Introduction to Algorithms
March 12, 2008
Massachusetts Institute of Technology
6.006 Spring 2008
Professors Srini Devadas and Erik Demaine
Quiz 1 Solutions
Quiz 1 Solutions
Problem 1.
Asymptotic workout
[15 points]
For each function
f
(
n
)
along the left side of the table, and for each function
g
(
n
)
across the
top, write
O
,
Ω
, or
Θ
in the appropriate space, depending on whether
f
(
n
) =
O
(
g
(
n
))
,
f
(
n
) =
Ω(
g
(
n
))
, or
f
(
n
) = Θ(
g
(
n
))
. If more than one such relation holds between
f
(
n
)
and
g
(
n
)
, write
only the strongest one. The first row is a demo solution for
f
(
n
) =
n
2
.
g
(
n
)
n
n
lg
n
n
2
n
2
Ω
Ω
Θ
n
1
.
5
Ω
Ω
O
√
2
n
Ω
Ω
Ω
f
(
n
)
n
√
lg
n
Ω
O
O
n
log
30
n
Ω
Θ
O
n
3
Ω
Ω
Ω
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6.006 Quiz 1 Solutions
Name
2
Problem 2.
Table of Speed
[20 points]
For each of the representations of a set of elements along the left side of the table, write down the
asymptotic running time for each of the operations along the top. For hashing, give the expected
running time assuming simple uniform hashing; for all other data structures, give the worstcase
running time. Give tight asymptotic bounds using
Θ
notation. If we have not discussed how to
perform a particular operation on a particular structure, answer for the most reasonable implemen
tation you can imagine.
Operation
Insert
Extractmin
Contains
Minimum
Θ(1)
Θ(
n
)
Θ(
n
)
Θ(
n
)
Unsorted
Insert at
Check every
Check every
Check every
linked list
beginning
element
element
element
Θ(
n
)
Θ(1)
Θ(
n
)
Θ(1)
Sorted
Walk down
Remove first
Walk down
Return first
linked list
list
element
list
element
Data
Θ(log
n
)
Θ(log
n
)
Θ(
n
)
Θ(1)
Structure
Min heap
Seen in
Seen in
Check every
Return first
class
class
element
element
Θ(log
n
)
Θ(
n
)
Θ(
n
)
Θ(
n
)
Max heap
Seen in
Check every
Check every
Check every
class
element
element
element
Hashsing with
Θ(1)
Θ(
n
)
Θ(1)
Θ(
n
)
chainig and
Seen in
Check every
Seen in
Check every
α
= 1
class
element
class
element
Definitions of operations:
•
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 Fall '08
 ErikDemaine
 Algorithms, Array data structure

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