320 CPSC Sample Final Examination
April 2009
Name:
Signature:
Student ID:
You have 2.5 hours to write the 8 questions on this examination. A total of 110 marks are available.
Justify all of your answers, except if the question
says not to.
Question
No notes or electronic equipment are allowed, except for
one 8.5 11 sheet of paper, handwritten.
2
Keep your answers short. If you run out of space for a
question, you have written too much.
4
The number in square brackets to the left of the question
number indicates the number of marks allocated for that
question. Use these to help you determine how much time
you should spend on each question.
6
Use the back of the pages for your rough work.
Marks
1
3
5
7
8
Total
UNIVERSITY REGULATIONS:
Each candidate should be prepared to produce, upon request, his/her library card.
No candidate shall be permitted to enter the examination room after the expiration of
one half hour, or to leave during the rst half hour of the examination.
CAUTION: candidates guilty of any of the following, or similar, dishonest practices shall
be immediately dismissed from the examination and shall be liable to disciplinary action.
1. Having at the place of writing, or making use of, any books, papers or memoranda,
electronic equipment, or other memory aid or communication devices, other than
those authorised by the examiners.
2. Speaking or communicating with other candidates.
3. Purposely exposing written papers to the view of other candidates. The plea of
accident or forgetfulness shall not be received.
Candidates must not destroy or mutilate any examination material; must hand in all
examination papers; and must not take any examination material from the examination
room without permission of the invigilator.
page 2 out of 11
[10] 1. Answer each of the following questions with true or false. Give a short justication
for each of your answers.
[5] a. 6n O(5n )
[5] b. 1.5n + n2 O(1.5n + n log n)
[18] 2. Short Answers
[3] a. What factor(s) does the average-case running time of SkipListSearch on a
given skip list (averaged over all keys one could search for) depend on?
[3] b. True or false: for small values of n (for instance n 800 or so), algorithm
DeterministicSelect is slower than the naive O(n log n) algorithm (the one that
sorts the array using quicksort or mergesort, and then returns the requested
element in constant time).
page 3 out of 11
[3] c. What is the main advantage of RandomizedQuickSelect over Select1?
[3] d. Name one reason why the choice of a random level for each node in a Skip
List helps us achieve a good average case running time.
[3] e. What is the main dierence, in terms of the running time of their operations,
between a binomial heap and a Fibonacci heap?
[3] f. When do we use amortized analysis?
page 4 out of 11
[9] 3. In class (or in previous courses you took), we learned about the following algorithms,
all of which take an array A as parameter.
(a) DeterministicSelect
(d) RandomizedQuicksort
(b) Heapsort
(e) RandomizedQuickSelect
(c) Insertion sort
(f) Quicksort
For each of the statements below, list all of the algorithms listed above for which
the statement is true (you do not need to justify your answer(s)).
[3] a. If we change the ordering of the elements in the array A, then the algorithms
running time may change by more than a constant factor (even if everything
else remains unchanged).
[3] b. Its average-case running time Ta (n) is much better than its worst-case running
time Tw (n), that is, Ta o(Tw ).
[3] c. The amount of storage used when the algorithm is executed, not counting the
input array A, is in o(n).
page 5 out of 11
[9] 4. For each of the following recurrence relations, determine whether or not the Master
Theorem discussed in class can be used. If it can be used, apply it to derive the
solution of the recurrence relation using O notation. If the Master Theorem can not
be used, explain why briey.
2T ( n) + n if n 2
[3] a. T (n) =
1
if n 1
[3] b. T (n) =
9T (n/3) + 2n2
1
[3] c. T (n) =
4T (n/2) + n2+odd(n)
1
if n 3
if n 2
if n 2 where odd(n) =
if n 1
1 if n is odd
0 if n is even
page 6 out of 11
[12] 5. Consider the following function:
Algorithm HappyNewYear(A, p, r)
//
// A is an array, p and r are positions in the array.
//
mid (p + r)/2
x Christmas(A[p]) + Christmas(A[mid]) + Christmas(A[r])
if (n 2) then
x x + HappyNewYear(A, p+1, r-1)
x x + HappyNewYear(A, p+1, r-1)
x x + HappyNewYear(A, p+1, r-1)
x x + HappyNewYear(A, p+1, r-1)
return x
[4] a. Let C (n) be the number of times that function Christmas will be called during
the execution of call the HappyNewYear(A,p,r), where n = r p + 1. Write
a recurrence relation for C (n).
