???????? ???????? Graphs

1
(No Transcript)
2
???????? ???????? Graphs Networks ?????
????? ???????? ?????? ??????????? 1289
????? ?. ???? ?????
3
????? ?????? ???????

• ????? ????? ???????? ???????? ????? ??????.
• ???? ???????? ???????? ???? ??????.
• ?????? ??????? ???????? Dijkstra??????
• ???? ??? ??? ????? ?? ???????.
• ?????? ??????? ???????? Floyed ??????
• ??????? ??? ????? ?? ???????.

4
????????? GRAPHSKS
???????? NETWORK
5
???????????? (Graphs)
????? ????????
?? ?????? ?????? ???? ??????? ??????? ??
??????? ??????? ??? ???? ??????? ??????? ??? ??
????? ????????
6
??? ?? ????? ??????? ??? ?????????
?????? ?????? ????? ?? ??????? ??????? ???
?????? ??? ?????? ????? ?????? ????????
?????????? ?????? ??????? ?? ???? ??? ?????
7
????? ???
??? ???? ????? ???? ?????? ?? ???????? ?????????
?? ???? ??????? ??? ??? ???? ?????? ????? ??????
????? ???? ?? ???? ?????? ?????? ?? ?????? ?????
????? ?????? ???? ?? ???? ?????? ??????? ??? ???
?????
8
???? ?? ????
??? ?????? ?? ???? ??????? ????????? ?? ???? ????
???????? ??? ??????? ???????? ?? ??????
(Vertices) ?????? ??????? ????? (Edges) ????? ??
???? ?????? ?? ??? ??????? ??? ??? ?????
???????? ??? ?????? ?????????
9
????? ???????? ?? ?????? ?????? ???

• ?????? ?? ??????? (Element)
• ?? ????? (Node) ????? ?????? V
• (Vertices)
• ?????? ?????? E (edge)
• ????? ???? ??? ?????? ???? ???? ??? ???
• G(V,E)

10
?????? ???????? Graphs Type
???? ????? ??????? ?? ???????? ??

• ???????? ????? ?????
• (UNDIRECTED GRAPHS)
• ???????? ???????
• (DIGRAPH )

11
????
???????? ????? ????? (UNDIRECTED GRAPHS)
???? ?????? ??? ???? ????? ???? ?? ???? ??? ???
??? ?? ????? ????? ?????
12
????? ??? ?? ???? ?????? ???? ???? ??? ????
?????? ?????? ????

??? ??? ??????
???? ????? (v1,v2) ?????? ????? (v2,v1)
13
v2
v3
v5
v1
v4
14
?????
???????? ??????? (DIGRAPHS)
?????? ???? ???? ??? ???? ?? ?????? ?????? ????
??? ??? ???? ?? ????? ????? ???? ??? ?????
??????? ?? ?? ?????? ??????? ?????? ???? ????
??? ??? ?????
15
v1
v2
v4
v3
v5
v7
v6
16
????? ????????Representation of Graphs
???? ??? ??? ?????? ???????? ????

• ??????? ??????? (Linked Representation)
• ?????? ?????? (Adjacency Matrices)

17
????
??????? ??????? (Linked Representation)
????? ?? ???? ????? ???????? ???????? ??????
????? ????? ?? ??? ?? ?????? ??? ????? ???
???????????? ???? ???? ?? ??????? ??????? ??
???? ?? ???? ??? ??? ?????? ?? ????? ??? ?????
18
???? ???????? ????????? ?????? ?? ?????? ?? ????
??? ????? ?? ????? ???? ??? ?? ?????? ?? ???
??? ?????? ???? ?? ???? ????? ?????? ??? ????
19
?? ??? ?? ?????? ?? ????? ???????? ??? ???
??????(??? ?????) ??? ???? ?? ??? ??????????? ??
?????? ?? ???? ??? ?? ????? ???? ????????? ??????
20
?????
??? ?????? ?????? ???????? ?????? ?????

