Prim's algorithm

Presentation
On
Prim’s Algorithm
Submitted to:
Mrs. Shano Solanki
Assistant Professor
Department of CSE
NITTTR Chandigarh
Submitted by:
Pankaj Thakur
ME(CSE) Modular
RN 171408

Sub Topics to be covered:
• Introduction
• Basics concepts
• Minimum Spanning Tree Problem
• Solution to Minimum Spanning Tree Problem
• Generic-MST
• Prim’s algorithm details
• Complexity Analysis of Prim’s Algorithm.
• Applications and latest research work based on Prim’s Algorithm
• Summary
• Queries.

Introduction
•Prim's algorithm is a greedy algorithm that finds a minimum spanning
tree for a weighted undirected graph.
•Developed in 1930 by Czech mathematician Vojtěch Jarník and later
rediscovered and republished by computer scientists Robert C. Prim in
1957 and E. W. Dijkstra in 1959.
•Therefore, it is also sometimes called the DJP algorithm, Jarník's
algorithm, the Prim–Jarník algorithm,or the Prim–Dijkstra algorithm.

Basic Concepts
Spanning Trees: A subgraph T of a undirected graph G = ( V, E ) is a
spanning tree of G if it is a tree and contains every vertex of G.
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
Graph
Spanning Tree 1 Spanning Tree 2 Spanning Tree 3
• Every connected graph has a spanning tree.
• May have multiple spanning tree.
• For example see this graph.

Basic Concepts Cont…..
Weighted Graph: A weighted graph is a graph, in which each edge has
a weight (some real number ) Example:
a
b
c
d
e
Weighted Graph
10
9
7 32
23

Basic Concepts Cont….
Minimum Spanning Tree in an undirected connected weighted graph is
a spanning tree of minimum weight. Example:
a
b
c
d
e
Weighted Graph
10
9
7 32
23
a
c
d
e
Spanning Tree 1,
w=74
10
9
32
23
a
c
d
e
Spanning Tree 2,
w=71
9
7 32
23
b b a
c
d
e
Spanning Tree 3,
w=72
10
7 32
23
b
(Minimum Spanning Tree)

Minimum Spanning Tree Problem
MST Problem : Given a connected weighted undirected graph G, design
an algorithm that outputs a minimum spanning tree (MST) of graph G.
• How to find Minimum Spanning Tree ?
• Generic solution to MST
Two
Algorithms
Kruskal’s
algorithm
Prim’s
algorithm

GENERIC-MST Algorithm
GENERIC-MST ( G, w )
1 A = ϴ
2 while A does not form a spanning tree
3 find an edge ( u, v ) that is safe for A
4 A = A U { ( u, v ) }
5 return A
•The idea is to start with an
empty graph and try to add
edges one at a time,
always making sure that
what is built remains
acyclic.
•Gives us an idea how to
grow a MST.
•An edge (u, v) is safe for A
if and only if A  {(u, v)} is
also a subset of some MST

PRIM’s Algorithm
• A special case of generic minimum-
spanning-tree algorithm and operates much
like Dijkstra’s algorithm.
• Edges in the set A always form a single tree.
• Greedy algorithm since at each step it adds
to the tree an edge that contributes the
minimum amount possible to the tree’s
weight.
•Connected graph G and the root r of the
MST to be drawn are inputs.
•During execution of the algorithm, all vertices
that are not in the MST reside in a min-priority
queue Q based on a key attribute.
•For each vertex v, the attribute v.key is the
minimum weight of any edge connecting v to
a vertex in the tree.
•The attribute v.Π names parent of v in the tree.
•Maintains the set A from GENERIC-MST as
A = { ( v, v.Π) : v ∈ V – {r} – Q }.
When the algorithm terminates, the min-priority
queue Q is empty.
The MST A for G is thus
A = { ( v, v.Π) : v ∈ V – {r} }.
MST-PRIM( G, w, r )
1 for each u∈V[G]
2 do key[u]←∞
3 Π[u]←NIL
4 key[r]←0
5 Q←V[G]
6 while Q is not Empty
7 do u←EXTRACT-MIN(Q)
8 for each v∈Adj[u]
9 do if v ∈ Q and w( u ,v ) < key[v]
10 then Π[v]←u
11 key[v]←w( u, v )

