Student Name:
COT 4400: Design and Analysis of Algorithms
Final Exam
Duration 90 minutes
MAX POINTS = 70 points
 The exam is open notes, open books
 There are six problems, for a total of 70 points
 Show your work, as partial credit will be given
10 points
Problem 1.
Consider a matrix-chain product with the sequence of dimensions <3, 10, 4, 15, 8>.
a) (7 pts) Fill out the empty spaces in the m and s tables below. Show your work.
b) (3 pts) Show the printing of an optimal parenthesization.
m
2
3
1 0
120
300
1
2
0
600
2
1
3
4
0
s
4
1
480
3
0
4
2
1
3
4
2
2
3
1
COT 4400: Design and Analysis of Algorithms
10 points
Problem 2.
Determine an optimal Huffman code (using binary strings) for a file F containing a set of characters
with the following frequencies:
m n p q r s
Frequency 30 9 15 3 7 25
t
11
7 points
a) Draw a Huffman tree based on the above frequencies. Show only the final tree.
3 points
b) Show the Huffman code of each character, based on the Huffman tree above.
m
Frequency 30
Huffman codewords
n
9
p
15
q
3
r
7
s
25
t
11
2
COT 4400: Design and Analysis of Algorithms
15 points
Problem 3.
Consider the following graph G:
A
B
C
D
E
F
G
H
I
(4 points) Show DFS applied to G.
(3 points) Show a classification of the edges
(4 points) Show DFS applied to GT.
(2 points) Circle the strongly connected components (SCCs) you obtained.
(2 points) Show the corresponding component graph.
3
COT 4400: Design and Analysis of Algorithms
10 points
Problem 4.
Find an MST using Prim’s algorithm, starting from the vertex a, for the following graph:
2
5
c
a
f
10
8
4
7
b
2
5
8
d
e
2
s
10
g
4
(3 points) Darken the edges of the resulting MST.
(2 points) List the order in which the vertices were extracted from the queue Q.
(5 points) Show a table with values for v.key and v. for each v V, at each iterative step.
4
COT 4400: Design and Analysis of Algorithms
15 points
Problem 5.
We apply (trace) the Bellman-Ford algorithm on the following graph, where vertex s is the source:
1
c
f
3
s
3
2
-5
g
(8 points)
Indicate the order of the edges that you used for relaxation:
Show the v.d and v. values for each vertex v  V, after each iteration of relaxations:
1
c
f
1
c
3
3
s
3
s
2
3
2
-5
-5
g
1
c
f
g
f
3
s
3
2
-5
g
(2 points) Darken the shortest-paths tree.
(5 points) Does the algorithm return TRUE or FALSE for our graph? If it returns FALSE, for which
edges (u, v) does it satisfy the inequality “v.d > u.d + w(u,v)” ?
5
COT 4400: Design and Analysis of Algorithms
10 points
Problem 6.
We are given a graph G=(V,E) which is undirected. Design an O(V+E) algorithm to determine if
the graph G is connected. Use pseudo-code to design the algorithm. Your algorithm should return
TRUE (if the graph G connected) or FALSE (otherwise). If you use an algorithm discussed in class,
you can call directly that algorithm.
Note: the graph is connected if there is a path between any two vertices u and v.
6