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HW 2

4.1

    • a) Starting from L:
L a b c d e f g h i j k l m W
0 (7, L) (14, L) (11, L) \infty \infty \infty \infty \infty \infty \infty \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) \infty \infty \infty \infty \infty \infty \infty \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (27, c) \infty \infty \infty \infty \infty \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) \infty (25, h) \infty \infty \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (41, e) (25, b) \infty \infty \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (41, e) (25, b) \infty \infty \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (41, e) (25, b) \infty (37, h) \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (34, f) (25, b) (43, f) (37, h) \infty \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (34, f) (25, b) (43, f) (37, h) (46, g) \infty \infty \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (34, f) (25, b) (43, f) (37, h) (46, g) \infty (52, j) \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (34, f) (25, b) (43, f) (37, h) (46, g) (52, i) (52, j) \infty
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (34, f) (25, b) (43, f) (37, h) (46, g) (54, i) (52, j) (57, k)
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (34, f) (25, b) (43, f) (37, h) (46, g) (54, i) (52, j) (57, k)
0 (7, L) (14, L) (11, L) (16, c) (23, d) (27, d) (34, f) (25, b) (43, f) (37, h) (46, g) (54, i) (52, j) (57, k)

L->c->d->f->g->k->W

  • b)
    - Starting from f:
    do the same thing as above
    the shortest path from f to L is f->d->c->L with 27 roughness and to W is f->g->k->W with 30 roughness.
    - Starting from i:
    do the same thing as above
    the shortest path from i to L is i->f->d->c->L with 43 roughness and to W is i->k->W with 20 roughness.
    - combine these together, we can get that 27+20 < 43 + 30, thus the shortest path is L->c->d->f->i->k->w with total roughness 27+16+20=63

    For a complete graph with three nodes, say a, b and c. The distance are 1, 2, and -999 seperatly for ab, ac, and bc. start from node a, we find the distance of ab and ac are 1 and 2, so we select b and add it to the visited node set. However, the actual shortest path from a to b is a -> c -> b with total cost -997.

4.2

    4.2_2_Both
    total cost of 59 4.2_2_C
    total cost of 61 4.2_2_D
    total cost of 57
    4.2_5
    by times -1 with all of the costs and find the MST.

4.3

    • a)
      4.3_2_a
    • b)
      4.3_2_b
    4.3_3
    4.3_6
    • a)
      4.3_8_a
      impossible
    • b)
      4.3_8_b
      possible
    4.3_9
    5 different ways.
    4.3_12
    because it start from a, then a's child is marked with a$^+$ and the child of child marked child$^+$ which is exacly as the same as a tree and a is the root.

4.4

    4.4_2_a&b
    4.4_4

impossible, at least 1 college want 7 Ph.D.s but there is only 6 universities.

4.4_8

4.5

    • a)
      North-West:
    1 2 3
    1 30 30
    2 10 20 30
    3 20 10 30
    40 40 10
    v1:5 v2:1 v3:-7
    u1:0 5 2(1) 0(-7) 30
    u2:-4 9 5 0(-3) 30
    u3:-7 4(12)+ 8 0 30
    40 40 10
    1 2 3
    1 10 20 30
    2 10 20 30
    3 20 10 30
    40 40 10
    v1:5 v2:2 v3:1
    u1:0 5 2 0(1) 30
    u2:-4 9 5(6)+ 0(5) 30
    u3:1 4 8(1) 0 30
    40 40 10
    1 2 3
    1 20 10 30
    2 30 30
    3 20 10 30
    40 40 10
    v1:5 v2:2 v3:1
    u1:0 5 2 0(1)+ 30
    u2:-3 9(8) 5 0(4) 30
    u3:1 4 8(1) 0 30
    40 40 10
    1 2 3
    1 10 10 10 30
    2 30 30
    3 30 30
    40 40 10
    v1:5 v2:2 v3:0
    u1:0 5 2 0 30
    u2:-3 9(8) 5 0(3)+ 30
    u3:1 4 8(1) 0(-1) 30
    40 40 10
    1 2 3
    1 10 20 30
    2 20 10 30
    3 30 30
    40 40 10
    v1:5 v2:2 v3:-3
    u1:0 5 2 0(-3) 30
    u2:-3 9(8) 5 0 30
    u3:1 4 8(1) 0(-4) 30
    40 40 10

    optimized

    • a)
      North-West:
    1 2 3 4
    1 40 40
    2 10 10 10 30
    3 30 50
    50 10 40 20

    optimize:

    1 2 3 4
    1 10 30 40
    2 20 10 30
    3 30 20 50
    50 10 40 20
    v1:4 v2:2 v3:5 v4:1
    u1:0 7(4) 2 5 0(1) 40
    u2:1 3 5(1) 4 0 30
    u3:1 4(3) 6(1) 3(4)+ 0 50
    50 10 40 20
    1 2 3 4
    1 10 30 40
    2 30 30
    3 20 10 20 50
    50 10 40 20
    v1:6 v2:2 v3:5 v4:2
    u1:0 7(4) 2 5 0(2)+ 40
    u2:3 3 5(-1) 4(2) 0(-1) 30
    u3:2 4 6(0) 3 0 50
    50 10 40 20
    1 2 3 4
    1 10 10 20 40
    2 30 30
    3 20 30 50
    50 10 40 20
    v1:6 v2:2 v3:5 v4:0
    u1:0 7(6) 2 5 0 40
    u2:3 3 5(-1) 4(2) 0(-3) 30
    u3:2 4 6(0) 3 0(-2) 50
    50 10 40 20

    optimized