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cover times, blanket times, and majorizing measures
Jian DingU. C. Berkeley
James R. LeeUniversity of Washington
Yuval PeresMicrosoft Research
random walks on graphs
By putting conductances {cuv} on the edges of the graph,we can get any reversible Markov chain.
hitting and covering
Hitting time: H(u,v) = expected # of steps to hit v starting at u
Commute time: κ(u,v) = H(u,v) + H(v,u)(metric)
Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
hitting and covering
Hitting time: H(u,v) = expected # of steps to hit v starting at u
Commute time: κ(u,v) = H(u,v) + H(v,u)(metric)
Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
pathcomplete graph
expander2-dimensional grid3-dimensional grid
complete d-ary tree
n2
n log nn log nn (log n)2
n log nn (log n)2/log d
orders of magnitude of some cover times
[coupon collecting][Broder-Karlin 88][Aldous 89, Zuckerman 90][Aldous 89, Zuckerman 90][Zuckerman 90]
hitting and covering
Hitting time: H(u,v) = expected # of steps to hit v starting at u
Commute time: κ(u,v) = H(u,v) + H(v,u)(metric)
Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
regular treesrandom graphs
discrete torus, lattices
[Aldous 91][Cooper-Frieze 08][Dembo-Peres-Rosen-Zeitouni 04]
asymptotically optimal bounds
hitting and covering
Hitting time: H(u,v) = expected # of steps to hit v starting at u
Commute time: κ(u,v) = H(u,v) + H(v,u)(metric)
Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
(1-o(1)) n ln n · tcov(G) · min(4n3/27, 2mn)
general bounds (n = vertices, m = edges)
[Feige’95, Alelinuas-Karp-Lipton-Lovasz-Rackoff’79][Feige’95, Matthews’88]
(conjecture: the completegraph is extremal)
electrical resistance
Hitting time: H(u,v) = expected # of steps to hit v starting at u
Commute time: κ(u,v) = H(u,v) + H(v,u)(metric)
Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
+u
v
Reff (u,v) = inverse of electrical current flowing from u to v
[Chandra-Raghavan-Ruzzo-Smolensky-Tiwari’89]:
If G has m edges, then for every pair u,v
κ(u,v) = 2m Reff (u,v)
(endows κ with specialgeometric properties)
computation
Hitting time: easy to compute in deterministic poly time by solving system of linear equations
H(u,u) = 0 H(u,v) = 1 + Ew»u H(w,v)
Cover time: easy to compute in deterministic exponential time
Approximations (deterministic, poly-time):
[Matthews’88, CRRST’89]
Augmented Matthews bound yields an O(log log n)2 approximation[Kahn-Kim-Lovasz-Vu’99]
For trees, there is an 1+² approximation for every ² > 0[Feige-Zeitouni’09]
maxu,v κ(u,v) yields an O(log n) approximation
Open question: Does there exist an O(1)-approximation for general graphs?[Aldous-Fill’94]
blanket times
Blanket times [Winkler-Zuckerman’96]:
¯-Blanket time is the expected first time T at which all the local times,
are within a factor of ¯.
blanket times, comparisons
Conjecture [Winkler-Zuckerman’96]: For every graph G and 0 < ¯ < 1, tblanket(G,¯) ³ tcov(G).
Proved for many special cases. True up to (log log n)2 by [KKLV’99]
Comparison of cover times:If G and G’ are two graphs on the same set of nodes andκG(u,v) · κG’(u,v) for all u,v 2 V, does it follow that
?
³ ³
main theorem
Talagrand introduced a functional on any metric space (X, d).
THEOREM: For any graph G,
where ³ denotes equivalence up to a universal constant.
°2(X ;d)
tcov(G) ³ [°2(G;p
· )]2
Some consequences:
- There is a deterministic O(1)-approximation to for any metric space, hence the same holds for tcov(G).
- Postively resolves the Winkler-Zuckerman blanket time conjectures.
- Bi-lipschitz stability. For instance, tcov(G) ³ tcov(G’) where G’ is a spectral sparsifier of G.
°2(X ;d)
main theorem
Talagrand introduced a functional on any metric space (X, d).
THEOREM: For any graph G, for any 0 < ¯ < 1,
where A . B denotes A · O(B).
