Introduction to Algorithms
Solution Set 9
CS 482, Spring 2008
(1)
There are many correct solutions; here is one. The graph has two vertices
s,t
and
n
triples of
vertices (
u
i
,v
i
,w
i
)
n
i
=1
, for a total of 3
n
+ 2 vertices. There are 5
n
edges, speciﬁed as follows. For
all
i
, the graph contains edges (
s,u
i
)
,
(
u
i
,v
i
)
,
(
v
i
,t
)
,
(
u
i
,w
i
)
,
(
w
i
,t
)
.
The subgraph consisting
of vertices
s,t,u
i
,v
i
,w
i
and the ﬁve edges between them will be denoted by
G
i
.
The unique minimum
s

t
cut is the one that separates
s
from all other vertices. This cut survives
the analogue of Karger’s algorithm if and only if none of the edges (
s,u
i
) is ever contracted. For
i
= 1
,
2
,...,n
, let
E
i
denote the event: “the algorithm picks edge (
s,u
i
) before it picks any other
edge of
G
i
.” We observe the following facts.
1. Pr(
E
i
) = 1
/
5
.
2. The events
{E
i

i
= 1
,
2
,...,n
}
are mutually independent.
3. The algorithm succeeds only if