Multiple Random Walks: Cover Times, Hitting
Times and Applications
Thomas Sauerwald
22 May 2015
(joint work with Robert Elsässer)
Outline
Introduction
Upper Bounds on the Multiple Cover Time
Lower Bounds on the Multiple Cover Time
Concrete Networks
Conclusion
Multiple Random Walks
Introduction
2
(Single) Random Walks
Random walk on an undirected, connected and finite graph G = (V , E):
X0 = u, X1 , X2 , . . .
Let P be the transition matrix of a lazy random walk on G
1
if u = v ,

2
1
if {u, v } ∈ E(G),
Puv = 2 deg(u)


0
otherwise.
π with πv =
deg(v )
2|E|
is the stationary distribution
Multiple Random Walks
Introduction
3
(Single) Random Walks
Random walk on an undirected, connected and finite graph G = (V , E):
X0 = u, X1 , X2 , . . .
Let P be the transition matrix of a lazy random walk on G
1
if u = v ,

2
1
if {u, v } ∈ E(G),
Puv = 2 deg(u)


0
otherwise.
π with πv =
deg(v )
2|E|
is the stationary distribution
Basic Quantities
Multiple Random Walks
Introduction
3
(Single) Random Walks
Random walk on an undirected, connected and finite graph G = (V , E):
X0 = u, X1 , X2 , . . .
Let P be the transition matrix of a lazy random walk on G
1
if u = v ,

2
1
if {u, v } ∈ E(G),
Puv = 2 deg(u)


0
otherwise.
π with πv =
deg(v )
2|E|
is the stationary distribution
Basic Quantities
t
mixing time: tmix ( 12 ) = min{t ∈ N : ∀u, v ∈ V : Pu,v
− πv ≤
Multiple Random Walks
Introduction
1
2
· πv }
3
(Single) Random Walks
Random walk on an undirected, connected and finite graph G = (V , E):
X0 = u, X1 , X2 , . . .
Let P be the transition matrix of a lazy random walk on G
1
if u = v ,

2
1
if {u, v } ∈ E(G),
Puv = 2 deg(u)


0
otherwise.
π with πv =
deg(v )
2|E|
is the stationary distribution
Basic Quantities
t
mixing time: tmix ( 12 ) = min{t ∈ N : ∀u, v ∈ V : Pu,v
− πv ≤
1
2
· πv }
(maximum) hitting time: thit = maxu,v ∈V Eu [ min{t : Xt = v } ]
Multiple Random Walks
Introduction
3
(Single) Random Walks
Random walk on an undirected, connected and finite graph G = (V , E):
X0 = u, X1 , X2 , . . .
Let P be the transition matrix of a lazy random walk on G
1
if u = v ,

2
1
if {u, v } ∈ E(G),
Puv = 2 deg(u)


0
otherwise.
π with πv =
deg(v )
2|E|
is the stationary distribution
Basic Quantities
t
mixing time: tmix ( 12 ) = min{t ∈ N : ∀u, v ∈ V : Pu,v
− πv ≤
1
2
· πv }
(maximum) hitting time: thit = maxu,v ∈V Eu [ min{t : Xt = v } ]
cover time: tcov = maxu∈V Eu min{t : ∪ts=0 Xt = V }
Multiple Random Walks
Introduction
3
(Single) Random Walks
Random walk on an undirected, connected and finite graph G = (V , E):
X0 = u, X1 , X2 , . . .
Let P be the transition matrix of a lazy random walk on G
1
if u = v ,

2
1
if {u, v } ∈ E(G),
Puv = 2 deg(u)


0
otherwise.
π with πv =
deg(v )
2|E|
is the stationary distribution
Basic Quantities
t
mixing time: tmix ( 12 ) = min{t ∈ N : ∀u, v ∈ V : Pu,v
− πv ≤
1
2
· πv }
(maximum) hitting time: thit = maxu,v ∈V Eu [ min{t : Xt = v } ]
cover time: tcov = maxu∈V Eu min{t : ∪ts=0 Xt = V }
Theorem [Feige]
For any graph G with n = |V | vertices, n ln n . tcov . n3 .
Multiple Random Walks
Introduction
3
Multiple Random Walks
k Random walks on G = (V , E) (independent and in parallel):
X01 = u, X11 , X21 , . . .
X02 = u, X12 , X22 , . . .
..
.
X0k = u, X1k , X2k , . . .
Multiple Random Walks
Introduction
4
Multiple Random Walks
k Random walks on G = (V , E) (independent and in parallel):
X01 = u, X11 , X21 , . . .
X02 = u, X12 , X22 , . . .
..
.
X0k = u, X1k , X2k , . . .
Basic Quantities
Multiple Random Walks
Introduction
4
Multiple Random Walks
k Random walks on G = (V , E) (independent and in parallel):
X01 = u, X11 , X21 , . . .
X02 = u, X12 , X22 , . . .
..
.
X0k = u, X1k , X2k , . . .
Basic Quantities
(k )
hitting time: thit = maxu,v ∈V E(u,...,u) min{t : ∪ki=1 Xti ⊇ {v }}
Multiple Random Walks
Introduction
4
Multiple Random Walks
k Random walks on G = (V , E) (independent and in parallel):
X01 = u, X11 , X21 , . . .
X02 = u, X12 , X22 , . . .
..
.
X0k = u, X1k , X2k , . . .