°2(X ;d)
h°2(G;
p· )
i 2. tcov(G) · tblanket(G;¯ ) . ¯
h°2(G;
p· )
i 2
Some consequences:
- There is a deterministic O(1)-approximation to for any metric space, hence the same holds for tcov(G).
- Postively resolves the Winkler-Zuckerman blanket time conjectures.
- Bi-lipschitz stability. For instance, tcov(G) ³ tcov(G’) where G’ is a spectral sparsifier of G.
°2(X ;d)
a fast randomized algorithm
THEOREM: If g is an n-dimensional Gaussian, then
D = diagonal degree matrixA = adjacency matrix of G
THEOREM: For m-edge graphs, there is an O(m polylog(m))-time randomized algorithm to compute an O(1)-approximation to the cover time.Uses [Spielman-Teng] and [Spielman-
Srivistava]
main theorem
THEOREM: For any graph G and δ 2 (0,1),
where ³ denotes equivalence up to a universal constant.
tcov(G) ³ [°2(G;p
· )]2 ³ ± tblanket(G;±)
°2(S;d)
Gaussian processes
Consider a Gaussian process {Xu : u 2 S} with E (Xu)=0 8 u 2 S
(i.e. every linear combination ®1X1 + + ®kXk is normal)
Such a process comes with a natural metric
d(s; t) =q
E jX s ¡ X tj2
transforming (S,d) into a metric space.
Equivalently, for S finite, consider S µ Rn, and the process Xu = hg, ui for u 2 Swhere g=(g1, …, gn) is an i.i.d. N(0,1) vector.
PROBLEM: What is E max { Xu : u 2 S } ?
Gaussian processes
PROBLEM: What is E max { Xu : u 2 S } ?
®
If random variables are “independent,”expect the union bound to be tight.
Gaussian concentration:
Expect max for k points is about
covering trees
Recursively partition into pieces of diameter j=0, 1, 2, …
4¡ jValue of this path iswhere dj is the sequence of degreesdown the path
Pj 4¡ j
plogdj
valc(T) = maxP
j 4¡ jp
logdj max over all root-leaf paths
°2(S;d) = minvalc(T) min over all covering trees
(S;d)
covering trees
valc(T) = maxP
j 4¡ jp
logdj max over all root-leaf paths
°2(S;d) = minvalc(T) min over all covering trees
(S;d)
packing trees
valp(T) = minP
j 4¡ jp
logdj min over all root-leaf paths
(S;d)
Main technical theorem:
maxT valp(T) ³ minT 0 valc(T0) = °2(S;d)
majorizing measure theorem
Majorizing measures theorem (Talagrand):
(Recalling that d(s; t) =q
E jX s ¡ X tj2.)
main theorem [Ding, L, Peres]
THEOREM: For any graph G and δ 2 (0,1),
where ³ denotes equivalence up to a universal constant.
tcov(G) ³ [°2(G;p
· )]2 ³ ± tblanket(G;±)
°2(S;d)
hints of a connection
Gaussian concentration:
KKLV concentration:
Here, Nt(w) denotes the number of visits to w when the random walk startedat u has returned to u for the (t deg(u))th time.
hints of a connection
- Trees + KKLV concentration suffice for upper bound
- [Barlow-Ding-Nachmias-Peres] prove the “Dudley version”
a problem on Gaussian processes
Gaussian process:
We need strong estimates on the size of this window as ε 0.(want to get a point there with probability at least 0.1)Problem: Majorizing measures handles first moments, but we need second moment bounds.
percolation on trees and the DGFF
First and second moments agree for
percolation on balanced trees
Problem: General Gaussian processes behaves nothing like percolation!
Resolution: Processes coming from the Isomorphism Theorem all arise from a “discrete Gaussian free field.”
percolation on trees and the DGFF
First and second moments agree for
percolation on balanced trees
For DGFFs, using electrical network theory, we show that it is possibleto select a subtree of the MM tree and a delicate filtration of theprobability space so that the Gaussian process can be coupled to apercolation process.
open questions
Holds for - complete graph - complete d-ary tree - discrete torus
QUESTION: Is there a deterministic, polynomial-time (1+²)-approximation to the cover time for every ² > 0 ?
QUESTION: Is the standard deviation of the time-to-cover bounded by the maximum hitting time?