1. G = (X, U), |X| = 5, |U| = 7. 2, 3, 4.
2. .
3. .
4. G = (X, U), ½X½ = 6; ½U½ = 10. , , , , . .
5. .
6. G = (X, U), ½X½ = 6; ½U½ = 18. .
7. , .
8. .
9. .
10. G .
, , , , .
()
. . .
1. R G.
, M. xk V(k) = 0, V(j) = M.
2. xj ¹ xk xj xi, ri , j. xi xk. xj xi
V(j) > V(i) + ri , j. (15.8)
, V(j) xj V(i) + ri , j. , V(j) .
3. . 1, 2 .
4. .
xj, xk, , .
, V(j). xj xk V(j), V(j) .
, G . 15.31 .
. 15.31. G
R , G . .
:
1 | 2 | 3 | 4 | 5 | 6 | 7 | ||||
1 | 0 | 4 | 5 | 9 | M | 2 | M | .
| ||
2 | 4 | 0 | M | M | M | 5 | 6 | |||
3 | 5 | M | 0 | 3 | 3 | M | 7 | |||
R = 4 | 9 | M | 3 | 0 | 8 | M | M | |||
5 | M | M | 3 | 8 | 0 | M | 1 | |||
6 | 2 | 5 | M | M | M | 0 | 3 | |||
7 | M | 6 | 7 | M | 1 | 3 | 0 |
1 V(1) = 0, V(2) = V(3) = = V(7) = M = ¥.
1) xk = x 1.
2) xi, xj (15.8).
i = 1, j = 2, V(1) = 0, V(2) = M, r1,2 = 4.
V(2) = M > V(1) + r 1,2 = 0 + 4 = 4. x 2 V(2)1 = 4.
, ri , j = M, (15.8) V(j) .
, x 3: V(3) = M > V(1) + r 1,3 = 0 + 5 = 5 V(3)1 = 5.
x 4: V(4) = M > V(1) + r 1,4 = 0 + 9 = 9 V(4)1 = 9.
x 6: V(6) = M > V(1) + r 1,6 = 0 + 2 = 2 V(6)1 = 2.
, x 1, .
x 2: V(2) = 4 < V(6) + r 2,6 = 2 + 5 = 7, , .
x 4: V(4) = 9 > V(3) + r 4,3 = 5 + 3 = 8, , V(4) 9 8 ( . 1.31).
x 6: V(6) = 2 < V(2) + r 2,6 = 4 + 5 = 9, .
, x 1.
x 5, x 3, x 4 x 7. x 7 , (15.8) .
x 5 x 4. V(5)41 = M > V(4)1 + r 4,5 = 8 + 8 = 16.
V(5)1 = 16 , x 3:
V(5)31= 16 > V(3)1 + r3,5 = 5 + 3 = 8.
(15.8) , V(5)1 = 16 V(5)2 = 8.
x 7 x 2, x 3, x 5, x 6.
V(7)21 = M > V(2)1 + r2,7 = 4 + 6 = 10,
V(7)51 = M > V(5)1 + r5,7 = 8 + 1 = 9,
V(7)31 = M > V(3)1 + r3,7 = 5 + 7 = 12,
V(7)61 = M > V(6)1 + r6,7 = 2 + 3 = 5.
, V(1) = 0, V(2) = 4, V(3) = 5, V(4) = 8, V(5) = 8, V(6) = 2, V(7) = 5.
, .
V(j) ─ V(i) = ri , j.
, x 7 x 1. V(7)1. x 7 ─ x 6. x 6 . x 6 - x1. , x 7 ─ x 6 ─ x 1, x 1 x 7 V(7)1 = 5.
, , , .
. 15.32 ,
. 15.32.
. 1 = 13. , 1 6 7 1 2 7 8 13. ( 3) = 8 (, 3 4 5 ).
, . .
|
|
s t.
, xi s . . , s .
:
l (xi) ─ xi.
( )
1. l (s) = 0. l (xi) = ¥ xi ¹ s ; p = s.
( )
2. xi Î (p), ,
l (xi) min[ l (xi); l (p) + r(p, xi)] (15.9)
( )
3. , l (xi *) =
= min[ l (xi)].
4. xi * p = xi *.
5.
) s t:
p = t, l (p) ─ , .
p ¹ t, . 2.
) s :
, , .
, . 2.
(. . 15.31).
