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Algorithms
CSE5811
MidTerm
Fall2006
Points 40
Time 70 min
1.
Describe what does the following algorithm produce, and analyze its asymptotic time
complexity. [For analyzing complexity, presume
n
= 2
k
, for some positive integer
k
.]
Algorithm
Unknown
(rational
a
, positive int
n
)
If
n
= = 1
return
a
Else
If
n
%2 = = 0 then
Return
Unknown
(
a
,
n
/2)*
Unknown
(
a
,
n
/2)
Else
Return
a
*
Unknown
(
a
, (
n
1)/2)*
Unknown
(
a
, (
n
1)/2)
End algorithm.
2.
For a weighted undirected graph
G
a
culprit
set
C
of arcs has the following property: if
all the elements of
C
are removed from
G
it becomes acyclic (tree), and a
minculprit
set
M
of arcs is such that the aggregate weight of arcs in
M
is minimum for all such sets
C
.
Write a greedy algorithm to find
M
given a graph
G
.
[Hint: First try to find appropriate
G
–
M
.]
3a.
The following is a directed graph
G
: [
a: b
,
d
], [
b: c
], [
c: a
], [
d: e
], [
e: f
], [
f: d
], where
[
a: b
,
d
] means there is a directed arc in
G
from
a
to
b
and from
a
to
d
. Draw
G
.
You are to find the strong connected components (SCCs) of the graph by running the
relevant algorithm. Show (i) the first DFSspanning tree of
G
starting with the node
a
,
along with the postorder traversal numbers; (ii) the postorder traversal numbers shown
on the nodes of the reverse graph
G’
; and (iii) the second set of DFS spanning trees on
the reverse graph
G’
, which identifies the SCCs of
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 Spring '12
 Dmitra
 Algorithms

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