Introduction to Algorithms
November 17, 2011
Massachusetts Institute of Technology
6.006 Fall 2011
Professors Erik Demaine and Srini Devadas
Quiz 2 Solutions
Quiz 2 Solutions
Problem 1.
[1 points] Write your name on top of each page.
Problem 2.
True/False
[28 points] (14 parts)
Circle (T)rue or (F)alse. You don’t need to justify your choice.
(a)
T F
[2 points] Computing
√
b
a
c
for an
n
bit positive integer
a
can be done in
O
(lg
n
)
iterations of Newton’s method.
Solution:
True. This is the bound obtained by Newton’s Method’s quadratic
convergence.
(b)
T F
[2 points] Suppose we want to solve a polynomial equation
f
(
x
) = 0
. While
our choice of initial approximation
x
0
will affect how quickly Newton’s method
converges, it will always converge eventually.
Solution:
False. Take e.g.
f
(
x
) =
x
3

2
x
+2
and
x
0
= 0
. Then
x
2
i
+1
= 1
and
x
2
i
= 0
for all
i
(that is, the approximations alternate between 0 and 1 without
ever converging).
(c)
T F
[2 points] Karatsuba’s integer multiplication algorithm always runs faster than
the gradeschool integer multiplication algorithm.
Solution:
False. Problem Set 5 has shown that the
O
(
N
2
)
algorithm runs faster
for small numbers.
(d)
T F
[2 points]
If we convert an
n
digit base256 number into base 2, the resulting
number of digits is
Θ(
n
2
)
.
Solution:
False.
log
256
n
=
log
2
n
=
log
2
n
log
256
8
. By converting a base256 number
to base 2, the number of digits is
2
multiplied by 8. For all
b
1
, b
2
= 1
, converting
a base
b
1
number to base
b
2
results in a linear increase or decrease in the number
of digits.
(e)
T F
[2 points]
In a weighted undirected graph
G
= (
V, E, w
)
, breadthfirst search
from a vertex
s
finds singlesource shortest paths from
s
(via parent pointers) in
O
(
V
+
E
)
time.
Solution:
False. Only in unweighted graphs.
(f)
T F
[2 points]
In a weighted undirected
tree
G
= (
V, E, w
)
, breadthfirst search
from a vertex
s
finds singlesource shortest paths from
s
(via parent pointers) in
O
(
V
+
E
)
time.
6
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6.006 Quiz 2 Solutions
Name
2
Solution:
True.
In a tree, there is only one path between two vertices, and
breadthfirst search finds it.
(g)
T F
[2 points] In a weighted undirected
tree
G
= (
V, E, w
)
,
depth
first search from
a vertex
s
finds singlesource shortest paths from
s
(via parent pointers) in
O
(
V
+
E
)
time.
Solution:
True.
In a tree, there is only one path between two vertices, and
depthfirst search finds it.
(h)
T F
[2 points] If a graph represents tasks and their interdependencies (i.e., an edge
(
u, v
)
indicates that
u
must happen before
v
happens), then the breadthfirst
search order of vertices is a valid order in which to tackle the tasks.
Solution:
No, you’d prefer depthfirst search, which can easily be used to pro
duce a topological sort of the graph, which would correspond to a valid task
order. BFS can produce incorrect results.
(i)
T F
[2 points] Dijkstra’s shortestpath algorithm may relax an edge more than once
in a graph with a cycle.
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 Fall '11
 ErikDemaine
 Graph Theory, Shortest path problem, Dijkstra

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