Basic Quantities
(k )
hitting time: thit = maxu,v ∈V E(u,...,u) min{t : ∪ki=1 Xti ⊇ {v }}
(k )
cover time: tcov = maxu∈V E(u,...,u) min{t : ∪ki=1 ∪ts=0 Xsi = V }
Multiple Random Walks
Introduction
4
Motivation and Challenges of Multiple Random Walks
Applications
many randomised algorithms employ multiple random walks
arise in stochastic or diffusive processes (component or analysis)
Page-Rank can be regarded as multiple random walks
Multiple Random Walks
Introduction
5
Motivation and Challenges of Multiple Random Walks
Applications
many randomised algorithms employ multiple random walks
arise in stochastic or diffusive processes (component or analysis)
Page-Rank can be regarded as multiple random walks
Mathematical Challenges
various configurations for start vertices
deal with short random walks
already for k = 2, hard(er) to apply linearity of expectations
(dependencies and interactions)
Multiple Random Walks
Introduction
5
Approaches for Speeding Up Cover Times
modified transition probabilities
locally computable transition probabilities with tcov . n2 log n
[Ikeda, Kubo, Okumoto, Yamashita]
non-backtracking random walk on high-girth expanders
[Alon, Benjamini, Lubetzky, Sodin]
Multiple Random Walks
Introduction
6
Approaches for Speeding Up Cover Times
modified transition probabilities
locally computable transition probabilities with tcov . n2 log n
[Ikeda, Kubo, Okumoto, Yamashita]
non-backtracking random walk on high-girth expanders
[Alon, Benjamini, Lubetzky, Sodin]
Neighborhood exploration
avoid visited vertices (or edges)
[Avin, Krishnamachari]
look-ahead: when visiting u, cover all neighbors of u
[Cooper, Frieze, Radzik]
Multiple Random Walks
Introduction
6
Approaches for Speeding Up Cover Times
modified transition probabilities
locally computable transition probabilities with tcov . n2 log n
[Ikeda, Kubo, Okumoto, Yamashita]
non-backtracking random walk on high-girth expanders
[Alon, Benjamini, Lubetzky, Sodin]
Neighborhood exploration
avoid visited vertices (or edges)
[Avin, Krishnamachari]
look-ahead: when visiting u, cover all neighbors of u
[Cooper, Frieze, Radzik]
Multiple Walks (this talk)
Multiple Random Walks
Introduction
6
Example
0
Multiple Random Walks
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Introduction
7
Example
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Multiple Random Walks
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Introduction
7
Example
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Example
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Introduction
7
Speed-up for Best-Case Start Vertices
Multiple Random Walks
Introduction
8
Speed-up for Best-Case Start Vertices
Multiple Random Walks
Introduction
8
Speed-up for Best-Case Start Vertices
k = 1: minu∈V Eu min{t : ∪ts=0 Xt = V } n2
Multiple Random Walks
Introduction
8
Speed-up for Best-Case Start Vertices
k = 1: minu∈V Eu min{t : ∪ts=0 Xt = V } n2
Multiple Random Walks
Introduction
8
Speed-up for Best-Case Start Vertices
k = 1: minu∈V Eu min{t : ∪ts=0 Xt = V } n2
i
h
n t
i
k = log n: minu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V } n
Multiple Random Walks
Introduction
8
Speed-up for Best-Case Start Vertices
k = 1: minu∈V Eu min{t : ∪ts=0 Xt = V } n2
i
h
n t
i
k = log n: minu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V } n
⇒
S (log n)
Multiple Random Walks
Introduction
8
Speed-up for Best-Case Start Vertices
k = 1: minu∈V Eu min{t : ∪ts=0 Xt = V } n2
i
h
n t
i
k = log n: minu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V } n
⇒
S
(log n)
minu∈V Eu min{t : ∪ts=0 Xt = V }
=
h
i
n t
i
minu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V }
Multiple Random Walks
Introduction
8
Speed-up for Best-Case Start Vertices
k = 1: minu∈V Eu min{t : ∪ts=0 Xt = V } n2
i
h
n t
i
k = log n: minu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V } n
⇒
S
(log n)
minu∈V Eu min{t : ∪ts=0 Xt = V }
=
h
i n
n t
i
minu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V }
Multiple Random Walks
Introduction
8
Speed-up for Stationary Start Vertices
Theorem [Broder, Karlin, Raghavan and Upfal]
For any graph G, the cover time for k stationary random walks satisfies
h
i |E| 2
E(π,...,π) min{t : ∪ki=1 ∪ts=0 Xti = V } .
· log3 n.
k
Multiple Random Walks
Introduction
9
Speed-up for Stationary Start Vertices
Theorem [Broder, Karlin, Raghavan and Upfal]
For any graph G, the cover time for k stationary random walks satisfies
h
i |E| 2
E(π,...,π) min{t : ∪ki=1 ∪ts=0 Xti = V } .
· log3 n.
k
Multiple Random Walks
Introduction
9
Speed-up for Stationary Start Vertices
Theorem [Broder, Karlin, Raghavan and Upfal]
For any graph G, the cover time for k stationary random walks satisfies
h
i |E| 2
E(π,...,π) min{t : ∪ki=1 ∪ts=0 Xti = V } .
· log3 n.
k
tcov n2 |E|2
Multiple Random Walks
Introduction
9
Speed-up for Stationary Start Vertices
Theorem [Broder, Karlin, Raghavan and Upfal]
For any graph G, the cover time for k stationary random walks satisfies
h
i |E| 2
E(π,...,π) min{t : ∪ki=1 ∪ts=0 Xti = V } .
· log3 n.
k
tcov n2 |E|2
⇒
S
(k )
Eπ min{t : ∪ts=0 Xti = V }
=
h
i
n t
i
E(π,...,π) min{t : ∪log
i=1 ∪s=0 Xs = V }
Multiple Random Walks
Introduction
9
Speed-up for Stationary Start Vertices
Theorem [Broder, Karlin, Raghavan and Upfal]
For any graph G, the cover time for k stationary random walks satisfies
h
i |E| 2
E(π,...,π) min{t : ∪ki=1 ∪ts=0 Xti = V } .
