CSE450/598 Design And Analysis of Algorithms
HW02 Grading Keys
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1. (10 pts) Exercise 3.2 (p.107)
Solution:
We can run either BFS or DFS on
G
to compute all connected components of
G
in
O
(
m
+
n
)
time.
G
does not contain a cycle if and only if none of its connected components contains a
cycle. Therefore we may assume that the graph is connected as we can work with each of the
connected components.
Starting from any node, we run BFS to compute a BFS tree
T
. If
T
contains all edges in
G
,
G
does not contain a cycle. Otherwise, there must be an edge (
u, v
) in
G
that is not used by
T
. We can ±nd the smallest common ancestor of
u
and
v
in
T
in linear time (If
u
and
v
are
on the same layer, we back o² one layer from both nodes at each step. If
u
has a higher layer,
we ±rst ±nd the ancestor of
u
which is on the same layer as
v
, then back o².).
Let it be
w
.
Then the
w
–
u
path, the edge (
u, v
), and the
v
–
w
path form a cycle of
G
.
It takes
O
(
m
+
n
) time to compute all connected components of
G
.
For each connected
component of
G
, our algorithm requires time proportional to the summation of the number
of nodes and number of edges in that component. Therefore the algorithm has
O
(
m
+
n
) time
complexity.
Grading Keys:
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 Spring '08
 GuoliangXue
 Algorithms, Graph Theory

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