PRIM’s Algorithm ( Steps 1-5 : Initialization )
MST-PRIM(G,w,r)
1 for each u∈V[G]
2 do key[u]←∞
3 Π[u]←NIL
4 key[r]←0
5 Q←V[G]
9 do if v∈Q and w(u,v)<key[v]
10 then Π[v]←u
11 key[v]←w(u, v)
u a b c d e f g h i
key[u] ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
Π[u] NIL NIL NIL NIL NIL NIL NIL NIL NIL
u a b c d e f G H i
key[u] 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
Q A b c d e f g h i
A EMPTY
After Steps 1-3
After Step 4
After Step 5
Initialization
Example Graph

Q a b c d e f g h i
key[u] 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
Q a b c d E f g h i
A
EM
PTY
PRIM’s Algorithm (Steps 6 to 11) ……
Before Step 6 Steps 6-11 (for u=a) After Step 6-11 (for u=a)
MST-PRIM(G,w,r)
1 for each u∈V[G]
2 do key[u]←∞
3 Π[u]←NIL
4 key[r]←0
5 Q←V[G]
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
u v
v∈Q AND
w(u,v) < Key[v]
Π[v]←u,
Key[v]←w(u,v)
a b YES
Π[b]←a,
Key[b]←4
a h YES
Π[h]←a,
Key[h]←8
Q a b c d e f g h i
key[u] 0 4 ∞ ∞ ∞ ∞ ∞ 8 ∞
Π[u] NIL a NIL NIL NIL NIL NIL a NIL
Q b c d e f g h i
A a

PRIM’s Algorithm (Steps 6 to 11) ……
MST-PRIM(G,w,r)
1 for each u∈V[G]
2 do key[u]←∞
3 Π[u]←NIL
4 key[r]←0
5 Q←V[G]
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Q a b c d e f g h i
key[u] 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
Q a b c d e f g h i
A
Q a b c d e f g h i
key[u] 0 4 ∞ ∞ ∞ ∞ ∞ 8 ∞
Q b c d e f g h i
A a
Before Step 6 Steps 6-11 (for u=a) After Step 6-11 (for u=a)
u v
v∈Q AND
w(u,v) < Key[v]
Π[v]←u,
Key[v]←w(u,v)
a b YES
Π[b]←a,
Key[b]←4
a h YES
Π[h]←a,
Key[h]←8

PRIM’s Algorithm (Steps 6 to 11, for u=b)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=b)
After Step 6-11 (for u=b)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
b c YES Π[c]←b, Key[c]←8
b h NO do nothing
b a NO do nothing
Q a b c d e f g h i
key[u] 0 4 8 ∞ ∞ ∞ ∞ 8 ∞
Π[u] NIL a b NIL NIL NIL NIL a NIL
Q c d e f g h i
A a b
Status of Q before using u=b
Q a b c d e f g h i
key[u] 0 4 ∞ ∞ ∞ ∞ ∞ 8 ∞
Q b c d e f g h i
A a

PRIM’s Algorithm (Steps 6 to 11, for u=b)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=b)
After Step 6-11 (for u=b)
Status of Q before using u=b
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
b c YES Π[c]←b, Key[c]←8
b h NO do nothing
b a NO do nothing
Q a b c d e f g h i
key[u] 0 4 8 ∞ ∞ ∞ ∞ 8 ∞
Q c d e f g h i
A a b
Q a b c d e f g h i
key[u] 0 4 ∞ ∞ ∞ ∞ ∞ 8 ∞
Q b c d e f g h i
A a

PRIM’s Algorithm (Steps 6 to 11, for u=c)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
2
7
c
8
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=c)
After Step 6-11 (for u=c)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
c i YES Π[i]←c, Key[i]←2
c f YES Π[f]←c, Key[f]←4
c d YES Π[d]←c, Key[d]←7
c b NO Do Nothing
Q a b c d e f g h i
key[u] 0 4 8 7 ∞ 4 ∞ 8 2
Π[u] NIL a b c NIL c NIL a c
Q d e f g h i
A a b c
Status of Q before using u=c
Q a b c d e f g h i
key[u] 0 4 8 ∞ ∞ ∞ ∞ 8 ∞
Q c d e f g h i
A a b
b

PRIM’s Algorithm (Steps 6 to 11, for u=i)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=i) After Step 6-11 (for u=i)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
i h YES Π[h]←i, Key[h]←7
i g YES Π[g]←i, Key[g]←6
i c NO Do Nothing
Q a b c d e f g h i
key[u] 0 4 8 7 ∞ 4 6 7 2
Π[u] NIL a b c NIL c i i c
Q d e f g h
A a b c i
Status of Q before using u=i
Q a b c d e f g h i
key[u] 0 4 8 7 ∞ 4 ∞ 8 2
Q d e f g h i
A a b c