· log3 n.
k
tcov n2 |E|2
⇒
S
(k )
Eπ min{t : ∪ts=0 Xti = V }
1
2
=
h
i k ·
log n t
log3 n
E(π,...,π) min{t : ∪i=1 ∪s=0 Xsi = V }
Multiple Random Walks
Introduction
9
Speed-up for Worst-Case Start Vertices
[Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ n, the speed-up is defined as
(1)
S (k ) =
tcov
(k )
tcov
Multiple Random Walks
Introduction
10
Speed-up for Worst-Case Start Vertices
[Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ n, the speed-up is defined as
(1)
maxu∈V Eu min{t : ∪ts=0 Xt = V }
tcov
(k )
=
S =
i
h
(k )
n t
i
tcov
maxu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V }
Multiple Random Walks
Introduction
10
Speed-up for Worst-Case Start Vertices
[Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ n, the speed-up is defined as
(1)
maxu∈V Eu min{t : ∪ts=0 Xt = V }
tcov
(k )
=
S =
i
h
(k )
n t
i
tcov
maxu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V }
“Many random walks are faster than one”
Multiple Random Walks
Introduction
10
Speed-up for Worst-Case Start Vertices
[Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ n, the speed-up is defined as
(1)
maxu∈V Eu min{t : ∪ts=0 Xt = V }
tcov
(k )
=
S =
i
h
(k )
n t
i
tcov
maxu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V }
“Many random walks are faster than one”
Questions:
Multiple Random Walks
Introduction
10
Speed-up for Worst-Case Start Vertices
[Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ n, the speed-up is defined as
(1)
maxu∈V Eu min{t : ∪ts=0 Xt = V }
tcov
(k )
=
S =
i
h
(k )
n t
i
tcov
maxu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V }
“Many random walks are faster than one”
Questions:
1. Is S (k ) ≤ k or S (k ) . k ?
2. How small can S (k ) be?
Multiple Random Walks
Introduction
10
Speed-up for Worst-Case Start Vertices
[Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ n, the speed-up is defined as
(1)
maxu∈V Eu min{t : ∪ts=0 Xt = V }
tcov
(k )
=
S =
i
h
(k )
n t
i
tcov
maxu∈V E(u,...,u) min{t : ∪log
i=1 ∪s=0 Xs = V }
“Many random walks are faster than one”
Questions:
1. Is S (k ) ≤ k or S (k ) . k ?
2. How small can S (k ) be?
3. How does S (k ) behave on natural topologies?
4. Can we relate S (k ) to quantities like mixing time or conductance?
Multiple Random Walks
Introduction
10
A Remarkable Example [Efremenko, Reingold]
Consider the following Markov chain (x → ∞):
1−
1
x
1
2x
a
1−
1
x
1−
1
x
c
b
1
2x
1
x
Multiple Random Walks
1
x
Introduction
11
A Remarkable Example [Efremenko, Reingold]
Consider the following Markov chain (x → ∞):
1−
1
x
1
2x
a
1−
1
x
1
x
1−
1
x
c
b
1
2x
1
x
(1)
tcov = Eb min{t : ∪ts=0 Xt = V } = 5x + o(x)
Multiple Random Walks
Introduction
11
A Remarkable Example [Efremenko, Reingold]
Consider the following Markov chain (x → ∞):
1−
1
x
1
2x
a
1−
1
x
1
x
1−
1
x
c
b
1
2x
1
x
(1)
tcov = Eb min{t : ∪ts=0 Xt = V } = 5x + o(x)
Multiple Random Walks
Introduction
11
A Remarkable Example [Efremenko, Reingold]
Consider the following Markov chain (x → ∞):
1−
1
x
1
2x
a
1−
1
x
1
x
1−
1
x
c
b
1
2x
1
x
(1)
tcov = Eb min{t : ∪ts=0 Xt = V } = 5x + o(x)
(2)
tcov = E(a,a) min{t : ∪2i=1 ∪ts=0 Xti = V } = 2.25x + o(x)
Multiple Random Walks
Introduction
11
A Remarkable Example [Efremenko, Reingold]
Consider the following Markov chain (x → ∞):
1−
1
x
1
2x
a
1−
1
x
1
x
1−
1
x
c
b
1
2x
1
x
(1)
tcov = Eb min{t : ∪ts=0 Xt = V } = 5x + o(x)
(2)
tcov = E(a,a) min{t : ∪2i=1 ∪ts=0 Xti = V } = 2.25x + o(x)
S (k ) > k possible!
Multiple Random Walks
Introduction
11
Outline
Introduction
Upper Bounds on the Multiple Cover Time
Lower Bounds on the Multiple Cover Time
Concrete Networks
Conclusion
Multiple Random Walks
Upper Bounds on the Multiple Cover Time
12
Bounds for small k
Baby Matthew Theorem [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ log n,
(k )
tcov ≤
Multiple Random Walks
e + o(1)
k
· thit · Hn .
Upper Bounds on the Multiple Cover Time
13
Bounds for small k
Baby Matthew Theorem [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ log n,
(k )
tcov ≤
e + o(1)
k
· thit · Hn .
For graphs with tcov thit · Hn ⇒ S (k ) & k for 1 ≤ k ≤ log n.
Multiple Random Walks
Upper Bounds on the Multiple Cover Time
13
Bounds for small k
Baby Matthew Theorem [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ log n,
(k )
tcov ≤
e + o(1)
k
· thit · Hn .
For graphs with tcov thit · Hn ⇒ S (k ) & k for 1 ≤ k ≤ log n.
Question: What happens for larger values of k ?
