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مسئله درخت پوشای مینیمال
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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
1
5 26
4
3
0
1
5 26
4
3
10
0
1
5 26
4
3
10
12
0
1
5 26
4
3
10
12
14
0
1
5 26
4
3
10
25
24
22
1812
16
28
14
0
1
5 26
4
3
10
12
14 16
0
1
5 26
4
3
10
12
14 16
22
25
0
1
5 26
4
3
10
12
14 16
22
16
( ) .
. .
. .
14
( ) 15
0
1
5 26
4
3
10
2524
22
1812
16
28
14
25
0
1
5 26
4
3
10
25
0
1
5 26
4
3
10
22
25
0
1
5 26
4
3
10
12
22
0
1
5 26
4
3
10
25
0
1
5 26
4
3
10
12
14 16
22
25
0
1
5 26
4
3
10
12
16
22
16
()
.
. .
n-1 . n
16
( ) 17
0
1
5 26
4
3
10
2524
22
1812
16
28
14
0
1
5 26
4
3
10
12
14
22
25
0
1
5 26
4
3
10
12
14 16
22
0
1
5 26
4
3
16
.
.
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
23
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