lecture-35 minimum spanning tree

8/3/2019 Lecture-35 Minimum Spanning Tree

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Minimum Spanning

Tree (MST)


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MotivationFor an electrical circuit certain pins have to begrounded. Finding the arrangement of wires

(connecting those pins) that uses the least amount of

wire.Let a G(V,E) be a graph such that (u, v) E and a

weight w(u,v) corresponding to wire needed to join u

and v.

We are looking for an acyclic subset T E that

connects all vertices and whose total weight w(T) is

minimized.

!Tvu

vuwTw),(

),()(


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MotivationSince T is acyclic and connects all the vertices, it mustform a tree.

Since it spans the graph, it is called a spanning tree.

MST or actually means Minimum Weight Spanning

Tree.


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Minimum Spanning Tree Problem: given a connected, undirected,

weighted graph:

14

10

3

6 4

5

2

9

15

8


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Minimum Spanning Tree Problem: given a connected, undirected,weighted graph, find a spanning tree using

edges that minimize the total weight

14

10

3

6 4

5

2

9

15

8


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Minimum Spanning Tree Which edges form the minimum spanning

tree (MST) of the following graph?

H B C

G E D

F

A

14

10

3

6 4

5

2

9

15

8


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Minimum Spanning TreeAnswer:

H B C

G E D

F

A

14

10

3

6 45

2

9

15

8


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Prims AlgorithmMST-Prim(G, w, r)

Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)

u = ExtractMin(Q);

for each v Adj[u]

if (v Q and w(u,v) < key[v])

p[v] = u;

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


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)

u = ExtractMin(Q);

for each v Adj[u]


p[v] = u;

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

1410

3

6 4

5

2

9

15

8

Run on example graph


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)

u = ExtractMin(Q);

for each v Adj[u]


p[v] = u;

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

g g g

g g g

g

g

1410

3

6 4

5

2

9

15

8

Run on example graph


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)

u = ExtractMin(Q);

for each v Adj[u]


p[v] = u;

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

g g g

0 g g

g

g

1410

3

6 4

5

2

9

15

8

Picka start vertex r

r


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)

u = ExtractMin(Q);

for each v Adj[u]


p[v] = u;

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

g g g

0 g g

g

g

1410

3

6 4

5

2

9

15

8

Blackvertices have been removed from Q

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)

u = ExtractMin(Q);for each v Adj[u]


p[v] = u;

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

g g g

0 g g

3

g

1410

3

6 4

5

2

9

15

8

Blackarrows indicate parent pointers

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

14 g g

0 g g

3

g

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

14 g g

0 g g

3

g

1410

3

6 4

5

2

9

15

8u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

14 g g

0 8 g

3

g

1410

3

6 4

5

2

9

15

8u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

10 g g

0 8 g

3

g

1410

3

6 4

5

2

9

15

8u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

10 g g

0 8 g

3

g

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

10 2 g

0 8 g

3

g

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

10 2 g

0 8 15

3

g

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

10 2 g

0 8 15

3

g

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

10 2 g

0 8 15

3

4

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

5 2 g

0 8 15

3

4

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

5 2 9

0 8 15

3

4

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

5 2 9

0 8 15

3

4

1410

3

6 4

5

2

9

15

8

u


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Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

5 2 9

0 8 15

3

4

1410

3

6 4

5

2

9

15

8

u


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R

eview: Prims AlgorithmMST-Prim(G, w, r)

Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

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

Whatis the hidden costin this code?


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Review: Prims Algorithm

MST-Prim(G, w, r)

Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

DecreaseKey(v, w(u,v));


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Review: Prims Algorithm

MST-Prim(G, w, r)

Q = V[G];

for each u Q

key[u] = g;

key[r] = 0;

p[r] = NULL;

while (Q not empty)



p[v] = u;

DecreaseKey(v, w(u,v));

How often is ExtractMin() called?

How often is DecreaseKey() called?


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lecture-35 minimum spanning tree

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