Algorithms CMSC-37000 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 (min-cost) 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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