Multiple Random Walks
Upper Bounds on the Multiple Cover Time
13
Bounds for small k
Baby Matthew Theorem [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ log n,
(k )
tcov ≤
e + o(1)
k
· thit · Hn .
For graphs with tcov thit · Hn ⇒ S (k ) & k for 1 ≤ k ≤ log n.
Question: What happens for larger values of k ?
Theorem [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
Let G be any regular graph. Then for any 1 ≤ k ≤ n,
h
i n log2 n · tmix
E(u1 ,...,uk ) min{t : ∪ki=1 ∪ts=0 Xti = V } .
k
Multiple Random Walks
Upper Bounds on the Multiple Cover Time
13
Bounds for small k
Baby Matthew Theorem [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
For any graph G and 1 ≤ k ≤ log n,
(k )
tcov ≤
e + o(1)
k
· thit · Hn .
For graphs with tcov thit · Hn ⇒ S (k ) & k for 1 ≤ k ≤ log n.
Question: What happens for larger values of k ?
Theorem [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
Let G be any regular graph. Then for any 1 ≤ k ≤ n,
h
i n log2 n · tmix
E(u1 ,...,uk ) min{t : ∪ki=1 ∪ts=0 Xti = V } .
k
Since tcov . n log n · tmix , the lower bound on S (k ) will be always o(k )!
Multiple Random Walks
Upper Bounds on the Multiple Cover Time
13
Improved Bound for large k
Theorem
For any graph G. Then for any 1 ≤ k ≤ n,
h
i thit · log n
+ tmix .
E(u1 ,...,uk ) min{t : ∪ki=1 ∪ts=0 Xti = V } .
k
Multiple Random Walks
Upper Bounds on the Multiple Cover Time
14
Improved Bound for large k
Theorem
For any graph G. Then for any 1 ≤ k ≤ n,
h
i thit · log n
+ tmix .
E(u1 ,...,uk ) min{t : ∪ki=1 ∪ts=0 Xti = V } .
k
Single Random Walk:
tmix
u1
?
1
?
2
?
3
?
4
?
5
?
6
2 · thit
0
Multiple Random Walks
Upper Bounds on the Multiple Cover Time
14
t
Improved Bound for large k
Theorem
For any graph G. Then for any 1 ≤ k ≤ n,
h
i thit · log n
+ tmix .
E(u1 ,...,uk ) min{t : ∪ki=1 ∪ts=0 Xti = V } .
k
Single Random Walk:
tmix
u1
?
1
?
2
3
?
?
4
?
5
?
6
2 · thit
0
Multiple Random Walks:
u1
?
1
?
2
u2
?
3
?
4
u3
?
5
?
6
0
2·
thit
k
Multiple Random Walks
t
Upper Bounds on the Multiple Cover Time
14
t
Outline
Introduction
Upper Bounds on the Multiple Cover Time
Lower Bounds on the Multiple Cover Time
Concrete Networks
Conclusion
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
15
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Multiple Random Walks
n
k
Lower Bounds on the Multiple Cover Time
· log n.
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
Multiple Random Walks
2·t
n
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
E[Y ]
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
k
n
2·t
E[Y ] ≥ · 1 −
n
2
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
k
n
2·t
E[Y ] ≥ · 1 −
n
2
1
& n · 4− 8 log n
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
k
n
2·t
E[Y ] ≥ · 1 −
n
2
1
& n · 4− 8 log n
√
n
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
k
n
2·t
E[Y ] ≥ · 1 −
n
2
1
& n · 4− 8 log n
√
n
Method of bounded independent differences yields
1
Pr Y ≥ · E [ Y ] .
2
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
k
n
2·t
E[Y ] ≥ · 1 −
n
2
1
& n · 4− 8 log n
√
n
Method of bounded independent differences yields
!
E [ Y ]2
1
Pr Y ≥ · E [ Y ] . exp −
2
8n
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
k
n
2·t
E[Y ] ≥ · 1 −
n
2
1
& n · 4− 8 log n
√
n
Method of bounded independent differences yields
!
E [ Y ]2
1
Pr Y ≥ · E [ Y ] . exp −
= o(1).
2
8n
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Simple Coupon-Collecting Lower Bound
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Proof:
Let t :=
1
16
0
·
n
k
· log n.
n
k
· log n
Define V := v ∈ V : Pr v ∈ ∪ts=0 Xs ≤
2·t
n
⇒ |V 0 | ≥ 21 n
Let Y be the no. unvisited vertices in V 0 after step t of X1 , X2 , . . . , Xk :
k
n
2·t
E[Y ] ≥ · 1 −
n
2
1
& n · 4− 8 log n
√
n
Method of bounded independent differences yields
!
E [ Y ]2
1
Pr Y ≥ · E [ Y ] . exp −
= o(1).
2
8n
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
16
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
p
4
n/tmix ,
17
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
p
4
n/tmix ,
Theorem [Broder, Karlin]
Let G be a regular graph and let ti be the first time when i vertices are
ti
visited. Then for any 0 ≤ i < n and 0 ≤ j ≤ n − i, u ∈ ∪s=0
Xs ,
Eu [ ti+j − ti ] ≥
Multiple Random Walks
nj
tmix i
1
·
−
− O(1).
2 n−i
n−i
Lower Bounds on the Multiple Cover Time
17
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
p
4
n/tmix ,
Theorem [Broder, Karlin]
Let G be a regular graph and let ti be the first time when i vertices are
ti
visited. Then for any 0 ≤ i < n and 0 ≤ j ≤ n − i, u ∈ ∪s=0
Xs ,
Eu [ ti+j − ti ] ≥
nj
tmix i
1
·
−
− O(1).
2 n−i
n−i
For j & i and tmix . n, time to visit j new states is at least j ·
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
n
n−i
17
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
p
4
n/tmix ,
k walks of length (n/k ) log n ≈ one single walk of length n log n.
