The Cover Time of Random Walks

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The Cover Time of Random Walks

• Uriel Feige
• Weizmann Institute

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Random Walks

• Simple graph.
• Move to a neighbor chosen uniformly at random.

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Random Walks
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Random Walks
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Random Walks
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Random Walks
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Random Walks
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Random Walks
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Random Walks
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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).

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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.

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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.

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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,Sv).
• Cov(s,G) H((s,s),(s,V))
• Corollary Cover time is rational and computable
  in exponential time.

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A detour - electrical networks

• Many analogies between random walks in graphs and
  electrical networks.
• Can help (depending on a persons background) in
  transferring intuition and theorems from one area
  to the other.

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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
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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.

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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.

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Series-parallel graphs

• RR1R2
• 1/R 1/R1 1/R2

R1
R2
R1
R2
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Fosters network theorem

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

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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.

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Spanning tree argument

• Arbitrary spanning tree AKLLR, CRRST
• Best spanning tree Feige 1995
• Lollipop graph

2n/3 clique
n/3 path
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Coupon collector

• The spanning tree upper bound gives
  Cov(clique)ltO(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.

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Relating cover time to hitting time Matthews
1988

• nth
  harmonic number

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Proof of Matthews bound

• Arbitrarily order all vertices but s.
• Let Pri denote the probability that i is the
  last vertex to be visited among 1, , i.
• For random permutation, Pri 1/i.

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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
  Matthews bound (applying it to subsets of
  vertices) works.

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Some special classes of graphs

• Order of magnitude of cover time
• Path n2
• Expanders n log n
• 2-dim grids n log2 n
• 3-dim grids n log n
• Full d-ary tree n log2 n / log d
• In many cases, much more is known.

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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.

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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 Ru,v.
• Upper bound on Cov(G) based on irregularity
  avg(deg) x avg(1/deg) of G.

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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).

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Approximating Cov(G)

• MaxC(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.

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Approximating Cov(s,G)

• Cov(s,G) might be much larger than maxH(s,v).
• key graph
• Chlamtac, Feige, Rabinovich 2003, 2005
• Cov(s,G) can be approximated within a ratio of
  O(log n approxCov(G)).

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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) gt 0, D(v,w) gt 0 imply D(u,w) gt 0.
• Induces order w,v, u,
• Partition order into homogeneous blocks.
• Upper bound Cov(s,G) by covering block after
  block.

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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).

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Techniques for approximating the cover time

• Systems of linear equations (hitting times).
• Using identities involving cover time (Aldous).
• Effective resistance (commute times, Fosters
  theorem, etc.).
• Spanning tree arguments and extensions.
• Matthews bounds and extensions.
• Graph partitioning (order induced by difference
  time).

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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.)

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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.