Algorithms CMSC37000 Final Exam. March 14, 2006
Instructor: L´
aszl´
o Babai
Show all your work
.
Do not use text, notes, or scrap paper.
When
describing an algorithm in pseudocode,
explain the meaning of your
variables
(in English). This exam contributes 30% to your course grade.
Take this problem sheet home.
1.
(4 points)
Given a digraph
G
by an array of adjacency lists, find an
adjacency list representation of the reverse digraph in linear time. De
scribe your algorithm is pseudocode.
2.
(12 points)
(Segmentation)
Consider the list of integers (1
,
2
, . . . , n
).
A
segment
is a contiguous sublist, (
i, i
+ 1
, . . . , j
) where
i
≤
j
.
A
segmentation
means splitting up the list into segments. For instance,
for
n
= 8, a possible segmentation is (1
,
2
,
3)
,
(4)
,
(5
,
6
,
7
,
8).
We are
given a triangular array
p
(
i, j
) (1
≤
i
≤
j
≤
n
) of
segment penalties
(real numbers); the “penalty” of segment (
i, . . . , j
) is
p
(
i, j
). The
cost
of the segmentation is the sum of the penalties of the segments. In the
example, the cost is
p
(1
,
3) +
p
(4
,
4) +
p
(5
,
8). Find, in
O
(
n
3
) time, the
optimal (mincost) segmentation. Give your algorithm in pseudocode.
Make sure you define your variables
. Name the method used.
3.
(8+4+5)
(a) Describe in pseudocode the algorithm that finds the median of
n
numbers using a linear number of comparisons.
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 Algorithms, Computational complexity theory, complexity class

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