CS 1501 Previous Final Exam
The following exam was given
in a previous term of CS 1501. Use it as
a guide for the types of questions that you may see on this term's final exam,
and for the content of some of the questions.
However, since the material is not covered in the same order from term
to term, there
may be some material on your final exam that is not covered in the exam below
(and vice versa). [For example, integer algorithms and encryption
are not on this exam – see the Quiz 2 Solution for
example problems on these topics]. To
ensure that you study all pertinent material, refer to the Online Syllabus,
the Online
Notes and your hand written notes.
Recall that the exam is cumulative for the ENTIRE TERM. However, significant emphasis will be placed
on the material since the midterm exam (below MIDTERM EXAM on the online
syllabus).
After giving the questions
below a good try, take a look at the solutions
to see how you did.
For all questions, be sure to show your work. Answers without work will not receive full
credit.
1)
(20
points – 2 points each) Fill in the Blanks. Complete the
statements with the MOST APPROPRIATE
word(s) and/or phrase(s).
a)
In order to save space in the hash
table storing dictionary entries for the LZW compression
algorithm, rather than storing the entire string in the hash table, we instead
store a(n) _________________________ and
a(n) _________________________ that together indicate the string with a constant
amount of space.
b)
During BFS and PFS, some
vertices are on the fringe, which means that __________________________
________________________________________________________.
c)
DFS on a graph with V vertices and E edges runs in time ___________________________ with an adjacency
list and in time __________________________ with an adjacency matrix.
d)
A graph G is said to be biconnected
if _____________________________________________________.
e)
The only difference
between Prim's MST algorithm using PFS and Djikstra's Shortest Path
algorithm using PFS is
______________________________________________________________.
f)
A sequence of N Inserts
followed by
g)
A sequence of N Inserts
followed by
h)
One rule for a valid
flow in a network graph is: For All (u,v) in V, f(u,v) = –f(v,u).
This means
____________________________________________________________________________.
i)
The Ford-Fulkerson
solution to the Network Flow problem involves finding a(n)
__________________ _________________________ for the network, updating the
residual network, and repeating until no
________________________________________ (same answer as first part) can
be found.
j)
We can prove a problem is
NP-complete either from scratch, or we can use reduction, by which
we ________________________________________________________________________________.
2)
(10
points – 2 points each) Indicate if each of the following
statements is TRUE or FALSE. For FALSE
statements, INDICATE WHY THEY ARE FALSE.
a)
An advantage of an adjacency
matrix representation of a graph is that all neighbors of a vertex
can be found in time Theta(V) (where V is the number of vertices in the
graph).
b)
NP-Complete problems are problems that are known to require exponential
run-times.
c) If I discover a true polynomial algorithm that solves the
Traveling Salesman Problem, I will have proved that P = NP.
d) The recursive solution to the Fibonacci Sequence is an
example of a algorithm that is both elegant and efficient in its run-time.
e) The dynamic programming solution to the Subset Sum
problem runs in polynomial time.
3) (16 points – 8 + 8) Consider the LZW compression algorithm that we discussed in lecture.
a)
Consider a file
with 3,000,000 letter
As in it (no other characters, except
for the end of file character). How large
a codeword size (in terms of bits) will be needed so that we DO NOT run out of
codewords in the process of compressing this file? Justify
your answer thoroughly for full credit. Hint: Trace it for the first few cycles to see the trend,
then use some math to determine the answer.
b)
Consider a file containing the following
codewords (stored in binary):
65 66 65
67 256 260
Trace the LZW
decoding process for the file (in the same way done in handout
lzw2.txt). Assume that the extended
ASCII set will use codewords 0-255. For
each step in the decoding, be sure to show all of the information indicated
below. Note: The ASCII value for 'A' is
65.
STEP CODEWORD INPUT STRING OUTPUT (CODE,
STRING) ADDED TO DICTIONARY
---- -------------- ------------- ----------------------------------
4) (8
points) An array representation of a min-heap
data structure is shown below. Draw
the resulting array after the operation Insert(20). Show your work.
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
|
15 |
25 |
20 |
30 |
60 |
55 |
35 |
45 |
50 |
80 |
|
5) (10
points – 8 + 2)
Consider
the graph below. Assume the vertices are
stored in alphabetical order, and that the edges
are stored in alphabetical order for each vertex.
a)
Complete
the table below, as it would look after a Breadth-First Search
Spanning Tree (starting from vertex A1) were created for the graph. val[] is the BFS number for the vertex, and dad[] is the parent vertex in the BFS tree. Show your work above or in the space below
the table for partial credit.
|
|
A1 |
B2 |
C3 |
D4 |
E5 |
F6 |
G7 |
H8 |
I9 |
J10 |
|
val |
|
|
|
|
|
|
|
|
|
|
|
dad |
|
|
|
|
|
|
|
|
|
|
b)
Identify
all of the articulation points in the graph.
6)
(10
points -- 8 + 2) Consider
the complete weighted graph below and an initial tour of the graph
ABCDEA of weight 25.
a)
Show ALL POSSIBLE edge swaps in the 2-OPT
neighborhood of this tour, listing the potential improvement (if any)
for each over the original tour.
b)
Indicate which of the possible tours above
is actually chosen, show the new resulting tour and list its total weight.
7) (8 points)
Consider the weighted graph below. The numbers are the edge capacities. S is the source vertex and
T is the sink vertex.
Using the Priority First Search implementation of the Ford-Fulkerson
algorithm, show EACH AUGMENTING PATH generated (in
the correct order that the paths are generated), the amount of flow for
each path, and the Maximum Flow for the graph.
For partial credit, be sure to SHOW YOUR WORK.
8)
(10
points) Consider an unweighted graph that may or may not be connected. Assume that
each VERTEX in the graph is storing a single, positive integer. Using a modification of the DFS algorithm for adjacency matrices, write C++ or Java code that will indicate and
write out the vertex with the minimum
data for each connected component in the
graph. For example, given the graph
below, the output of your code should be:
C.C. 1: Minimum
Element B, 15
C.C. 2: Minimum
Element K, 15
C.C. 3: Minimum Element
D, 5
Below is some (modified) code from the text with comments that
you should use as a starting point.
Data
that is already initialized for you to use:
M[][] – adjacency matrix for the graph
label[] – char array that gives the letter for each vertex (1 is
A, 2 is B, etc)
data[] – int array storing the integer data for each vertex
val[] – array to determine if a vertex has been visited or not
void
search() // Main search function – all
of your output should be done
{ // from here.
id = 0;
// Assume id is a global variable
for (int k = 1; k <= V; k++) // V is the number of vertices in the graph
val[k] = unseen; // Initialize all vertices to unseen
//
}
int
find_min(int k) // Recursive DFS, but
now with a return int that
{ // is the index of the minimum data
for that call.
val[k] = ++id; // val set to DFS number of vertex. This number
// is only important in
determining what has been visited
}
9)
(8
points – 4 + 4) Consider a weighted graph with V
vertices and E edges
a)
State
and thoroughly justify the minimum and maximum values for E
in terms of V.
b)
State,
in terms of V and E, the run-time of Prim's MST algorithm using PFS
when using an adjacency list and when using an adjacency matrix (2
answers required).