Final Exam (December 15, 2000)
Fall 2000
1.
True, False, or Maybe
Indicate whether each of the following statments is always true, sometimes true, always
false, or unknown. Some of these questions are deliberately tricky, so read them carefully.
Each correct choice is worth
+1
, and each incorrect choice is worth
−
1
.
Guessing will hurt
you!
(a) Suppose SMARTALGORITHM runs in
Θ(
n
2
)
time and DUMBALGORITHM runs in
Θ(2
n
)
time
for all inputs of size
n
. (Thus, for each algorithm, the bestcase and worstcase running
times are the same.) SMARTALGORITHM is faster than DUMBALGORITHM.
True
False
Sometimes
Nobody Knows
(b) QUICKSORT runs in
O
(
n
6
)
time.
True
False
Sometimes
Nobody Knows
(c)
⌊
log
2
n
⌋ ≥ ⌈
log
2
n
⌉
True
False
Sometimes
Nobody Knows
(d) The recurrence
F
(
n
) =
n
+ 2
√
n
·
F
(
√
n
)
has the solution
F
(
n
) = Θ(
n
log
n
)
.
True
False
Sometimes
Nobody Knows
(e) A Fibonacci heap with
n
nodes has depth
Ω(log
n
)
.
True
False
Sometimes
Nobody Knows
(f) Suppose a graph
G
is represented by an adjacency matrix. It is possible to determine
whether
G
is an independent set without looking at every entry of the adjacency matrix.
True
False
Sometimes
Nobody Knows
(g) NP
n
=
coNP
True
False
Sometimes
Nobody Knows
(h) Finding the smallest clique in a graph is NPhard.
True
False
Sometimes
Nobody Knows
(i) A polynomialtime reduction from X to 3SAT proves that X is NPhard.
True
False
Sometimes
Nobody Knows
(j) The correct answer for exactly three of these questions is “False”.
True
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 Spring '09
 A
 Computational complexity theory, Convex hull, Conjunctive normal form, disjunctive normal form, NPcomplete, Algebraic normal form

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