1
4
2
3
21
?????
??? ????? ?????? ?? ???? ???????

1. ????? ???????
2. ????? ???????

22
??????? ??????
????? ???????
2
1
1
1




3
3
2
2
4
4
4
3
1
NULL
2
NULL
NULL
3
NULL
4
23
?????? ?????? (Adjacency Matrices)
?????
?? ????? ?? ?????? ?????? ??????? ???? ????????
????? ???????? ??? ??? ?? ?????? ??????? ??????
A(aij)?? ?????? ?? ????? n x n ??? ???? ??
?????? aij ???? ?????
24
??? ??? ???? ???? ?? ????? i ??? ????? j ?????
0 ??? ?? ??? ???? ???? ???? ??? ???????
????? ??? ??????? ??????? ???? ?????? ??????
aij 1 if there is an edge from i to j
0 if there is no edge
25
?????
??? ????? ?????? ???????? ?????? ??????
v2
v1
v4
v3
v5
v7
v6
26
?????
???? ????? ?????? ?????? ??????? ?????? ???????
V7 V6 V5 V4 V3 V2 V1
0 0 0 0 0 1 0 V1
0 0 1 0 0 0 0 V2
0 1 0 0 0 0 1 V3
0 0 1 0 1 0 0 V4
1 0 0 0 0 0 0 V5
0 0 0 0 0 0 0 V6
0 1 0 0 0 0 0 V7
27
????? ?? ?????? ?????? ????? ????? ??? ????????
0 ? 1 ????? ????? ???? ????? ???????? ????????
28
???? ?????? ??????? ???????? ?? ?? ???? ??
?????? ???????? ?????? ??????
????? ??? ?????? ???????
?????? ??? ?????? n ????? ?????? ?? ??????
?????? ????? ??? ???? ????? ????? 1 ?? ??????
??????
29
????? ??????? ????? ?????? ???????
??? ?????? ??? ?????? n ????? ???? ?? ??????
?????? ????? ??? ???? ???? ????? 1 ?? ??? ????
?????? ??? ???? ???? ????? 1 ??? ?????? ???????
?? ?? ???? ?? ??????.
30
?????
????? ?????? ??????? ???????? ?? ???? ?? ????
?? ?????? ???????? ????? ?????? ?????? ????
?????? ???????
buildadjm(intadjMAX, int n)
31
void buildadjm(int adjMAX, int n)
int i,j for(i0iltni)
for(j0jltnj)
coutltlt"Enter 1 if there is an edge from "ltlt i ltlt"
ltlt "to" ltlt" "ltlt j ltlt"
otherwise enter 0 "ltltendl
cingtgtadjij
32
?? ??????
?? ???? ?????? indegree ??? ???? ??? ???? ????
????? 1 ?? ?? ???? ?? ?????? ?????? ????????
??? ?????? ???? ???? ???? ??????? ??????? ???
????? ???? ???? ????????
33
??? ???????
Int indegree (intadjMAX,intx,int n)
int i, count 0 for(i0iltni)
if( adjix 1) count
return(count)
34
?? ??????
?? ???? ?????? outdegree ??? ???? ??? ????
???? ????? 1?? ?? ?? ?? ?????? ?????? ????????
??? ?????? (?????) ???? ????? ??? ??? ??????
35
??? ???????
int outdegree(int adjMAX,int x,int n)
int i, count 0 for(i0iltni)
if( adjxi 1) count
return(count)
36
?? ??????
???? ???? ???????? ??????? ???????
void main() int adjMAXMAX, n, i
coutltlt"Enter the number of nodes in graph
maximum ltltMAXltltendl cingtgtn
buildadjm(adj,n)
37
for(i0iltni) coutltlt"The indegree of
the node "ltltiltlt" is ltltindegree(adj,i,n)ltltendl
coutltlt"The outdegree of the nodeltltiltlt"is ltltout
degree(adj,i,n)ltlt endl
38

• ??? ????????
• ???? ??? ???????? ???? ????? ?????? ??????
• ?? ???? ?????? ??????buildadjm???
• ??? ???? ?? ???? ??????? ?????? ??????
  ??????? ???????? ??? ???? ?? ??????
• ?? ???? ?????? ??????...........
  indegree?outdegree??? ???????.

