1. [10] Suppose we have 14 integers: 10, 100, 30, 130, 80, 50, 140, 20, 60, 70, 120, 40, 90, 110. Please create a 2-3 tree by inserting one integer at a time in the order shown. Please draw the resulting 2-3 tree after each insertion for the last five insertions.

Ans:

10 10, 100

30

10 100

30

10 100, 130

30, 100

10 80 130

30, 100

10 50, 80 130

30, 100

10 50, 80 130, 140

30, 100

10, 20 50, 80 130, 140

30, 100

10, 20 50, 80 130, 140

60

30 100

10, 20 130, 14050 80

Insert 70

60

30 100

10, 20 130, 14050 70, 80

Insert 120

60

30

10, 20 50 70, 80 120 140

100, 130

Insert 40

60

30

10, 20 70, 80 120 140

100, 130

40, 50

Insert 90 30

60, 100

80 130

70 9010, 20 40, 50 120 140

Insert 110 30

60, 100

80 130

70 9010, 20 40, 50 110, 120 140

2.[10] Suppose we have 14 integers: 10, 190, 100, 30, 130, 80, 180, 50, 140, 20, 200, 60, 160, 120. Please create a 2-3-4 tree by inserting one integer at a

time in the order shown. Please draw the resulting 2-3-4 tree after each insertion for the last five insertions.

10 10 190 10 100 190

100

10 30 190

100

10 30 130 190

100

10 30 80 130 190

100

10 30 80 130 180 190

30 100

10 130 180 19050 80

30 100 180

10 20 19050 80 130 140

30 100 180

10 20 190 20050 80 130 140

30 100 180

10 20 190 20050 60 80 130 140

30 100 180

10 20 190 20050 60 80 130 140 160

100

30 140 180

10 20 50 60 80 120 130 190 200160

40

20 80

10 30 60 90

50 70 100

3-1(插入100)

3.[10] Suppose we have 14 integers: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140. Please create an AVL tree by inserting one integer at a time in the order shown. Please draw the resulting AVL tree after each insertion for the last five insertions.

3-2 (插入 110)

40

20 80

10 30 60 100

50 70 11090

3-3 (插入 120)

80

40 100

20 60 90 110

10 50 12030 70

3-4 (插入 130)

80

40 100

20 60 90 120

10 50 13030 70 110

3-5 (插入 140)

80

40 120

20 60 100 130

10 50 11030 70 90 140

4. [10] Suppose we have 14 integers: 10, 20, 30, 40, 50, 80, 100, 120, 130, 160, 180, 29, 48, 68. Please create a hash table of size 17, with linear probing and the hash function h(x) = x%17, by inserting one integer at a time in the order shown. Please draw the resulting hash table after each insertion for the last two insertions.• 4-1(紅色字代表插入的數字 )

• 4-2(紅色字代表插入的數字 )

Index Value0 291 1202 483 20456 407 1608910 1011 13012 8013 3014 18015 10016 50

Index Value0 291 1202 483 204 6856 407 1608910 1011 13012 8013 3014 18015 10016 50

5.[10] Suppose we have 14 integers: 10, 20, 30, 40, 50, 80, 100, 120, 130, 160, 180, 29, 48, 68. Please create a hash table of size 17, with double hashing and the hash functions h1(x) = x%17 and h2(x) = 7 – (x%7), by inserting one integer at a time in the order shown. Please draw the resulting hash table after each insertion for the last two insertions.

• 5-1(紅色字代表插入的數字 )• 5-2(紅色字代表插入的數字 )

Index Value0 481 1202 293 20456 407 1608910 1011 13012 8013 3014 18015 10016 50

Index Value0 481 1202 293 204 6856 407 1608910 1011 13012 8013 3014 18015 10016 50

6. [10] Suppose we have a 2-3 tree shown in Figure 1. Please answer the following questions:

a) [2.5] Please insert 45 into Figure 1 and show the resulting 2-3 tree.Ans:

30, 35

37, 50

39 70, 90

38 4010, 20 33, 34 10060 8036

30, 35

37, 50

39 70, 90

38 40, 4510, 20 33, 34 10060 8036

b) [2.5] Please insert 15 into Figure 1 and show the resulting 2-3 tree.

