prims algorithm for algorithms for cs st

Greedy Algorithm
Prim’s Algorithm for Computing MSTs

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
e
d
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
10
9
e
d
Weighted
Graph
7 32
23

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

PRIM’s Algorithm ( Steps 1-5 :
Initialization )
MST-PRIM(G,w,r)
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
1 for each u∈V[G]
2 do key[u]←∞
3 Π[u]←NIL Initialization
4 key[r]←0
5 Q←V[G]
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)
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)
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)
MST-PRIM(G,w,r)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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)
……
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)
MST-PRIM(G,w,r)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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

u=b)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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

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
u=c)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
7
c
8
2
7
6
1 2
g
h
4 14
10
9
Example Graph
Steps 6-11(for u=c)
After Step 6-11 (for u=c)
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

u v
v∈Q AND
w( u, v ) < Key[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
u=c)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
2
7
c
8
7
6
1 2
g
h
4 14
10
9
Example Graph
Steps 6-11(for u=c)
After Step 6-11 (for u=c)
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
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

u=i)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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

u=i)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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

Q d e g h
A a b c i f
8
u=f)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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
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

Q d e g h
A a b c i f
8
u=f)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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
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

u v
v∈Q AND
w( u, v ) < Key[v]
g h YES Π[h]←g, Key[h]←1
g i NO Do Nothing
g f NO Do Nothing
u=g)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
4 14
10
9
Example Graph
Steps 6-11(for u=g)
After Step 6-11 (for u=g)
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

u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
h a NODo Nothing
h b NODo Nothing
h i NODo Nothing
h g NODo Nothing
u=h)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
4 14
10
9
Example Graph
Steps 6-11(for u=h)
After Step 6-11 (for u=h)
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

u v
v∈Q AND
w( u, v ) < Key[v]
Then Π[v]←u,
Key[v]←w(u,v)
h a NODo Nothing
h b NODo Nothing
h i NODo Nothing
h g NODo Nothing
u=h)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
4 14
10
9
Example Graph
Steps 6-11(for u=h)
After Step 6-11 (for u=h)
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

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
u=d)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
4 14
10
9
Example Graph
Steps 6-11(for u=d)
After Step 6-11 (for u=d)
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

u v
v∈Q AND
w( u, v ) < Key[v] Then Π[v]←u,
Key[v]←w(u,v)
e d NODo Nothing
e f NODo Nothing
u=e)
10 then Π[v]←u
11 key[v]←w(u,v)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
4 14
10
9
Example Graph
Steps 6-11(for u=e)
After Step 6-11 (for u=e)
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

u=e)
d
f
e
i
4
8
11
a
8 7
c
b
2
7
6
1 2
g
h
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
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 )
MST-PRIM( G, w, r )

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]
9
do if v∈Q and w(u,v)<key[v]
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)

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

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

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