Theorem [Broder, Karlin]
Let G be a regular graph and let ti be the first time when i vertices are
ti
visited. Then for any 0 ≤ i < n and 0 ≤ j ≤ n − i, u ∈ ∪s=0
Xs ,
Eu [ ti+j − ti ] ≥
nj
tmix i
1
·
−
− O(1).
2 n−i
n−i
For j & i and tmix . n, time to visit j new states is at least j ·
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
n
n−i
17
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
p
4
n/tmix ,
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
Multiple Random Walks
n
k
Lower Bounds on the Multiple Cover Time
· log n.
17
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
p
4
n/tmix ,
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
n
k
· log n.
Corollary
Let G be a regular graph with tmix ≤ n1− . Then for any 1 ≤ k ≤ n,
(k )
tcov ≥ kn · log n.
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
17
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
p
4
n/tmix ,
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
n
k
· log n.
Corollary
Let G be a regular graph with tmix ≤ n1− . Then for any 1 ≤ k ≤ n,
(k )
tcov ≥ kn · log n.
Unnatural condition
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
17
A Lower Bound for small k
Proposition
Let G be a regular graph with tmix . n1− . Then for any 1 ≤ k ≤
(k )
tcov & kn · log n.
p
4
n/tmix ,
Proposition
(k )
Let G be any graph. Then for any log n ≤ k ≤ n, tcov &
n
k
· log n.
Corollary
Let G be a regular graph with tmix ≤ n1− . Then for any 1 ≤ k ≤ n,
(k )
tcov ≥ kn · log n.
Unnatural condition
(k )
Question: Does tcov &
n
k
Multiple Random Walks
· log n hold for every k and every graph G?
Lower Bounds on the Multiple Cover Time
17
General Bound
Conjecture: Does S (k ) . k hold for every graph G?
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
18
General Bound
Conjecture: Does S (k ) . k hold for every graph G?
Proposition [Efremenko, Reingold]
Let G = (V , E) be any graph. Then for any 1 ≤ k ≤ n, S (k ) . k log n.
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
18
General Bound
Conjecture: Does S (k ) . k hold for every graph G?
Proposition [Efremenko, Reingold]
Let G = (V , E) be any graph. Then for any 1 ≤ k ≤ n, S (k ) . k log n.
Proof:
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
18
General Bound
Conjecture: Does S (k ) . k hold for every graph G?
Proposition [Efremenko, Reingold]
Let G = (V , E) be any graph. Then for any 1 ≤ k ≤ n, S (k ) . k log n.
Proof:
k
By definition of tcov
, for all pairs u, v ∈ V ,
(k )
1
2tcov
Xs ≥
Pru v ∈ ∪s=0
4k
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
18
General Bound
Conjecture: Does S (k ) . k hold for every graph G?
Proposition [Efremenko, Reingold]
Let G = (V , E) be any graph. Then for any 1 ≤ k ≤ n, S (k ) . k log n.
Proof:
k
By definition of tcov
, for all pairs u, v ∈ V ,
(k )
1
2tcov
Xs ≥
Pru v ∈ ∪s=0
4k
(k )
By the Markov property, thit ≤ 4k · (2 · tcov ),
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
18
General Bound
Conjecture: Does S (k ) . k hold for every graph G?
Proposition [Efremenko, Reingold]
Let G = (V , E) be any graph. Then for any 1 ≤ k ≤ n, S (k ) . k log n.
Proof:
k
By definition of tcov
, for all pairs u, v ∈ V ,
(k )
1
2tcov
Xs ≥
Pru v ∈ ∪s=0
4k
(k )
(k )
By the Markov property, thit ≤ 4k · (2 · tcov ), and so tcov . k log n · tcov
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
18
General Bound
Conjecture: Does S (k ) . k hold for every graph G?
Proposition [Efremenko, Reingold]
Let G = (V , E) be any graph. Then for any 1 ≤ k ≤ n, S (k ) . k log n.
Proof:
k
By definition of tcov
, for all pairs u, v ∈ V ,
(k )
1
2tcov
Xs ≥
Pru v ∈ ∪s=0
4k
(k )
(k )
By the Markov property, thit ≤ 4k · (2 · tcov ), and so tcov . k log n · tcov
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
18
Bottlenecks
Conductance
Φ(G) :=
min
∅(S⊆V :
vol(S)≤|E|
|E(S, V \ S)|
vol(S)
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
19
Bottlenecks
Conductance
Φ(G) :=
min
∅(S⊆V :
vol(S)≤|E|
|E(S, V \ S)|
vol(S)
Multiple Random Walks
(P
=2·
min
u∈S,v 6∈S
∅(S⊆V :
πS ≤1/2
Lower Bounds on the Multiple Cover Time
πu · Pu,v
πS
19
)
Bottlenecks
Conductance
Φ(G) :=
min
∅(S⊆V :
vol(S)≤|E|
|E(S, V \ S)|
vol(S)
Multiple Random Walks
(P
=2·
min
u∈S,v 6∈S
∅(S⊆V :
πS ≤1/2
Lower Bounds on the Multiple Cover Time
πu · Pu,v
πS
19
)
Bottlenecks
Conductance
Φ(G) :=
Φ
min
∅(S⊆V :
vol(S)≤|E|
|E(S, V \ S)|
vol(S)
(P
=2·
min
u∈S,v 6∈S
∅(S⊆V :
πS ≤1/2
πu · Pu,v
πS
1
√
n
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
19
)
Bottlenecks
Conductance
Φ(G) :=
Φ
min
∅(S⊆V :
vol(S)≤|E|
|E(S, V \ S)|
vol(S)
1
√
n
Φ
Multiple Random Walks
(P
=2·
min
u∈S,v 6∈S
∅(S⊆V :
πS ≤1/2
πu · Pu,v
πS
1
n
Lower Bounds on the Multiple Cover Time
19
)
Bottlenecks
Conductance
Φ(G) :=
Φ
min
∅(S⊆V :
vol(S)≤|E|
|E(S, V \ S)|
vol(S)
1
√
n
Φ
Multiple Random Walks
(P
=2·
min
u∈S,v 6∈S
∅(S⊆V :
πS ≤1/2
1
n
πu · Pu,v
πS
Φ
Lower Bounds on the Multiple Cover Time
1
n2
19
)
Bottlenecks
Conductance
Φ(G) :=
Φ
min
∅(S⊆V :
vol(S)≤|E|
|E(S, V \ S)|
vol(S)
1
√
n
Φ
(P
=2·
min
u∈S,v 6∈S
∅(S⊆V :
πS ≤1/2
1
n
πu · Pu,v
)
πS
Φ
1
n2
Some bottlenecks slow down single and multiple random walks, others
only affect multiple random walks.
