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AMS303/HW2/HW2.md
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2022-05-23 05:57:24 -04:00

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# HW 2
## 4.1
- 4)
- 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 $27+16+20=63$
- 9)
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
- 2)
![4.2_2_Both](./4.2_2_both.png "4.2_2_Both")
total cost of 59
![4.2_2_C](./4.2_2_c.png "4.2_2_C")
total cost of 61
![4.2_2_D](./4.2_2_d.png "4.2_2_D")
**total cost of 57**
- 5)
![4.2_5](./4.2_5.png "4.2_5")
by times -1 with all of the costs and find the MST.
## 4.3
- 2)
- a)
![4.3_2_a](./4.3_2_a.png "4.3_2_a")
- b)
![4.3_2_b](./4.3_2_b.png "4.3_2_b")
- 3)
![4.3_3](./4.3_3.png "4.3_3")
- 6)
![4.3_6](./4.3_6.png "4.3_6")
- 8)
- a)
![4.3_8_a](./4.3_8_a.png "4.3_8_a")
impossible
- b)
![4.3_8_b](./4.3_8_b.png "4.3_8_b")
possible
- 9)
![4.3_9](./4.3_9.png "4.3_9")
5 different ways.
- 12)
![4.3_12](./4.3_12.png "4.3_12")
- 21)
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
- 2)
![4.4_2_a&b](./4.4_2_a&b.png "4.4_2_a&b")
- 4)
![4.4_4](./4.4_4.png "4.4_4")
- 5)
impossible, at least 1 college want 7 Ph.D.s but there is only 6 universities.
- 8)
![4.4_8](./4.4_8.png "4.4_8")
## 4.5
- 4)
- 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
- 6)
- 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
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