Ans:

c) [2.5] Please insert 32 into Figure 1 and show the resulting 2-3 tree.Ans:

39 70, 90

38 4033, 34 10060 8036

37

30 50

15 35

10 20

39 70, 90

38 40 10060 8036

37

33 50

30 35

10, 20 32 34

d) [2.5] Please insert 25 into Figure 1 and show the resulting 2-3 tree. Ans:

39 70, 90

38 4033, 34 10060 8036

37

30 50

20 35

10 25

7. [10] Suppose we have a 2-3 tree shown in Figure 1. Please answer the following questions:

a) [5] Please delete 34 from Figure 1 and show the resulting 2-3 tree.Ans:

30, 35

37, 50

39 70, 90

38 4010, 20 33, 34 10060 8036

30, 35

37, 50

39 70, 90

38 4010, 20 33 10060 8036

b) [5] Please delete 50 from Figure 1 and show the resulting 2-3 tree. Note that a deleted node is replaced by the least number in its right subtree.

Ans:

30, 35

37, 60

39 90

38 4010, 20 33, 34 10070, 8036

8.[10] Suppose we have a 2-3-4 tree shown in Figure 2. Please answer the following questions:

[2.5] Please insert 45 into Figure 2 and show the resulting 2-3-4 tree.[2.5] Please insert 18 into Figure 2 and show the resulting 2-3-4 tree.[2.5] Please insert 31 into Figure 2 and show the resulting 2-3-4 tree.[2.5] Please insert 16 into Figure 2 and show the resulting 2-3-4 tree.

37 50

70 90

39

3860 80 100

40 45

a

30 35

10 15 20

32 33 34

36

37 50

70 90

39

3860 80 100

40

b

15 30 35

18 2036

10

32 33 34

37 50

70 90

39

3860 80 100

40

c

30 33 35

31 32 36

10 15 20

34

37 50

70 90

39

3860 80 100

40

d

15 30 35

16 2036

10

32 33 34

9.[10] Suppose we have a 2-3-4 tree shown in Figure 2. Please answer the following questions:[5] Please delete 33 from Figure 2 and show the resulting 2-3-4 tree.

[5] Please delete 35 from Figure 2 and show the resulting 2-3-4 tree. Note that a deleted node is replaced by the least number in its right subtree.

37 50

70 90

39

3860 80 100

40

a

30 35

10 15 20

32 34

36

37 50

70 90

39

3860 80 100

40

b

30 34

10 15 20

32 33

36

10.[10] Suppose we have an undirected graph shown in Figure 3. Please answer the following questions:[2.5] Please show the adjacency matrix for this graph (ignore the weights).[2.5] Please show the adjacency list for this graph (ignore the weights). Note that the nodes in a linked list should be alphabetically ordered. [2.5] Please find a minimum spanning tree for this graph starting from node a using the algorithm introduced in the class. Note that a node that is alphabetically smaller should be selected earlier. Hint: The addition of an edge should not cause a cycle.[2.5] Please using the breadth-first search algorithm to visit the graph starting from node a. Note that a node that is alphabetically smaller should be visited earlier. Please indicate the order of each node in the traversal.

a b c d e f g h i

a 0 1 1 1 0 0 0 0 0

b 1 0 1 0 0 0 0 0 0

c 1 1 0 1 1 0 0 0 0

d 1 0 1 0 0 1 0 1 0

e 0 0 1 0 0 1 1 0 1

f 0 0 0 1 1 0 1 0 0

g 0 0 0 0 1 1 0 0 0

h 0 0 0 1 0 0 0 0 0

i 0 0 0 0 1 0 0 0 0

a.

• b.

• C.

• D. a b c d e f h g i

a

cb d

h

e f

i g

11.[10] Suppose we have a directed graph shown in Figure 4. Please answer the following questions:[2] Please show the adjacency matrix for this graph (ignore the weights).[2] Please show the adjacency list for this graph (ignore the weights). Note that the nodes labeled with smaller numbers should appear earlier in a linked list. [6] Find the shortest distances from node 0 to all the other nodes using the algorithm introduced in the class. Note that the nodes labeled with smaller numbers should be selected earlier.

A.0 1 2 3 4 5 6

0 0 1 1 1 0 0 0

1 0 0 1 0 1 0 0

2 0 0 0 1 1 0 0

3 0 0 1 0 1 1 0

4 0 0 1 0 0 1 1

5 0 0 0 0 1 0 0

6 0 0 0 0 0 1 0

• B.

• C.0 1 2 3 4 5 6

Step 1 0 0 2 4 6

Step 2 0 1 0 2 4 6 5

Step 3 0 1 2 0 2 4 6 5

Step 4 0 1 2 4 0 2 4 6 5 10 6

Step 5 0 1 2 4 3 0 2 4 6 5 9 6

Step 6 0 1 2 4 3 6 0 2 4 6 5 8 6

Step 7 0 1 2 4 3 6 5 0 2 4 6 5 8 6

Step 5 0 1 2 4 6 0 2 4 6 5 8 6

Step 6 0 1 2 4 6 3 0 2 4 6 5 8 6

Step 7 0 1 2 4 3 6 5 0 2 4 6 5 8 6