quiz2 - Introduction to Algorithms Massachusetts Institute...

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 grade-school 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 base-256 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 base-256 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 ) , breadth-first search
from a vertex s finds single-source 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 ) , breadth-first search
from a vertex s finds single-source shortest paths from s (via parent pointers) in
O ( V + E ) time.
6

6.006 Quiz 2 Solutions Name 2
Solution: True. In a tree, there is only one path between two vertices, and
breadth-first 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 single-source 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
depth-first 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 breadth-first
search order of vertices is a valid order in which to tackle the tasks.
Solution: No, you’d prefer depth-first 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 shortest-path algorithm may relax an edge more than once
in a graph with a cycle.