PRIM’s Algorithm (Steps 6 to 11, for u=i)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=i)
After Step 6-11 (for u=i)
u v
v∈Q AND
w( u, v ) < Key[v] Then Π[v]←u,
Key[v]←w(u,v)
i h YES Π[h]←i, Key[h]←7
i g YES Π[g]←i, Key[g]←6
i c NO Do Nothing
Q a b c d e f g h i
key[u] 0 4 8 7 ∞ 4 6 7 2
Q d e f g h
A a b c i
Status of Q before using u=i
Q a b c d e f g h i
key[u] 0 4 8 7 ∞ 4 ∞ 8 2
Q d e f g h i
A a b c

8
PRIM’s Algorithm (Steps 6 to 11, for u=f)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=f) After Step 6-11 (for u=f)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
f d NO Do nothing
f e YES
Π[e]←f,
Key[e]←10
f g YES Π[g]←f, Key[g]←2
f c NO Do nothing
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 7 2
Π[u] NIL a b c f c f i c
Q d e g h
A a b c i f
Status of Q before using u=f
Q a b c d e f g h i
key[u] 0 4 8 7 ∞ 4 6 7 2
Q d e f g h
A a b c i
i

8
PRIM’s Algorithm (Steps 6 to 11, for u=f)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=f)
After Step 6-11 (for u=f)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
f d NO Do nothing
f e YES
Π[e]←f,
Key[e]←10
f g YES Π[g]←f, Key[g]←2
f c NO Do nothing
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 7 2
Q d e g h
A a b c i f
Status of Q before using u=f
Q a b c d e f g h i
key[u] 0 4 8 7 ∞ 4 6 7 2
Q d e f g h
A a b c i
i

PRIM’s Algorithm (Steps 6 to 11, for u=g)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=g)
After Step 6-11 (for u=g)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
g h YES Π[h]←g, Key[h]←1
g i NO Do Nothing
g f NO Do Nothing
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 1 2
Π[u] NIL a b c f c f g c
Q d e h
A a b c i f g
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 7 2
Q d e g h
A a b c i f
Status of Q before using u=g

PRIM’s Algorithm (Steps 6 to 11, for u=h)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=h)
After Step 6-11 (for u=h)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
h a NO Do Nothing
h b NO Do Nothing
h i NO Do Nothing
h g NO Do Nothing
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 1 2
Q d e
A a b c i f g h
Status of Q before using u=h
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 7 2
Q d e h
A a b c i f g

PRIM’s Algorithm (Steps 6 to 11, for u=h)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=h)
After Step 6-11 (for u=h)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
h a NO Do Nothing
h b NO Do Nothing
h i NO Do Nothing
h g NO Do Nothing
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 1 2
Q d e
A a b c i f g h
Status of Q before using u=h
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 7 2
Q d e g h
A a b c i f

PRIM’s Algorithm (Steps 6 to 11, for u=d)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=d)
After Step 6-11 (for u=d)
u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
d c NO Do Nothing
d f NO Do Nothing
d e YES Π[e]←d, Key[e]←9
Q a b c d e f g h i
key[u] 0 4 8 7 9 4 2 1 2
Π[u] NIL a b c d c f g c
Q e
A a b c i f g h d
Status of Q before using u=d
Q a b c d e f g h i
key[u] 0 4 8 7 10 4 2 7 2
Q d e
A a b c i f h

PRIM’s Algorithm (Steps 6 to 11, for u=e)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Steps 6-11(for u=e)
After Step 6-11 (for u=e)
u v
v∈Q AND
w( u, v ) < Key[v] Then Π[v]←u,
Key[v]←w(u,v)
e d NO Do Nothing
e f NO Do Nothing
Q a b c d e f g h i
key[u] 0 4 8 7 9 4 2 1 2
Q
A a b c i f g h d e
Status of Q before using u=e
Q a b c d e f g h i
key[u] 0 4 8 7 9 4 2 1 2
Q e
A a b c i f g h d

PRIM’s Algorithm (Steps 6 to 11, for u=e)
d
f
ei
4
8
11a
8 7
cb
2
7
6
1 2gh
4 14
10
9
Example Graph
Q a b c d e f g h i
key[u] 0 4 8 7 9 4 2 1 2
Q
A a b c i f g h d e
MST
MST-PRIM( G, w, r )
1 for each u∈V[G]
2 do key[u]←∞
3 Π[u]←NIL
4 key[r]←0
5 Q←V[G]
9 do if v ∈ Q and w( u ,v ) < key[v]
10 then Π[v]←u
11 key[v]←w( u, v )