G :
1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
1 | 0 | 4 | 5 | 9 | 0 | 2 | 0 | . |
2 | 4 | 0 | 0 | 0 | 0 | 5 | 6 | |
3 | 5 | 0 | 0 | 3 | 3 | 0 | 7 | |
R = 4 | 9 | 0 | 3 | 0 | 8 | 0 | 0 | |
5 | 0 | 0 | 3 | 8 | 0 | 0 | 1 | |
6 | 2 | 5 | 0 | 0 | 0 | 0 | 3 | |
7 | 0 | 6 | 7 | 0 | 1 | 3 | 0 |
*, .
xs, , . x 1. , x 1, ¥, p = x 1.
1. L(x 1) = 0*. L(xi) = ¥, " xi ¹ x 1, p = x 1.
.
2. , x 1 G(x 1) = { x 2, x 3, x 4, x 6}, (15.9) .
l (x 2) min[ ¥; 0*+ 4 ] = 4.
l (x 3) min[ ¥; 0*+ 5 ] = 5.
l (x 4) min[ ¥; 0*+ 9 ] = 9.
l (x 6) min[ ¥; 0*+ 2 ] = 2.
3. , :
Min [ 4, 5, 9, ¥, 2, ¥ ]
x 2 x 3 x 4 x 5 x 6 x 7
, x 6.
4. x 6 l (x 6) = 2*, p = x 6.
5. , . 2.
2. , x 6 G(x 6) = { x 2, x 7}, (15.9) :
l (x 2) min[ 4; 2*+ 5 ] = 4.
l (x 7) min[ ¥; 2*+ 3 ] = 5.
x 2 , , x 7 l (x 7) = ¥ l (x 7) = 5.
3. , :
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Min[ 4, 5, 9, ¥, 5]
x 2 x 3 x 4 x 5 x 7
, x 2.
4. x 2 l (x 2) = 4*, p = x 2.
5. , . 2.
2. , x 2 G(x 2) = { x 7}, (15.9) .
l (x 7) min[ 5; 4*+ 6 ] = 5.
x 7 , l (x 7) = 5.
3. , :
Min [ 5, 9, ¥, 5 ]
x 3 x 4 x 5 x 7
l (x 3) = l (x 7) = 5 . x 3.
4. x 3 l (x 3) = 5*, p = x 3.
5. , . 2.
2. , x 3 G(x 3) = { x 4, x 5, x 7} (1.9) :
l (x 4) min[ 9; 5*+ 3 ] = 8,
l (x 5) min[ ¥; 5*+ 3 ] = 8,
l (x 7) min[ 5; 5*+ 7 ] = 5.
x 7 , , x 4 x 5 .
3. , , :
Min [ 8, 8, 5 ]
x 4 x 5 x 7
, x 7.
4. x 7 l (x 7) = 5*, p = x 7.
5. , . 2.
2. , x 7 G(x 7) = { x 5}, (15.9) .
l (x 5) min[ 8; 5*+ 1 ] = 6.
3. , :
Min [ 8, 6 ]
x 4 x 5
x 5.
4. x 5 l (x 5) = 6*, p = x 5.
5. , . 2.
2. , x 3 x 4: G(x 1) = { x 4}. (15.9):
l (x 4) min[ 8; 6*+ 8 ] = 8.
x 4 , .
3. :
Min [ 8, ]
x 4
4. x 4 l (x 4) = 8*, p = x 4.
5. , .
, x 1 x 2, x 3, x 6 :
x 1 x 2(4*), x 1 x 3(5*), x 1 x 6(2*),
x 4, x 5, x 7 :
x 1 x 3 x 4(8*);
x 1 x 6 x 7 x 5(6*);
x 1 x 6 x 7(5*).
xs xi, .. , :
l (xs) + r (xs, xj) = l (xi),
.. , , xs, r (xs, xj) , l (xi).
: l (x 4)= 8. x 4 x 1, x 1 x 4:
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l (x 1) = l (x 4)─ r (x 3, x 4) = 8 3 = 5 = l (x 3) - r (x 1, x 3) = 5 5 = 0.
, x 1 x 4: x 1 x 3 x 4.