39

• ??????indegree ???? ??? ????
• ???? ????? 1 ?? ?? ???? ?? ??????
• ?????? ???????? ??? ?????? ?????
• ???? ???? ?????? ??????? ??? ?????
• ???? ???? ????????.

40

• ??????outdegree ???? ???
• ???? ???? ????? 1 ?? ?? ??
• ?? ?????? ?????? ???????? ???
• ?????? ???? ????? ??? ??? ??????.

41
?????
??????? NETWORKS
???? ?????? ??? ???? ???? ????? Diagram ?????
?? ???? ????? A ????? ????? F?????? ???? ??????
??????? ??????? ?? ????? Nods ????? ??? ????
?? ?????? Edges
42
(?? ???? ??? ??? ?????? ?? ???? ) ???? ???? ??
????? ???? ????? ????? ???? ??? ??? ??????
43
??? ????? ???? ??????? SHORTEST PATH
?? ????? ???? ??????? ?? ?????? ?????? ????? ???
???? ??? ??? ?? ?????? ?? ?????? ???? ???? ?????
????? ?????? ?? ?????
44
????? ?????? ?????? ?????? ?????? ?? ???? ???
??? ?? ???? ????? ???? ??? ????? ???? ???????
??? ??? ??????? ?? ???? ??? ???? ????? ???????
??? ?????? ??????? ????? ?? ???? ????????
???????? ??? ?????? ????????
45
??????? ???? ??????? ?????? ???? ?????????

• ??????? ??????
• ???????? ???????? ???????? (Dijkstra)
• ??????? ???????
• ???????? ???????? ????? (Floyd)

46

• ??????? ??????
• ???????? ???????? ???????? Dijkstra

???? ?????? ?? ??????????? ?????? ??????? ?? ???
??? ???? ?? ?????? ???? ???????? Dijkstra ???
???? ??? ?????????? ??? ????? ??????? ?? ???
????? ????? ????? ?????? ?? ??????
47
???? ???????? Dijkstra ?? ??? ????? ???? ????
?????? (source) ?? ?????? ?????? ????? ??? ???
??? ???? ????? ?????? ?? ?????? ????? ??? ?????
??????? ?????? ?? ????? ???????? ??? ?????
?????? ?????? ?????
48
???????? ???? ???? ???????? Dijkstra ??? ?????
????? ??? ????? ?????? ?? ???? ??????? ??? ????
??????? ??? ?????? ?????? ?? ????? ???
?????????? ?? O(n2)
49

• ?????? ????? ?????? (Adjacency Matrix Cost)
• ?????? ????? ( Distance Matrix)
• ?????? ??????? (Path Matrix)
• ?????? ??????? (Include Matrix)

????? ??? ?????????? ??? ???? ??????? ???
50
???????? ?????? ?????? ????? ??????
(Adjacency Matrix Cost)
?? ????? ?????? ???? ????? ??????
51
A
B
D
C
52
???? ????? ????? ?????? ???????? ?????? ??????
??? ???
D C B A  
A TO D A TO C A TO B 0 A
B TO D B TO C 0 0 B
C TO D 0 0 0 C
0 D TO C 0 0 D
???? ??????? ?? ???????? ???? C ???????
int COSTNN
53
???????? ??????? ?????? ????? (Distance
Matrix)

• ?????? ??? ???? ????? ?? ???? ????????
• ??? ?????? J ???? ?? ??????
• DISTANCEJ
• ???? ??????? ?? ???????? ???? C ???????
• int DISTANCENN