Complexity Analysis of Prim’s algorithm
MST-PRIM(G,w,r)
1 for each u∈V[G]
2 do key[u]←∞
3 Π[u]←NIL
4 key[r]←0
5 Q←V[G]
10 then Π[v]←u
11 key[v]←w(u,v)
O(V), if Q is implemented as min-heap.
Body of while loop is
executed |V| times.
Takes O( lg V) times.
Takes O( V lg V)
times.
Executed O(E) times total.
Constant
Takes O(lg V) times.
O(E lg V)
Total time: O(VlgV + ElgV) = O(ElgV)

1
2
6
3
7
5
4
10
28
14
16
25
24
18
12
Example 2 Continue…..
22

Applications
• Design of a network
(telephone network, computer network, electronic circuitry, electrical wiring
network, water distribution network, cable TV network)
• A less obvious application is that the minimum spanning tree can be used
to approximately solve the travelling salesman problem.
• Finding airline routes.
• To create high quality mazes
• Routing algorithms
• Study of molecular bonds in Chemistry
• Cartography
• Geometry
• Clustering
•Tour/Travel Management

Latest research papers based on Prim’s Algorithm
Exploring the parallel implementations of the three classical MST algorithms
( N. R. Latha; G. Shyamala; G. R. Prasad 2017 International Conference on Inventive Communication and
Computational Technologies (ICICCT))
Joint reconfiguration of feeders and allocation of capacitor banks in distribution systems using a multi-start
strategy
( Márcio M. Montsutsumi; Jose N. Melchor; Leonardo H. Macedo; Rubén Romero, 2017 IEEE PES Innovative Smart
Grid Technologies Conference - Latin America (ISGT Latin America) )
Distributed minimum spanning tree based information exchange policy for distributed systems
( Taj Alam; Zahid Raza, 2016 Fourth International Conference on Parallel, Distributed and Grid Computing (PDGC) )
Generating spanning tree of non-regular graphic sequences through a variant of Prim's algorithm
( Prantik Biswas; Abhisek Paul; Paritosh Bhattacharya, 2015 International Conference on Circuits, Power and Computing
Technologies [ICCPCT-2015] )
The transmission time analysis of IPTV multicast service in SDN/OpenFlow environments
( Pornnipa Rattanawadee; Natchaphon Ruengsakulrach; Chaiyachet Saivichit, 2015 12th International Conference on
Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON) )
Optimization of the Connection Topology of an Offshore Wind Farm Network
( Ouahid Dahmani; Salvy Bourguet; Mohamed Machmoum; Patrick Guérin; Pauline Rhein; Lionel Jossé, IEEE Systems
Journal )
Multiagent-Based Distribution Automation Solution for Self-Healing Grids
( Markus Eriksson; Mikel Armendariz; Oleg O. Vasilenko; Arshad Saleem; Lars Nordström, IEEE Transactions on
Industrial Electronics )
Cost-minimum network planning in large wind farm using revised prim's algorithm
Ichiro Kousaka; Daisuke Eguchi; Daiki Yamashita; Yosuke Nakanishi; Ruichi Yokoyama; Kenji Iba ISGT 2014
Prime Object Proposals with Randomized Prim's Algorithm
Santiago Manen; Matthieu Guillaumin; Luc Van Gool, 2013 IEEE International Conference on Computer Vision
Prim's algorithm based P2MP energy-saving routing design for MiDORi
Akiko Hirao; Yuki Nomura; Haruka Yonezu; Hidetoshi Takeshita; Daisuke Ishii; Satoru Okamoto; Naoaki Yamanaka, The
10th International Conference on Optical Internet (COIN2012)

References
• Coremen, Leiserson, Rivest and Stein: Introduction to algorthms, PHI
• Horowitz, Sahni and Rajsekaran: Fundamentals of Computer Algorithms,
Galgotia.
• www.mathworld.wolfram.com
• IEEE xplore

Summary
Prim's algorithm is a greedy algorithm, and is a special
case of generic minimum-spanning-tree algorithm and
operates much like Dijkstra’s algorithm, that finds a
minimum spanning tree for a weighted undirected
graph and is mainly used for a dense graph i.e. a graph
with lots of edges.

Prim's algorithm

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