1. G = (X, U). G .
: G:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
1 | 0 | 10 | M | M | M | M | 3 | 6 | 12 | . |
2 | 10 | 0 | 18 | M | M | M | 2 | M | 13 | |
3 | M | 18 | 0 | 25 | M | 20 | M | M | 7 | |
R=4 | M | M | 25 | 0 | 5 | 16 | 4 | M | M | |
5 | M | M | M | 5 | 0 | 10 | M | 23 | M | |
6 | M | M | 20 | 16 | 10 | 0 | 17 | 15 | 9 | |
7 | 3 | 2 | M | 4 | M | 17 | 0 | M | 24 | |
8 | 6 | M | M | M | 23 | 15 | M | 0 | 5 | |
9 | 12 | 13 | 7 | M | M | 9 | 24 | 5 | 0 |
1.1. x 1. V(1) = 0, : V(2) = V(3) = = V(9) = ¥ = M.
1.2. , M. : x 2, x 7, x 8, x 9.
1.3. , , (1.8):
V(2) = ¥ > 0 + 10 = 10; V(7) = ¥ > 0 + 3 = 3; V(8) = ¥ > 0 + 6 = 6;
V(9) = ¥ > 0 + 12 = 12.
x 3, x 4, x 5, x 6 .
x 1, .
1.4. :
x 2:
- V(2) > V(7) + r27 10 > 3 + 2 ─ , , : V(2)1 = 5,
- V(2) > V(9) + r29 10 > 1 + 13 ─ , , ;
x 7:
- V(7) > V(2) + r27 3 > 10 + 2 ─ , , ;
- V(7) > V(9) + r79 3 > 12 + 24 ─ , , .
x 8:
- V(8) > V(9) + r89 6 > 12 + 5 ─ , , ;
x 9:
- V(9) > V(2) + r29 12 > 10 + 13 ─ , , ,
- V(9) > V(7) + r79 12 > 3 + 24 ─ , , ,
- V(9) > V(8) + r89 12 > 6 + 5 ─ , , : V(9)1 = 11.
1.5. : .
, x 1.
2.1. x 3, x 2, x 4, x 6 x 9. x 4 x 6 , x 2 x 9:
- V(3) > V(2) + r23 ¥ > 5 + 18 - , , : V(3)1 = 23,
- V(3) > V(9) + r39 23 > 11 + 7 ─ , , : V(3)2 = 18.
2.2. x 4, x 3, x 5, x 6 x 7. x 5 x 6 , x 3 x 7:
- V(4) > V(3) + r34 ¥ > 18 + 25 ─ , , : V(4)1 = 43,
- V(4) > V(7) + r47 43 > 3 + 4 ─ , , : V(4)2 = 7.
2.3. x 5, x 4, x 6 x 8. x 6 , x 4 x 8:
- V(5) > V(4) + r45 ¥ > 7 + 5 ─ , , : V(5)1 = 12,
- V(5) > V(8) + r58 12 > 6 + 23 ─ , , .
2.4. x 6, x 3, x 4, x 5, x 7, x 8 x 9:
- V(6) > V(3) + r36 ¥ > 18 + 20 ─ , , : V(6)1 = 38,
- V(6) > V(4) + r46 38 > 7 + 16 ─ , , : V(6)2 = 23,
- V(6) > V(5) + r56 23 > 12 + 10 ─ , , : V(6)3 = 22,
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- V(6) > V(7) + r67 22 > 3 + 17 ─ , , : V(6)4 = 20,
- V(6) > V(8) + r68 20 > 6 + 15 ─ , , .
- V(6) > V(9) + r69 20 > 11 + 9 ─ , , ,
3. .
, .
V(j) ─ V(i) = ri , j.
, x 6 x 1. V(6). x 6 ─ x 7. x 7 . x 6 - x 1. , x 6 x 7 ─ x 1, x 1 x 6 V(6) = 20.
2. G = (X, U). G .
: G:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
1 | 0 | 10 | 0 | 0 | 0 | 0 | 3 | 6 | 12 | . |
2 | 10 | 0 | 18 | 0 | 0 | 0 | 2 | 0 | 13 | |
3 | 0 | 18 | 0 | 25 | 0 | 20 | 0 | 0 | 7 | |
R=4 | 0 | 0 | 25 | 0 | 5 | 16 | 4 | 0 | 0 | |
5 | 0 | 0 | 0 | 5 | 0 | 10 | 0 | 23 | 0 | |
6 | 0 | 0 | 20 | 16 | 10 | 0 | 17 | 15 | 9 | |
7 | 3 | 2 | 0 | 4 | 0 | 17 | 0 | 0 | 24 | |
8 | 6 | 0 | 0 | 0 | 23 | 15 | 0 | 0 | 5 | |
9 | 12 | 13 | 7 | 0 | 0 | 9 | 24 | 5 | 0 |
*, .
