# minimal spanning tree problem

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• Minimal Spanning Tree Problem

:

• MST

3

MST

• L.Euler 1736 .

. Konigsberg

()

4

• . n-1 .

5

• . .

.

=40

6

a

c

e

d

b

2

45

9

6

4

5

5

• G .

7

• 1925 .

.

8

• . G .

=15

9

a

c

e

d

b

2

44 5

• MST

:

( ) .

10

• :

1956))1957))1965)())

11

Joseph Kruskal

Robert C. Prim

Otakar Boruvka

• T T .

T T . n > 0 G

n . T 1

12

{0,5} 10

{2,3} 12

{1,6} 14

{2,6} 16

{1,2} 16

{3,6} 18

{4,3} 22

{4,6} 24

{4,5} 25

{0,1} 28

0

1

5 26

4

3

10

25

24

22

1812

16

28

14

• 13() 0

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28

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• ()

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n-1 . n

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2524

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14 16

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• .

.

18

• n=10 n=25 n=50 n=75 n=100 n=125

Prim's algorithm 27 41 150 400 960 1800

Boruvka's algorithm 27 35 40 42 60 84

Kruskal's algorithm 27 30 32 34 36 40

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Prim's algorithm Boruvka's algorithm Kruskal's algorithm

19

• MST

MST .

)

. ( :

20

• MST

1 e Xe :

0

Min z =

S.t. = n-1

(,) S-1 ,

Xe 0,1 e E

21

• 22

a

b

c

d

2 4

4

3 2

Min z = 2Xab + 3Xac + 4Xbd + 4Xbc + 2Xcd

S.t. Xab + Xbd + Xcd + Xac + Xbc= 3

Xab 1

Xac 1

Xbd 1

Xcd 1

Xbc 1

Xab + Xbc + Xac 2

Xbc + Xcd + Xbd 2

Xe 0,1 e E

• Network flows: theory, algorithms, and applications I Ravindra K. Ahuja Thomas L.

Magnantl James B. Orlin.

On the History of the Minimum Spanning Tree Problem / R.L.Graham Pavol Hell

Comparing minimum spanning tree algorithms / Igor Podsechin

Tampereen lyseon lukio Tietotekniikka

Networks in Action / Gerard Sierksma Diptesh Ghosh

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