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
19
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
There are |S| ≤ n/2 and u ∈ S s.t. Pru ∪ts=0 Xs ⊆ S ≥ 1 − t ·
u
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
Φ
2
S
V \S
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
There are |S| ≤ n/2 and u ∈ S s.t. Pru ∪ts=0 Xs ⊆ S ≥ 1 − t ·
q
Start all k walks from u and let t = 12 · kn · Φ1
u
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
Φ
2
S
V \S
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
There are |S| ≤ n/2 and u ∈ S s.t. Pru ∪ts=0 Xs ⊆ S ≥ 1 − t ·
q
Start all k walks from u and let t = 12 · kn · Φ1
Φ
2
The probability that a random walk leaves S is
r
r
1
n 1 Φ
1
n
≤ ·
· ·
= ·
·Φ
2
k Φ 2
4
k
u
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
S
V \S
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
There are |S| ≤ n/2 and u ∈ S s.t. Pru ∪ts=0 Xs ⊆ S ≥ 1 − t ·
q
Start all k walks from u and let t = 12 · kn · Φ1
Φ
2
The probability that a random walk leaves S is
r
r
1
n 1 Φ
1
n
≤ ·
· ·
= ·
·Φ
2
k Φ 2
4
k
By Markov, w.p. ≥
1
2
the no. visited nodes in S is
Multiple Random Walks
u
Lower Bounds on the Multiple Cover Time
S
V \S
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
There are |S| ≤ n/2 and u ∈ S s.t. Pru ∪ts=0 Xs ⊆ S ≥ 1 − t ·
q
Start all k walks from u and let t = 12 · kn · Φ1
Φ
2
The probability that a random walk leaves S is
r
r
1
n 1 Φ
1
n
≤ ·
· ·
= ·
·Φ
2
k Φ 2
4
k
By Markov, w.p. ≥
≤ 2k ·
1
·
2
1
2
r
the no. visited nodes in S is
!
r
n
1
n 1
·Φ ·
·
·
k
4
k Φ
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
u
S
V \S
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
There are |S| ≤ n/2 and u ∈ S s.t. Pru ∪ts=0 Xs ⊆ S ≥ 1 − t ·
q
Start all k walks from u and let t = 12 · kn · Φ1
Φ
2
The probability that a random walk leaves S is
r
r
1
n 1 Φ
1
n
≤ ·
· ·
= ·
·Φ
2
k Φ 2
4
k
By Markov, w.p. ≥
≤ 2k ·
1
·
2
1
2
r
the no. visited nodes in S is
!
r
n
1
n 1
n
·Φ ·
·
·
=
k
4
k Φ
4
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
u
S
V \S
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Proof:
There are |S| ≤ n/2 and u ∈ S s.t. Pru ∪ts=0 Xs ⊆ S ≥ 1 − t ·
q
Start all k walks from u and let t = 12 · kn · Φ1
Φ
2
The probability that a random walk leaves S is
r
r
1
n 1 Φ
1
n
≤ ·
· ·
= ·
·Φ
2
k Φ 2
4
k
By Markov, w.p. ≥
≤ 2k ·
1
·
2
1
2
r
the no. visited nodes in S is
!
r
n
1
n 1
n
·Φ ·
·
·
= < |S|.
k
4
k Φ
4
Multiple Random Walks
Lower Bounds on the Multiple Cover Time
u
S
V \S
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Theorem
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
(k )
tcov
.
thit · log n
Multiple Random Walks
k
n·
+ tmix .
log2 n
Φ2
k
+
log n
Φ
Lower Bounds on the Multiple Cover Time
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Theorem
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
(k )
tcov
.
thit · log n
k
n·
+ tmix .
log2 n
Φ2
k
+
log n
Φ
Corollary
r
Multiple Random Walks
log2 n
1
(n)
. tcov .
Φ
Φ2
Lower Bounds on the Multiple Cover Time
20
Relating Conductance to Multiple Cover Time
Proposition
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
r
n 1
(k )
tcov &
· .
k Φ
Theorem
Let G = (V , E) be any graph and 1 ≤ k ≤ n. Then,
(k )
tcov
Corollary
.
thit · log n
k
n·
+ tmix .
log2 n
Φ2
k
+
log n
Φ
Cover time of n multiple random walks captures rapid mixing
r
Multiple Random Walks
log2 n
1
(n)
. tcov .
