Advanced Analysis of Algorithms  Homework IV (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
{
[email protected]
}
1
Problems
1. Problem
26
.
2(9)
on Page
664
of [CLRS01].
Solution:
Let
G
=
(
V,E
)
denote the undirected graph and let
s
∈
V
denote an arbitrary vertex. Transform
G
into a
flow network
G
′
as follows:
(i) Replace each undirected edge
(
u,v
)
∈
E
with two directed edges
(
u,v
)
and
(
v,u
)
, each with capacity
1
.
(ii) Let
s
be the source vertex of the flow network.
Run the maxflow algorithm treating each vertex
u
∈
V
− {
s
}
as the sink
t
, recording the value of the maximum
flow (
mf
u
). The edgeconnectivity of
G
is then
min
u
∈
V
−{
s
}
mf
u
. The crucial observation is that any cut of
G
, the
vertex
s
is always on one side and hence it suffices to consider only the cases in which
s
is the source vertex in
G
′
. It
is straightforward to see that a directed cut in
G
′
corresponds to a cut in
G
with the same capacity and vice versa.
a50
2. Problem
26
.
3(3)
on Page
668
of [CLRS01].
Solution:
Without loss of generality, assume that

L
 ≤ 
R

. Any augmenting path has the following structure:
s
→
L
→
R
→
L...
→
R
→
t
In other words, the path enters
L
from
s
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 .
 Algorithms, Flow network, Maximum flow problem, Maxflow mincut theorem, Zij, ) E. Let

Click to edit the document details