Notes on Complexity Theory
Last updated: October, 2011
Lecture 14
Jonathan Katz
1
Randomized Space Complexity
1.1
Undirected Connectivity and Random Walks
A classic problem in
RL
is
undirected connectivity
(
UConn
). Here, we are given an
undirected
graph and two vertices
s, t
and are asked to determine whether there is a path from
s
to
t
. An
RL
algorithm for this problem is simply to take a “random walk” (of suﬃcient length) in the graph,
starting from
s
. If vertex
t
is ever reached, then output 1; otherwise, output 0. (We remark that
this approach does
not
work for
directed
graphs.) We analyze this algorithm (and, speciﬁcally, the
length of the random walk needed) in two ways; each illustrates a method that is independently
useful in other contexts. The ﬁrst method looks at random walks on
regular
graphs, and proves a
stronger result showing that after suﬃciently many steps of a random walk the location is close to
uniform over the vertices of the graph. The second method is more general, in that it applies to
any (nonbipartite) graph; it also gives a tighter bound.
1.1.1
Random Walks on Regular Graphs
Fix an undirected graph
G
on
n
vertices where we allow selfloops and parallel edges (i.e., integer
weights on the edges). We will assume the graph is
d
regular and has at least one selfloop at every
vertex; any graph can be changed to satisfy these conditions (without changing its connectivity) by
adding suﬃciently many selfloops. Let
G
also denote the (scaled) adjacency matrix corresponding
to this graph: the (
i, j
)th entry is
k/d
if there are
k
edges between vertices
i
and
j
. Note that
G
is
symmetric
(
G
i,j
=
G
j,i
for all
i, j
) and
doubly stochastic
(all entries are nonnegative, and
all rows and columns sum to 1). A
probability vector
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 Fall '08
 staff
 Graph Theory, Matrices, loop, Orthogonal matrix, Multigraph

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