Φ
Φ2
Lower Bounds on the Multiple Cover Time
20
Outline
Introduction
Upper Bounds on the Multiple Cover Time
Lower Bounds on the Multiple Cover Time
Concrete Networks
Conclusion
Multiple Random Walks
Concrete Networks
21
1D Grid
tmix n2
thit n2
tcov n
2
Multiple Random Walks
2D Grid
tmix n
thit n log n
tcov n log2 n
Concrete Networks
3D Grid
tmix n2/3
thit n
tcov n log n
22
1D Grid
tmix n2
thit n2
tcov n
2
Hypercube
tmix log n log log n
thit n
2D Grid
tmix n
thit n log n
Multiple Random Walks
tmix n2/3
thit n
tcov n log2 n
tcov n log n
Expander Graph
Binary Tree
tmix log n
thit n
tcov n log n
3D Grid
tmix n
thit n log n
tcov n log n
tcov n log2 n
Concrete Networks
22
Speed-up for Expander Graphs
S (k )
n
1
Multiple Random Walks
n
Concrete Networks
k
23
Speed-up for Expander Graphs
S (k )
n
1
(k )
tcov .
thit ·log n
k
n
k
+ tmix
Multiple Random Walks
Concrete Networks
23
Speed-up for Expander Graphs
S (k )
n
1
(k )
tcov .
thit ·log n
k
+ tmix .
n
k
n·log n
k
Multiple Random Walks
Concrete Networks
23
Speed-up for Expander Graphs
S (k )
n
1
(k )
thit ·log n
k
(k )
n
k
tcov .
tcov &
+ tmix .
n
k
n·log n
k
· log n (since tmix . log n log log n)
Multiple Random Walks
Concrete Networks
23
Speed-up for Hypercubes
S (k )
n
n
log log n
n
log log n
1
Multiple Random Walks
n
Concrete Networks
k
24
Speed-up for Hypercubes
S (k )
n
n
log log n
n
log log n
1
(k )
tcov .
thit ·log n
k
n
k
+ tmix
Multiple Random Walks
Concrete Networks
24
Speed-up for Hypercubes
S (k )
n
n
log log n
n
log log n
1
(k )
tcov .
thit ·log n
k
+ tmix .
n·log n
k
Multiple Random Walks
n
k
+ log n log log n
Concrete Networks
24
Speed-up for Hypercubes
S (k )
n
n
log log n
n
log log n
1
tcov .
thit ·log n
k
(k )
tcov
n
k
(k )
&
+ tmix .
n·log n
k
n
k
+ log n log log n
· log n (since tmix . log n log log n)
Multiple Random Walks
Concrete Networks
24
Speed-up for Hypercubes
S (k )
n
n
log log n
n
log log n
1
n
k
tcov .
thit ·log n
k
(k )
tcov
&
n
k
(n)
tcov
& log n log log n (due to coupon-collecting argument)
(k )
+ tmix .
n·log n
k
+ log n log log n
· log n (since tmix . log n log log n)
Multiple Random Walks
Concrete Networks
24
Speed-up for Hypercubes
S (k )
n
n
log log n
n
log log n
1
n
k
tcov .
thit ·log n
k
(k )
tcov
&
n
k
(n)
tcov
& log n log log n (due to coupon-collecting argument)
(k )
+ tmix .
n·log n
k
n1+
+ log n log log n
· log n (since tmix . log n log log n)
Multiple Random Walks
Concrete Networks
24
Speed-up for Cycles
S (k )
n
log n
1
Multiple Random Walks
n
Concrete Networks
k
25
Speed-up for Cycles
S (k )
n
log n
1
n
k
Direct Analysis
[Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
Multiple Random Walks
Concrete Networks
25
Speed-up for 2D Grids
S (k )
n
log3 n
log2 n
1 log2 n
Multiple Random Walks
n
Concrete Networks
k
26
Speed-up for 2D Grids
S (k )
n
log3 n
log2 n
1 log2 n
(k )
tcov .
thit ·log n
k
n
k
+ tmix
Multiple Random Walks
Concrete Networks
26
Speed-up for 2D Grids
S (k )
n
log3 n
log2 n
1 log2 n
(k )
tcov .
thit ·log n
k
+ tmix .
n·log2 n
k
Multiple Random Walks
n
k
+n
Concrete Networks
26
Speed-up for 2D Grids
S (k )
n
log3 n
log2 n
1 log2 n
tcov .
(k )
thit ·log n
k
(n)
tcov
diam(G)2
log n
&
+ tmix .
n·log2 n
k
Multiple Random Walks
n
k
+n
[Carne]
Concrete Networks
26
Speed-up for 2D Grids
S (k )
n
log3 n
log2 n
1 log2 n
tcov .
(k )
thit ·log n
k
(n)
tcov
diam(G)2
log n
&
+ tmix .
=
n·log2 n
k
n
+n
n
log n
Multiple Random Walks
k
[Carne]
Concrete Networks
26
Speed-up for 3D Grids
S (k )
n
n1/3 log2 n
n1/3 log n
1
Multiple Random Walks
n1/3 log n
Concrete Networks
k
n
27
Speed-up for 3D Grids
S (k )
n
n1/3 log2 n
n1/3 log n
1
(k )
tcov .
thit ·log n
k
n1/3 log n
k
n
+ tmix
Multiple Random Walks
Concrete Networks
27
Speed-up for 3D Grids
S (k )
n
n1/3 log2 n
n1/3 log n
n1/3 log n
1
(k )
tcov .
thit ·log n
k
+ tmix .
n·log n
k
Multiple Random Walks
k
n
+ n2/3
Concrete Networks
27
Speed-up for 3D Grids
S (k )
n
n1/3 log2 n
n1/3 log n
n1/3 log n
1
tcov .
thit ·log n
k
(k )
tcov
n
k
(k )
&
+ tmix .
n·log n
k
k
n
+ n2/3
· log n (since tmix . n2/3 )
Multiple Random Walks
Concrete Networks
27
Speed-up for 3D Grids
S (k )
n
n1/3 log2 n
n1/3 log n
n1/3 log n
1
tcov .
