The Cover Time of Random Walks

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The Cover Time of Random Walks. Uriel Feige Weizmann Institute. Random Walks. Simple graph. Move to a neighbor chosen uniformly at random. Random Walks. Random Walks. Random Walks. Random Walks. Random Walks. Random Walks. Random Walks. Hitting time and its variants.

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## The Cover Time of Random Walks

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### The Cover Time of Random Walks

Uriel Feige

Weizmann Institute

Random Walks
• Simple graph.
• Move to a neighbor chosen uniformly at random.
Hitting time and its variants

Random variables associated with a random walk. Here we shall only deal with their expectations.

Hitting timeH(s,t). Expected number of steps to reach t starting at s.

Commute time. Symmetric.

C(s,t) = C(t,s) = H(s,t) + H(t,s).

Difference time. Anti-symmetric.

D(s,t) = -D(t,s) = H(s,t) - H(t,s).

Cover time

Cov(s,G). The expected number of steps it takes a walk that starts at s to visit all vertices.

Cov(G). Maximum over s of Cov(s,G).

What characterizes the cover time of a graph?

How large might it be? How small?

Special families of graphs.

Deterministic algorithms for estimating the cover time for general graphs.

Computing the hitting time

System of n linear equations.

H(t,t) = 0.

H(v,t) = 1 + avg H(N(v),t).

Compute all hitting times to t by one matrix inversion. (Related approach computes hitting times for all pairs [Tetali 1999].)

Applies to arbitrary Markov chains.

Corollary: Hitting time is rational and computable in polynomial time.

Reducing cover time to hitting time

Markov chain M on states (v,S).

v - current vertex.

Step in G from u to v corresponds to step in M from (u,S) to (v,S+{v}).

Cov+(s,G) = H((s,{s}),(s,V))

Corollary: Cover time is rational and computable in exponential time.

A detour - electrical networks

Many analogies between random walks in graphs and electrical networks.

Can help (depending on a person’s background) in transferring intuition and theorems from one area to the other.

Effective Resistance
• Every edge – a resistor of 1 ohm.
• Voltage difference of 1 volt between u and v.

R(u,v) – inverse of electrical current from u to v.

v

_

+

u

Understanding the commute time

Theorem[Chandra, Raghavan, Ruzzo, Smolensky, Tiwari 1989]: For every graph with m edges and every two vertices u and v,

C(u,v) = 2mR(u,v)

Proof: by comparing the respective systems of linear equations, for random walks and for electrical current flows.

Easy useful principles

Removing an edge – increases is resistance to be infinite.

Adding/removing an edge anywhere in the graph can only reduce/increase effective resistance.

Contracting an edge – reduces its resistance to 0.

Contracting an edge anywhere in the graph can only reduce effective resistance.

Series-parallel graphs

R=R1+R2

1/R =1/R1 + 1/R2

R1

R2

R1

R2

Foster’s network theorem

For every connected graph on n vertices, the sum of effective resistances taken over all neighboring pairs of vertices is n-1.

Relating cover time to commute time

[Aleliunas, Karp, Lipton, Lovasz, Rackoff 1979] Cover time is upper bounded by sum of commute times along edges of a spanning tree.

Spanning tree argument

Arbitrary spanning tree [AKLLR, CRRST]:

Best spanning tree [Feige 1995]:

Lollipop graph:

2n/3 clique

n/3 path

Coupon collector

The spanning tree upper bound gives Cov(clique)<O(n2). Too pessimistic.

Covering a clique is almost like throwing balls in bins at random, until every bin has a ball. Hence

Observe that H(u,v) = n-1. Covering requires a ln n overhead.

Proof of Matthews bound

Arbitrarily order all vertices but s.

Let Pr[i] denote the probability that i is the last vertex to be visited among {1, …, i}.

For random permutation, Pr[i] = 1/i.

Lower bound on cover time

[Feige 1995]:

Proof: either there is a pair of vertices that witness the lower bound through their mutual hitting times, or a generalization of the Matthew’s bound (applying it to subsets of vertices) works.

Some special classes of graphs

Order of magnitude of cover time:

Path n2

Expanders n log n

2-dim gridsn log2 n

3-dim gridsn log n

Full d-ary treen log2 n / log d

In many cases, much more is known.

Regularity and cover time

[Kahn, Linial, Nisan, Saks 1989]: the cover time on regular graphs is at most 4n2.

[Coppersmith, Feige, Shearer 1996]: every spanning tree has resistance at most 3n/d.

[Feige 1997]: cover time at most 2n2.

Worse example known (necklace): 15n2/16.

Irregular graphs

[Coppersmith, Feige, Shearer 1996]: every graph has a spanning tree of resistance at most O(n avg(1/deg)).

Proof: random spanning tree. Uses the fact that fraction of spanning trees that use edge (u,v) is exactly R[u,v].

Upper bound on Cov+(G) based on irregularity avg(deg) x avg(1/deg) of G.

Spanning tree - without return

[Feige 1997] (proof essentially, by induction):

• In every graph there is a vertex s with
• Path is the most difficult tree to cover (starting at the middle).
Approximating Cov(G)

Max[C(u,v)] approximates Cov(G) within a factor of log n.

Augmented Matthews lower bound (AMLB):

[Kahn, Kim, Lovasz, Vu 2000]: AMLB approximated Cov(G) within a factor of O((log log n)2), and can be efficiently approximated within a factor of 2.

Approximating Cov(s,G)

Cov(s,G) might be much larger than max[H(s,v)].

key graph

[Chlamtac, Feige, Rabinovich 2003, 2005]:

Cov(s,G) can be approximated within a ratio of O(log n approx[Cov(G)]).

Tools used in proof

Cycle identity for reversible MC:

H(u,v)+H(v,w)+H(w,u) = H(u,w)+H(w,v)+H(v,u)

Transitivity of difference time:

D(u,v) > 0, D(v,w) > 0 imply D(u,w) > 0.

Induces order …w,…v, …u,…

Partition order into homogeneous blocks.

Upper bound Cov(s,G) by covering block after block.

Full d-ary trees

Cover time known in great detail [Aldous].

The technique:

Compute return time to root r (easy).

Compute expected number of returns to root during cover (recursive formula).

Multiply the two to get Cov+(r,T).

Techniques for approximating the cover time
• Systems of linear equations (hitting times).
• Using identities involving cover time (Aldous).
• Effective resistance (commute times, Foster’s theorem, etc.).
• Spanning tree arguments and extensions.
• Matthew’s bounds and extensions.
• Graph partitioning (order induced by difference time).
Open questions

Deterministic approximation of Cov(G) and of Cov(s,G).

(Conjecture: PTAS on trees soon.)

Extremal problems. Which (regular) graphs have the largest/smallest cover times?

(Conjectures exist.)