54
???????? ??????? ??????
??????? (Path Matrix)
?????? ??????? ???? ??? ??? ?? ???? ??????? ???
?? path j ????? ??? ?????? ??????? ????????
????? j ?? ???? ??????? ???? ??????? ?? ????????
???? C ??????? int PATHN
55
???????? ??????? ?????? ???????
(Include Matrix)
??? ?????? ?????? ?????? ????? ??? ???????
???????? ? ??? ?? ??? ???? ????? (????????) ???
???? ??????? ????? ??? ?????? ?? ????? ??????????
???? ?? ???? ????????
56
??? ???? ???? ?? ?? ?????? ??? j ??? ????
??????? Included ????? True ???? ??????? ??
???????? ???? C ??????? bool
INCLUDEN
57
??? ???????? ???????? ?????? ???? ???????
Const int N50 Typedef bool Boolean Type N
Int cost NN Int DISTANCE N Int PATH
N Bool INCLUDE N Void SHORTPATH ( int
SOURCE ) Boolean Type Included Int I , j
ReadMat ()
58
Initialize ( SOURCE ) Do FINDMIN (J)
INCLUDEJ true For
(i0iltN i) If ( ! INCLUDEDJ
) UPDATE ( I, J ) PRINT () While
ALL( Included ) PrintPath ()
59
?????? ????????? ?? ???????? ????????

• ?????? ReadMAT
• ??? ??? ?? ???? ??? ??????
• ????? ?????? ????? ??????

60
???? ????? ??? ??? ????? ??? ????? ?????? 0 ??
?????? ???
???? ??? ???? ?????? (Vi ,Vi) ?? ?? ????
??????. ????? ??? ?? ???? ????.
61

• ???? Initialize
• ??? ??? ?? ??? ?????? ????? ????? ??????????
• INCLUDE SOURCE ???? ?? ???????
• ??? ???? ????? true INCLUDE i ???
• ?????? ???????? ????? false

62

• ?????? FINDMIN
• ??? ???? ???? ???? ??????? ?? ???? ???????
• ??? ?????? J
• ????? ???? ?????? J ??? ???? ???????
• ??? ??? ????? ???? Included j
• ????? ?????? ?????? true

63
??????? ??? ??? ??? ????? ???? ??????? ?? ????
??????? ??? ???? j ???? ??????? ???? ??
????????? Distance Matrix
64

• ?????? UPDATE
• ?????? ?????? ???????? Distance Matrix
• ????????? Path Matrix ?? ???? ????? ???? ???
• i ??? ?? (INCLUDE i false)
• ???? ??? ???? ???? ??????? ????????
• ??? ?????? ??? ??????

65
??? ??? ?????? ???? ??? ?????? ??? ??????
???????? ?? ???? ???????
66

• ?????? ALL
• ? ?????? ????? ??????
• ? ???? ?????? true ????? ???? ??? ????
• ??????? ?? ???????? Include Matrix
• ????? true
• ????? ??? ?????? ?????? false ???? ???

67

• ?????? SHORT PATH
• ??????? ????? ??? ?? ??????
• ??????? ?????? ???????
• ???????? ??? ???? ??????? ???? ???? ??????
• ??????? ??? ??????? ???? ???? ????
• ??? ????? ??? ?????? ??? ????? ???? ???????
  ?????? ??? ?????

68

• ???? ??????? PRINT
• ?????? ??? ??????
• ?????? ??? ???????????
• DISTANSE , INCLUDE , PATH
• ??? ?? ???? ?? ????? ??????????

69

• ?????? PRINTPATH
• ?????? ??? ???????
• ?????? ???? ??????? ??? ???? ??????
• ????? ?????? ????????

70
?????? ???? ??????? ??? ?????? ???????? ????????
????????
???????? ?????? ????? ?????? ??????? ?? ?????
?????? ??? ??? ???? ??????? ?? ?????? 1???
??????3
71
5 4 3 2 1  
200 310 2985 800 0 1
0 612 410 0 800 2
0 1421 0 410 2985 3
400 0 1421 612 310 4
0 400 0 0 200 5
72
???????

