Notes on Complexity Theory
Last updated: October, 2011
Lecture 15
Jonathan Katz
1 Randomized Space Complexity
1.1 Undirected Connectivity and Random Walks
1.1.1 Markov Chains
We now develop some machinery that gives a diﬀerent, and somewhat more general, perspective
on random walks. In addition, we get better bounds for the probability that we hit
t
. (Note that
the previous analysis calculated the probability that we end at vertex
t
. But it would be suﬃcient
to pass through vertex
t
at any point along the walk.) The drawback is that here we rely on some
fundamental results concerning Markov chains that are presented without proof.
We begin with a brief introduction to (ﬁnite, timehomogeneous) Markov chains. A sequence of
random variables
X
0
,...
over a space Ω of size
n
is a
Markov chain
if there exist
{
p
i,j
}
such that,
for all
t >
0 and
x
0
,...,x
t

2
,x
i
,x
j
∈
Ω we have:
Pr[
X
t
=
x
j

X
0
=
x
0
,...,X
t

2
=
x
t

2
,X
t

1
=
x
i
] = Pr[
X
t
=
x
j

X
t

1
=
x
i
] =
p
j,i
.
In other words,
X
t
depends only on
X
t

1
(that is, the transition is
memoryless
) and is furthermore
independent of
t
. We view
X
t
as the “state” of a system at time
t
. If we have a probability
distribution over the states of the system at time
t
, represented by a probability vector
p
t
, then the
distribution at time
t
+ 1 is given by
P
·
p
t
(similar to what we have seen in the previous section).
Similarly, the probability distribution at time
t
+
‘
is given by
P
‘
·
p
t
.
A ﬁnite Markov chain corresponds in the natural way to a random walk on a (possibly directed
and/or weighted) graph. Focusing on undirected graphs (which is all we will ultimately be interested
in), a random walk on such a graph proceeds as follows: if we are at a vertex
v
at time
t
, we move
to a random neighbor of
v
at time
t
+ 1. If the graph has
n
vertices, such a random walk deﬁnes
the Markov chain given by:
p
j,i
=
‰
k/
deg(
i
) there are
k
edges between
j
and
i
0
otherwise
.
We continue to allow (multiple) selfloops; each selfloop contributes 1 to the degree of a vertex.