1. l (x 1) = 0*. l (xi) = ¥, " xi ¹ x 1, p = x 1.
x 1. l (xi) = ¥.
2. (p) = (x 1) = { x 2, x 7, x 8, x 9}. x 2 (15.9) :
l (x 2) min[ l (x 2), l (x 1) + r(x 1, x 2)] = min[¥, 0* + 10] = 10.
, x 1: l (x 7) = 3, l (x 8) = 6, l (x 9) = 12.
3. , : Min[ x 2, x 7, x 8, x 9, x 3, x 4, x 5, x 6] = min[10, 3, 6, 12, ¥, ¥, ¥, ¥] = 3 ─ x 7.
4. x 7 l(x 7) = 3*, p = x 7.
5. , . 2.
2. (p) = (x 7) = { x 2, x 4, x 6, x 9} ─ . (1.9) :
l (x 2) = min[ l (x 2), l (x 7) + r(x 7, x 2)] = min[10, 3* + 2] = 5.
l (x 4) = 7, l (x 6) = 17, l (x 9) = 12.
3. Min[ x 2, x 4, x 6, x 8, x 9, x 3, x 5] = min[5, 7, 17, 6, 12, ¥, ¥] = 5 ─ x 2.
4. x 2 l (x 2) = 5*, p = x 2.
5. . 2, .. .
2. (p) = (x 2) = { x 1, x 3, x 7, x 9}, x 3 x 9 . (15.9) l (x 3) = min[¥, 5* + 18] = 23, l (x 9) = min[12, 5* + 13] = 12.
3. Min[ x 3, x 4, x 8, x 6, x 9, x 5] = min[23, 7, 6, 17, 12, ¥] = 6 ─ x 8.
4. x 8 l (x 8) = 6*, p = x 8.
5. . 2, . .
2. (p) = (x 8) = { x 9, x 6, x 5}, x 5 x 9 . (15.9) :
l (x 5) = min[¥, 6* + 23] = 29, l (x 9) = min[12, 6* + r(x 8, x 9)] = 11.
3. Min[ x 5, x 9, x 3, x 4, x 5] = min[29, 11, 23, 7, 29] = 7 ─ x 4.
4. x 4 l (x 4) = 7*, p = x 4.
5. . 2.
2. (p) = (x 4) = { x 3, x 6, x 5}. l (x 3) = 23, l (x 6) = 17, l (x 5) = 12.
3. Min[ x 5, x 6, x 9, x 3] = min[12, 17, 11, 23] = 11.
4. x 9 l (x 9) = 11*, p = x 9.
5. . 2.
2. (p) = (x 9) = { x 3, x 6} ( ).
(x6) = min[ l (x 6), l (x 9) + r(x 6, x 9)] = [17, 11* + 9] = 17, l (x 3) = 18.
3. Min[ x 5, x 6, x 3] = min[12, 17, 18] = 12.
4. x 5 l (x 5) = 12*, p = x 5.
5. . 2.
2. (p) = (x 5) = { x 6}.
l (x 6) = min[ l (x 6), l (x 5) + r(x 5, x 6)] = [17, 12* + 10] = 17.
3. Min[ x 6, x 3] = min[17, 18] = 17.
4. x 6 l (x 6) = 17*, p = x 6.
5. . 2.
.
2. (p) = (x 3) = Æ.
l (x 3) = min[18] = 18, p = x 3. .
, x 1 :
x 1 x 2(5*), x 1 x 3(18*), x 1 x 4(7*), x 1 x 5(12*), x 1 x 6(17*), x 1 x 7(3*), x 1 x 8(6*), x 1 x 9(11*);
x 1 x 7 x 2 x 3(18*), x 1 x 7 x 2(5*), x 1 x 8(6*), x 1 x 8 x 9(11*), x 1 x 7 x 6(17*), x 1 x 7 x 4 x 5(12*), x 1 x 7 x 4(7*).
, .. , .
: x 1 x 3(18*), 18 x 9 { x 9 ─ x 3}:
x 3 x 9(11*), 11 x 8 { x 8 ─ x 9}.
x 3 x 9 x 8(6*), 6 x 1 { x 1 ─ x 8}.
x 3 x 9 x 8 x 1 18.
G . 15.33.
. 15.33. G
1. ?
2. ?
3. ?
4. ?
5. ?
6. ?
7. ?
8. ?
9. ?
10. ?