FUNDAMENTAL ALGORITHMS MIDTERM
SOLUTIONS
You
must
omit at least 10 points.
Mark on your booklet which part or
parts of problems you are omitting. Maximum score: 140.
What is slight can still be great, if it is written in a natural,
flowing and easy style  and at the same time bears the mark of
sound composition.
– letter to Wolfgang Mozart from his father, 1778
1. (35) These questions concern a maxheap
A
with auxilliary structures
length[A]
,
Left[i]
,
Right[i]
and
Parent[i]
.
(a) (5) Give the maxheap property. (That is, what is the relationship
between the values
A
[
i
] that one needs for a heap.)
Solution:
A[Parent[i]]
≥
A[i]
(b) (5) Suppose
length[A]
=100 and the values
A
[
i
] are distinct.
What are the possible
i
for which
A
[
i
] is the second largest value?
(You must give all possible
i
.)
Solution:
The biggest is the root and the second biggest is one
of its children, so
i
= 2
,
3 are the possibilities.
(c) (5) Suppose
length[A]
=100 and the values
A
[
i
] are distinct.
What are the possible
i
for which
A
[
i
] is the smallest value? (You
must give all possible
i
.)
Solution:
It must be a leaf so 51
≤
i
≤
100.
(d) (10) Describe the algorithm
MAXHEAPINSERT(A,key)
which adds
to heap
A
a new entry with value
key
. (Either a description in
words or a pseudocode program would be fine.)
Solution:
First increment heapsize and place key in the last po
sition. Set
x
= heapsize. Now while
x
negationslash
= 1 and
A
[
x
] is less than
A
[
parent
[
x
]], interchange those two and reset
x
→
parent
[
x
].
(e) (10) How long (in worst case) does the above algorithm take as
a function of
n
=
length[A]
? Give a
reason
for your answer.
Solution:
You climb up the tree to the top so it takes Θ(lg
n
)
steps.
2. (35) These questions concern
QUICKSORT
. We will take
PARTITION(A,p,r)
as
given
. (This procedure rearranges
A
[
p
···
r
], placing all entries less
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than
A
[
r
] to the left and all entries greater than
A
[
r
] to the right. You
are
not
being asked to write it!!)
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 Spring '09
 Naver
 Algorithms, LG, questions concern

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