• ??? ?? ??? ???? ???? ?? ?????? (?)
• ??? ??????(?) ??? ???? ??????
• ?? ???? ? ??????? ? ????? ?? ??????? .

73

• ??? ??? ???????? ??? ?????? Initialize
• ???? ??? ????? ??????? ??????? ?? ???
• ???? ???? ??? ?????? ?? ?????? ??? ????
• ???? ????? ????? ?? ??? C ????? ???
• ??? ?????? ?????? MAXINT ??? ?????
• 32767 .

74
?? ??????
???? ???????? ???????? ?????? ??? ?????? ??????
? ???? ????? ??? ????? ?? ???????? cost ???? ????
?????? ????? cost 2,3 ?????? 410 ???? ???? ??
??? ?????? ???? ??? ??? ???????? 2 ? 3 ?? 410
75
??? ??? ??? ??????? ??????? initialize ?????
???? ????? ?????????? ????????? Distance
?Path ? Included ??? ???
76

• ????
• ??????? ???????? DISTANCE ??? ???
• ??? ?????? 1 ?? ???? ??????? ??? ?????? ?????
• ?? DISTANCE 1 ????? ???
• ??? ?????? 2 ?? ???? ???????
• ?? DISTANCE 2 ????? 800

77

• ??? ?????? 3 ?? ???? ???????
• ?? DISTANCE 3 ????? 2985
• ??? ?????? 4 ?? ???? ???????
• ?? DISTANCE 4 ????? 310
• ??? ?????? 5 ?? ???? ???????
• ?? DISTANCE 5 ????? 200

