minimal spanning tree problem

of 24/24
ن الرحیم رحم بسم ا

Post on 16-Nov-2015

19 views

Category:

Documents

0 download

Embed Size (px)

DESCRIPTION

مسئله درخت پوشای مینیمال

TRANSCRIPT

  • 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

  • 24