[8] b. Prove using the guess and test method that the solution of the recurrence
relation you gave in part (a) is in O(2n ).
page 7 out of 11
[20] 6. Skip Lists
[4] a. When a new key is inserted in a skip list, we choose the level of its node
randomly, where for each i the probability that the node has level i is pi1 .
Explain briey how this is done. Recall that p is a constant between 0 and 1
that is chosen beforehand (it is easier to think of p = 1/2).
[3] b. Consider now the following skip list:
51
16
62
3
40
69
71
80
Show the pointers followed by the Search algorithm discussed in class, when
the key to search for is 73. Assume that the current value of MaxLevel is 3.
page 8 out of 11
[5] c. Show the skip list obtained after the following sequence of operations has
been performed on the skip list from part (b): inserting key 44 with level = 3,
deleting key 80, deleting key 16, inserting key 72 with level = 2, and deleting
key 44. Hint: it is faster if you do them all at once.
[8] d. Suppose now that we have a skip list whose keys are intervals, sorted according
to their left endpoint, like the following skip list (its current MaxLevel is 3):
8
6
[2,6]
6
[3,4]
8
4
[4,7]
7
[5,8]
8
We have augmented that skip list by storing in each pointer P the maximum
right endpoint of the intervals stored between the node containing P and the
node immediately preceding the node the pointer points to. For instance, the
second pointer at level 2 contains the value 6 because 6 is the maximum right
endpoint amongst the two intervals [2, 6] and [3, 4]
Explain how you would update the values associated with the pointers when
a new key is inserted.
page 9 out of 11
[15] 7. The CEO of a software company wants to keep his developers happy by giving them
a bonus, but does not want to spend too much money. He thus wants to select as
few developers as possible (these will get a bonus, and be happy), chosen so that
every other developer likes at least one of those that get a bonus (and hence will be
happy for him/her).
The CEOs problem can be modeled using a graph G = (V, E ): each node in the
graph is a developer, and an edge (u, v ) means that developer u likes developer v .
The CEO is looking for a minimum subset W of the set V of vertices, where for
every vertex u W , there is an edge (u, w) where w W . This is called a minimum
/
dominating set for the graph G.
[9] a. Write a greedy algorithm that nds a subset W of V (it does need to be
the subset with the fewest elements possible) with that property. Your mark
will depend in part on the criterion you use to decide which vertex to add to
W . You do not need to give pseudo-code, but you should indicate what data
structure you are using to store the vertices that your algorithm has not dealt
with yet.
[3] b. What is the running time of the algorithm you described in part (a)?
[3] c. The minimum dominating set problem does not satisfy the greedy choice property. Knowing this, what can you conclude about the algorithm you gave in
part (a)?
page 10 out of 11
[17] 8. Bill and Simon have been arguing about where to have lunch after the CPSC 320 nal
examination, and decide to solve the decision by playing a sequence of Poker games.
The rst person to win n games gets to choose where they will have lunch. Simon is
a slighly better poker player, and has a 51% chance of winning any individual Poker
game.
Let P (i, j ) be the probability that Bill will be the rst person to win n games, given
that he only needs to win another i games (that is, Bill has won n i games so far),
and that Simon only needs to win another j games (that is, she has won n j games
so far). It can be proved that
P (i, j ) = 0.49P (i 1, j ) + 0.51P (i, j 1)
(1)
Bill is worried about his chances of winning, and asks you to use dynamic programming to compute the probability P (n, n) that he will win n games before Simon
does.
[3] a. State how big a table you need to answer Bills question using dynamic programming, and the meaning of each entry in the table.
[3] b. Give one possible order that you can use to compute the entries in the table
from part (a).
[3] c. List all of the base cases that will be needed when you are computing the
entries in the table from part (a). That is, list the cases where you can not use
equation (1) to compute P (i, j ), and the value of P (i, j ) for each of them.
page 11 out of 11
[8] d. Write pseudo-code for an algorithm that uses dynamic programming to determine the probability that Bill will win n Poker games before Simon does.