78
????? ???? ???????? DISTANCE ??? ???
Distance 0 800 2985 310 200
79
????? ??????? ???????? PATH ??? ???? ????
??????? ?? ??? ???? ???? ?? ????? ???? ???????
??? ?????? ?????? . ????? ???? ???????
????????PATH ?????? ?????? 1(?? ???? ???????)
???????? PATH1 ??? ????? ????? ???? . Path
0 1 1 1 1
80
????? ??????? ???????? INCLUDE ??? ??? ????
??????? ????? false ???????? ???? ??????INCLUDE
1 ??? ?? ?????? 1 ???? ???? ????????????
????? true???? ???? ?? ?????? 1 ????? ?? ????
?? ???? ??????? .
81
????? ???? ??? ????? ?????????? ?????????
DISTANCE ? INCLUDE ? PATH ??? ???
?????? 1 2 3 4 5
Distance 0 800 2985 310 200
Path 0 1 1 1 1
Include true false false false false
82
??? ??????? ??????? ?? ????? ?????????? ???
????? ??????? ????????? ?????? ???? ???? ???
????????? ????? ???? ??????? ?? ???? ??????? ???
???? ??? ????????? ?????? ???? ?? ????? ????
???????? ??????? ???????? ?? ???? ???????
83
???? ??? ?? ?????? ??????? ?? ?????????? ?????
????????? ??????? ??? ????????? ??????
???? ?? ??? ?????? 5 ?? ?????? 1 ?? ????? ??
????? ?????? ?????? ??? ?????? 5 ????? ?? ????
??? ???? ???????
84
????? ???? INCLUDE 5 ?????? ?????? true ?????
?????? 1 ?????? ?????? ???? ???? ?????? ??? ???
??? ?????? ?? ?????????? ????? ??????? ?? ????
??????? (??? ??????1) ?????? ??? ?????? ??????
??? ???? ??????? ???????? PATH ??? ??
85
????? ???? ??????? ???????? ??????
??????? ??? ?????? 5 ??? ???
?????? 1 2 3 4 5
Distance 200 1000 1821 400 0
Path 5 1 4 5 0
Include true false false true true
86
???? ?????? ????? ????? ?? ?????? 5 ??? ?????? 3
?? ???? ?????? 2
?????? 1 2 3 4 5
Distance 200 1000 1410 400 0
Path 5 1 2 5 0
Include true true false true true
87
????? ??? ???
?????? 1 2 3 4 5
Distance 200 1000 1410 400 0
Path 5 1 2 5 0
Include true true true true true
88
???????
?? ????? ?????? 2 ?????? ???? ????? ???????? ??
????????? ?????? ????? ?? ??????? ?????? ??????
?? ????????? ?????? ?? (false , 1 , 800) ????
???? ?? ??? ???? ??????? ?? ?????? 1 ??? ?????? 2
??? ??? ?????? ?? 800
89
??? ???? ?????? 1 ?? ?????? ???? ?? ?????????
?????? ??? ?????? 2 ?? ?????? 1 ????? ??????
false ?? ?????? 2 ?? ???? ???? ???????? ??????
???? ??????? ??? ??? ??????
90
?????? ??????? ???? ?????? ??? ?????? 3 ??? ??
?????? 1 ?????? ?????? ??? ????? 2985 ?? ?? ????
?????? 2 ?????? ??? ????? 800 ?? ?????? 1 ?? ??
???? ?????? 4 ?????? ?????? ?? ??? ?????? ??
310
91
???? ?? 310 ???? ?? 800 ???? ???? ?????? 4 ???
???? ??????? ????? ???? ???? DISTANCE 3
????? ( 3101421 1731 )
92
??? ???? ??????? PATH 4 1 ?? ?? ??????
??? ?????? 3 ?? ???? ?????? 4 ???? ?? PATH 4
1 ?? ?? ?????? ??? 4 ?? ???? ?????? 1 ( ????
??????? ) ???? ??? ??? ?????? ?????? 4 ??????
??? ?????? 3
93
????? ??????? ?????? ?????? ????????
INCLUDE ?????? ?????? true ?? ?? ???????
????????? ?????? ??? ???
94
?????? 1 2 3 4 5
Distance 0 800 1731 310 200
Path 0 1 4 1 1
Include true false false true true
95
???? ?? ??? ??? ???????? INCLUDE ??? ?????? ????
???????? (??? ?? ???? ???? ?????? false) ???
????? ????? ?????????? ??? ???? ???? ??? ?????
??? ???????? ?????? ?????? true
96
???? ?? ?????? ??????? ?? ?????????? ?????? ??
??? ????? ?? 1 ??? 2 ??? 3 ???? ?? ??? ????? ??
?????? 1 ??? 4 ?????? ???? ???? ??? ??????? ??
1 ??? 3 ????? 800410 1210
97
??? ??? ?? ????? ?????? ????? ?? 1 ??? 4 ?? ???
3 ????? ??? ????? ????????? ??????? ???
????????? ?? ???? ??????? UPDATE
98
??? ??? ??? ????????? ?? ??? ?????? ????? ????
???? PATH 3 ?? 4 ??? 2 ?????? DISTANCE3 ??
1731??? 1210 ?????? ???? INCLUDE 2 ?? false
??? true
99
??? ???? ?? ?????? 2 ?? ???? ???? ???????? ?????
???? ??????? ????????? ??????
?????? 1 2 3 4 5
Distance 0 800 1210 310 200
Path 0 1 2 1 1
Include true true true false true
100
??? ???????? ??????? ??? ???? ??????? ????
?????? ??? ?????? 3 ( ?? ???? ??????? ) ???
????? ??PATH 3 2 ?? ?? ?????? ??? 3 ?? ??
2 ??? ?? ????? ?????? ??? 2 ??? ?????? ??? ??
??? 1 ??? PATH 2 1
101
???? ?? 1 ?? ???? ??????? ??? ???? ??????? ?? 1
??? 2 ??? 3 ??????? ???? ??? ?????
?????? ???????? DISTANCE ??????? DISTANCE 3
?????? (1210) ??? ??? ?????
102
?????? 1 2 3 4 5
Distance 0 800 1210 310 200
Path 0 1 2 1 1
Include true true true true true
103
?????? ?????????? ???? ?????? ??????? ?????
??????
104
????? ???? ????? ???? ?????? ??? ?????? 1 ?
?????? 3
??????????
1210 410800 3 2 1
2985 3 1
3233 1421612800 3 4 2 1
1731 1421310 3 4 1
2021 1421400200 3 4 5 1
1622 410612400200 3 2 4 5 1
1332 410612310 3 2 4 1
105
?????? ??????
???? ???? ????? ???? ?????????? ??? ?????? 1 ?
?????? 3 ???? ?? ????? ?? ?????? 1 ??? ?????? 2
??? ?????? 3 ?? ???? ??????? ?????? ?? ?????
???? ??????? ??? ??? ??????? ?? ?? ??????
?????????? 1 2 3 800 410
1210
106
??????? ???????
???????? ????? Floyd
??? ???????? ????? ?????? ??? ???????? Dijkstra
??? ???? ??? ?????????? ??? ???? ?????? ????
??????? ??? ?????? ???????? ?? ?????? ??????
107
?? ???? ?????? ???? ??? ?????? ????? ?????
??????? ?????? ??????? ??????? ????? ?????
?????? ????? ?? ???? ????? ?????? ?? ??? ????
??????? ?? ???????? ??? ??? ??????? ??? ?? ??????
?????? ????? ???? ?????? ????? ??????? ???????
??? ?? ??? ???? ??????? ????? ??? ?????????? ??
O(n3)
108

