

⁴

i

²

d

⁷

e

¹⁰

Priority queue Q :

w(c,b) = 8 w(c,i) = 2 w(c,f) = 4 w(c,d) = 7

b not in Q f not in Q

EXAMPLE :: ITERATION #7

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

i

²

d

⁷

e

¹⁰

Priority queue Q :

w(i,h) = 7 w(i,g) = 6 w(i,c) = 2

h not in Q g not in Q c not in Q

EXAMPLE :: ITERATION #8

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

d

⁷

e

⁹

Priority queue Q :

w(d,c) = 7 w(d,f) = 14

w(d,e) = 9

c not in Q f not in Q

EXAMPLE :: ITERATION #9

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14

9

10

e

⁹

Priority queue Q :

w(e,d) = 9 w(e,f) = 10

d not in Q f not in Q

RESULT #1 VS RESULT #2

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14 9

10

a

b c

i

h g f

d

e 8

8

11 4

7

4 2

7 6

1 2

14 9

10

PSEUDOCODE

1.

MST-Prim(G, w, r)

2.

for each u  V do

3.

key[u] = 

4.

parent[u] = NIL

5.

key[r] = 0

6.

Q = all vertices of G

7.

While Q ≠  do

8.

u = EXTRACT-MIN(Q)

9.

for each v adjacent to u do

10.

if v  Q and w(u,v) < key[v] then

11.

parent[v] = u

12.

key[v] = w(u,v)

Initialization :

•Set all vertices’ key to 

except the root, so that it will be the first vertex processed.

•Parent of each vertex is set to NIL.

•Min-priority queue Q contains all vertices.

IMPLEMENTATION



The performance of Prim’s MST depends on how we implement the min-priority queue Q



Suppose we implement Q using a min-heap:



Store vertices’ id on the heap’s array



Store vertices’ key on a separate array



Store vertices’ location on a separate array



Additionally, we need an array parent to

store the resulting tree

IMPLEMENTATION



Example:

vertex a b c d e f g h i

key 0 4 8     8 

parent NIL a b NIL NIL NIL NIL a NIL locatio

n -1 -1 2 3 4 5 6 1 7

h

⁸

c

⁸

d

^

e

^

f

^

g

^

i

^

idx 1 2 3 4 5 6 7 8 9

heap h c d e f g i

h

c d

e f g i

-1 means the vertex is not in

the heap

TIME COMPLEXITY



The performance of Prim’s MST depends on how we implement the min-priority queue Q



If Q is implemented using min-heap :

(V = #vertices , E = #edges)



Initialization phase :



The main loop has V iterations :

^{While Q ≠} u = EXTRACT-MIN(Q)^{ do} for each v adjacent to u do

if v  Q and w(u,v) < key[v]

then parent[v] = u

key[v] = w(u,v) V

V.lg V E

E E

= DECREASE-KEY  E lg(V) for each u  V do

key[u] = 

parent[u] = NIL key[r] = 0

Q = all vertices of G

V V V 1 V