thit ·log n
k
(k )
tcov
n
k
(k )
(n)
&
tcov &
+ tmix .
n·log n
k
k
n
+ n2/3
· log n (since tmix . n2/3 )
diam(G)2
log n
Multiple Random Walks
Concrete Networks
27
Speed-up for 3D Grids
S (k )
n
n1/3 log2 n
n1/3 log n
n1/3 log n
1
tcov .
thit ·log n
k
(k )
tcov
n
k
(k )
(n)
&
tcov &
+ tmix .
n·log n
k
k
n
+ n2/3
· log n (since tmix . n2/3 )
diam(G)2
log n
=
n2/3
log n
Multiple Random Walks
Concrete Networks
27
Speed-up for Vertex Transitive Graphs
S (k )
n
tcov
diam2
tcov
diam2
· log n
·
1
d log n
1
Multiple Random Walks
tcov
diam2
Concrete Networks
k
n
28
Speed-up for Vertex Transitive Graphs
S (k )
n
tcov
diam2
tcov
diam2
· log n
·
1
d log n
1
(k )
tcov .
thit ·log n
k
tcov
diam2
k
n
+ tmix
Multiple Random Walks
Concrete Networks
28
Speed-up for Vertex Transitive Graphs
S (k )
n
tcov
diam2
tcov
diam2
· log n
·
1
d log n
1
(k )
tcov .
S
(k )
thit ·log n
k
+ tmix .
tcov ·log n
k
tcov
diam2
+ d diam2 log n
k
n
[Babai]
= O(k log n)
Multiple Random Walks
Concrete Networks
28
Speed-up for Vertex Transitive Graphs
S (k )
n
tcov
diam2
tcov
diam2
· log n
·
1
d log n
1
(k )
tcov .
S
(k )
(n)
tcov
thit ·log n
k
+ tmix .
tcov ·log n
k
tcov
diam2
+ d diam2 log n
k
n
[Babai]
= O(k log n)
&
diam2
log n
[Carne]
Multiple Random Walks
Concrete Networks
28
Binary Trees
S (k )
n
√
n log2 n
log2 n
1 log2 n
Multiple Random Walks
n
Concrete Networks
k
29
Binary Trees
S (k )
n
√
n log2 n
log2 n
1 log2 n
(k )
tcov .
thit ·log n
k
n
k
+ tmix
Multiple Random Walks
Concrete Networks
29
Binary Trees
S (k )
n
√
n log2 n
log2 n
1 log2 n
(k )
tcov .
thit ·log n
k
+ tmix .
n·log2 n
k
Multiple Random Walks
n
k
+n
Concrete Networks
29
Binary Trees
S (k )
n
√
n log2 n
log2 n
1 log2 n
(k )
tcov .
(k )
tcov
&
thit ·log n
k
q
n
k
·
+ tmix .
1
Φ(G)
=
n·log2 n
k
n
k
+n
n
√
k
Multiple Random Walks
Concrete Networks
29
Binary Trees
S (k )
n
√
n log2 n
log2 n
1 log2 n
(k )
tcov .
(k )
tcov
(k )
&
tcov .
thit ·log n
k
q
n
√
k
n
k
·
+ tmix .
1
Φ(G)
=
n·log2 n
k
n
k
+n
n
√
k
· log5 n (specific analysis)
Multiple Random Walks
Concrete Networks
29
Binary Trees vs. 3D Grids
S (k )
S (k )
n
n
√
n
1/3
n log2 n
2
log n
n1/3 log n
1
n1/3 log n
Multiple Random Walks
n
k
log2 n
Concrete Networks
1 log2 n
n
30
k
Outline
Introduction
Upper Bounds on the Multiple Cover Time
Lower Bounds on the Multiple Cover Time
Concrete Networks
Conclusion
Multiple Random Walks
Conclusion
31
Summary and Open Problems
General Conjectures [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
S (k ) & log k
S (k ) . k
Multiple Random Walks
Conclusion
32
Summary and Open Problems
General Conjectures [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
S (k ) & log k
S (k ) . k
Both conjectures hold in the router-router model (a.k.a.
Propp machine)! [Dereniowski, Kosowski, Pajak, Uznanski]
Multiple Random Walks
Conclusion
32
Summary and Open Problems
General Conjectures [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
S (k ) & log k
S (k ) . k
Both conjectures hold in the router-router model (a.k.a.
Propp machine)! [Dereniowski, Kosowski, Pajak, Uznanski]
Conjecture [Efremenko, Reingold]
Maximum cover time of k multiple walks is attained for an appropriate
start vertex common to all k walks.
Multiple Random Walks
Conclusion
32
Summary and Open Problems
General Conjectures [Alon, Avin, Koucký, Kozma, Lotker and Tuttle]
S (k ) & log k
S (k ) . k
Both conjectures hold in the router-router model (a.k.a.
Propp machine)! [Dereniowski, Kosowski, Pajak, Uznanski]
Conjecture [Efremenko, Reingold]
Maximum cover time of k multiple walks is attained for an appropriate
start vertex common to all k walks.
Load Balancing
tight analysis of a natural protocol based on multiple random walks
load tokens
negatively correlated multiple random walks
qualitative relationship based on hitting-set probabilities
Multiple Random Walks
Conclusion
32
The End
Further Directions
Can we exploit multiple cover time for clustering applications?
Can we extend our results to random walks of varying lengths?
What happens on dynamic graphs?
Multiple Random Walks
Conclusion
33
The End
Further Directions
Can we exploit multiple cover time for clustering applications?
Can we extend our results to random walks of varying lengths?
What happens on dynamic graphs?
Multiple Random Walks
Conclusion
33