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

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

Hitting time H(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).Cov+(G). Cover and return to start.

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.

S – vertices already visited.

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.

_

u

v

+

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:

3)1(2)( nnmGCov

2n/3 cliquen/3 path

27/4)( 3nGCov

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.

nnKCov n ln)(

Relating cover time to hitting time [Matthews 1988]

nth harmonic number

)],(max[)( 1 vuHhGCov n

)],(min[)( 1 vuHhGCov n

n

in ni

h1

ln1

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.

1

1)Pr()(*,),(

n

iiiHGsCov

)],([min/)(*,),( 1

1

1vuHhiiHGsCov svn

n

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.

nnoGsCov ln))1(1(),(

Some special classes of graphs

Order of magnitude of cover time:Path n2

Expanders n log n2-dim grids n log2 n3-dim grids n log nFull d-ary tree n 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).

)]),(max[)(min2(2

1),( vsDTRmGsCov

27/2),( 3nGsCov

)],(min[)(min2),( svHTRmGsCov

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.

)]},([min||{lnmax)( , vuCSGCov SvuS

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.)

Additional topics

Some results (e.g., correspondence with effective resistance) extend to reversible Markov chains.

Some results (e.g., Matthews’ bounds) extend to arbitrary Markov Chains.

This talk referred only to expected cover time. More known (and open) on full distribution of cover time.