COP 3503 Summer 2005
Exam #2
7/6/05
TA: ____________________
Name: __________________
I. Sorting
1) (5 pts) If the array below were sorted using Merge-Sort, what would the array look
like after the tenth call to the Merge method, in the middle of the sort? (Hint: A total of
15 calls are made to the Merge method during the entire run of Merge-Sort on this array.)
index 0
value 19
1
4
2
8
3
5
4
15
5
3
6
1
7
7
8
14
9 10
11 9
11
13
12
2
13
17
14
6
15
12
1
2
3
4
5
6
7
8
9
11
12
13
14
15
Answer:
index 0
value
10
2) (8 pts) To show students intuitively that Quick Sort's average case running time on an
array of n unsorted elements is O(nlgn) time, rather than do the whole proof we did in
class, some teachers will show a simplified proof where it is assumed that the array is
always split by partition into two sides, one with length n/4 and the other with length
3n/4, where n is the original size of the array being partitioned. Letting T(n) be the
running time of Quick Sort under this assumption, we can write down the following
recurrence relation for Quick Sort:
T(n) = T(n/4) + T(3n/4) + n.
(Note: The last term should be O(n), and technically, we should have (n-1)/4 and 3(n-1)/4
as the two inputs to the function T, since the partition element is not included in any of
the recursive calls. But, I have made these simplifications to make this question easier.)
Use the iteration technique for one step to get an expression for T(n) of the following
form:
T(n) = aT(n/16) + bT(3n/16) + cT(9n/16) + dn, where a, b, c and d are positive integers.
Find the values of a, b, c and d.
3) (8 pts) What is the minimum number of comparisons necessary to sort 12 values?
(Hint: 12! is approximately 4.79x108, log210 = 3.32 to three significant digits, and
log24.79 = 2.26 to three significant digits. Note: You will not get full credit on this
question if you manually multiply powers of 2. You will only get full credit if you utilize
all the values in the hint and properly apply log rules.)
4) (6 pts) Show each iteration of Radix sort when sorting the following values:
Original List
1823
6254
6853
2424
3824
1665
2399
6364
2491
6129
1 sort
10 sort
100 sort
sorted
1665
1823
2399
2424
2491
3824
6129
6254
6364
6853
5) (10 pts) Use iteration (not the Master Theorem) to solve the following recurrence for a
closed form big-O bound:
T(n) = 3T(n/3) + n
6) (3 pts) Verify your result from question 5 by utilizing the Master Theorem to solve the
recurrence.
II. Heaps
7) (10 pts) Show the end result of running Make Heap on the following set of random
values that do not yet form a heap. Please use the array-based representation of a heap
shown in class where the minimal element of the heap is stored in index 1.
index 1
value 25
2
13
3
18
4
6
5
19
6
22
7
11
8
17
9
1
10
3
11
7
12
9
4
5
6
7
8
9
10
11
12
Place your answer here:
index 1
value
2
3
Use the space below to trace through the algorithm:
8) (10 pts) Show the end result of maintaining a heap, if we start with an Empty heap H
and perform the following operations on it:
H.insert(10)
H.insert(5)
H.insert(7)
H.insert(12)
H.insert(2)
H.insert(4)
H.insert(6)
H.insert(8)
H.insert(9)
H.deleteMin()
H.deleteMin()
Please draw your heap H as a binary tree instead of showing the end result in the array
representation.
III. 2-4 Trees
9) (5 pts) Given a 2-4 Tree of height 3, what is the minimum and maximum possible
number of nodes the tree could have?
10) (5 pts) Show the result of inserting the value 18 into the 2-4 Tree below:
10,
/
1, 2
20,
|
11, 12, 13
30
|
22, 25
\
33
11) (5 pts) Show the result of deleting the value 15 from the 2-4 Tree below:
20
/
10
/
5
/ \
2 8
\
30
\
/
\
17
24
40
/ \ / \
/ \
15 18 22 27 31 43
IV. AVL Trees
12) (5 pts) Show the result of inserting the value 13 into the AVL Tree below:
20
/
10
/
5
\
30
\
40
\
15
/
12
\
17
13) (5 pts) Show the result of deleting the value 5 from the AVL Tree below:
20
/
10
/
\
5
15
/ \
12 17
\
30
/
25
\
27
\
40
/ \
35 50
\
38
V. Huffman Coding
14) (10 pts) Given the following frequency of characters in a file, determine a valid
Huffman coding for that file:
Character
A
B
C
D
E
F
G
H
Frequency
14
5
71
7
42
24
6
10
Huffman Code
Please show your final Huffman tree to earn full-credit for the question.
15) (5 pts) For the encoding you gave in question 14, determine the number of bits saved
when coding the file specified in that question over a fixed-length encoding where 3 bits
were originally used to code for each character.
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