• ????? ??? ?????????? ??? ???? ??????? ???
• ?????? ????? ?????? (Adjacency Matrix Cost)
• ?? ????? ?????.
• ?????? ??????? (Path Matrix)
• ????? ??? ???????? ??? ????? ??????? ??????
  ????? ????????? ????? ??????? ?? ??? ??????
• I ? J ????? ??? ??? ? PATH IJ

109

• ?????? P
• ??? ????? ???????? ?? ????? ?????????? ???
  ???

??? ?? ??? COST IJ ?????? ????? ?
COSTIJPIJ ??? ???? JI ( ?? ????? ?????
) ????? ??? ??? ??????? ?????? MAXINT ?? ?? ??
?????
110
???? ?? ?????????? ??????? ?? ?????? ????????
??? ??? ?????? K ???? ??? PIJ
MINIMUM(PIKPKJ,PIJ ) ???? ?????
???? ??? ???? ??????? ?? ?????? I ??? ?????? J
??? ?????? ???? ?? ???? ????? ???? ?? J ??? ???
?? ???? ?????? ????? ??? N-1
111
???? ?????? ???? ??????? ?????? ?????? ?? ?????
???????? ???????? ????? Floyd
?????
112
?????
???? ??? ?????? ????? ??????

• ??????? ???????? ??? ????? ??????? ??????
  ??????.
• ????? 8 ??? ??? ??? ???? ?? ???? ?????.
• ??? ?? ??? ???? ?????? ?????
• 5 (A,B,C,D,E)
  

113
???? ??? 5 ??????? ((P1, P2, P3, P4, P5
E D C B A
18 8 10 23 8 A
8 48 10 8 23 B
7 19 8 10 10 C
20 8 19 48 8 D
8 20 7 8 18 E
114
??????? ???????? P1

• ???? ??? ?????? ????? ????? ????? ??????
• ??? ????? ?????? ?? ????? ????? ??????
• ????? ?? ??? ???? ???????.
• ??? ??? ??????? ??? ?? ?????? ??? ?? ????
• ?????? ????? ???????.

115
E D C B A
18 8 10 23 8 A
8 48 10 8 23 B
7 19 8 10 10 C
20 8 19 48 8 D
8 20 7 8 18 E
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18 8 10 23 8 A
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