1. Mixing Time in Robotic Explorations
Chang He 1
and Shun Yang 2
1
Department of Mathematics, Centre College 2
Department of Mathematics, Carleton College
Introduction
We use a Markov model to evaluate the mixing time of a point robot
R that navigates autonomously by randomly moving in parallel to
the coordinate axes in an open domain B, which is a collection of
rectangular rooms having internal walls restricting the motion of the
robot, as shown in Figure 1. We call such domain a gallery. For a
given gallery with s sides and a bottleneck ratio of , we find that the
mixing time for the robot to explore the entire gallery has an upper
bound of O(s2
/ ). We also give exact bounds of the mixing time in
some special gallery configurations.
Model and Definitions
Figure 1: A possible path of a robot in a well-defined gallery
Motion of the Robot: Each step the robot takes consists of a pair
of ordered moves (hi, vi). Define the horizontal move hi as a mapping
from point p → p , where p is a random point on the largest
continuous horizontal segment in the gallery containing the starting
point p. The vertical move vi is determined in the same manner. We
use a finite Markov chain to model this process with the gallery as the
state space and each rectangular room as a state. We denote the
corresponding transition matrix by P.
Mixing Time tmix: Mixing time is the time it takes for a Markov
Chain to get close to its stationary distribution. Let
d(t) := max
x∈Ω
||Pt
(x, ·) − π||TV , where · TV is the total variation
distance between the long-term distribution of P and its stationary
distribution π. The mixing time is defined as
tmix(δ) := min{t : d(t) ≤ δ}, where δ is a substantially small value.[1]
Coupling Time tcoup: The coupling of two Markov chains Xt, Yt
with X0 = x and Y0 = y but the same transition matrix P is defined
as the process (Xt, Yt)∞
t=0. We define coupling time,
tcoup := min{t : Xt = Yt}, as the first time the two chains meet, and
after which the two chains have exactly the same trajectories.[1]
Relaxation Time trel: Let λ2 be the second largest eigenvalue of P,
i.e. λ2 = max{|λ| : λ = 1}, then relaxation time trel := 1
1−λ2
. We will
use the fact that tmix is bounded from above by trel.[1]
Comb Gallery and Snake Gallery
ε
Figure 2: A 6-comb gallery
ε
Figure 3: A 6-snake gallery
We consider galleries with parallel internal walls that have same-size
bottleneck openings. The comb gallery has all openings on internal
walls aligned with no overlapping in height; the snake gallery has no
two adjacent openings aligned. We make the additional assumption
that the first and last room of a snake gallery are directly connected
by an external bottleneck for it to be called ouroboric.
Comb Gallery: The transition matrix P for the comb gallery can
be written as
P = (1 − ) · I + /N · J, (1)
where I is the identity matrix and J is the matrix of all ones. The
second largest eigenvalues for P is 1 − , meaning that
trel = O(1/ ), and so is tmix.
Ouroboric Snake: The block transition matrix Pn for the nth
room is in the form of
Pn =
n − 1 n n + 1
n /2 1 − /2
. (2)
Since P is circulant, the kth
eigenvector rk has the form [3]
rk = c · (1, e−2πik/N
, . . . , e−2πik(N−1)/N
) , (3)
where k = 0, 1, . . . , N − 1 and c is a constant. Then
trel = N2
/8π2
= O(N2
/ ), and so is tmix.
Non-ouroboric Snake: We introduce a random point Y into
the model that evolves through the same Markov chain as the robot
does and design a coupling algorithm that matches the height of Y
and R. The coupling time can be approximated through the
recurrence relation
t(n + 1) − 2t(n) + t(n − 1) + 1/ = 0, (4)
with boundary conditions
t(0) = t(1) + 2/ , t(N) = 0. (5)
Therefore, tcoup = O(N2
/ ), and so is tmix.
Lego Gallery
We also consider an N-Lego gallery that has N connected rectangular
rooms with the same size bottleneck openings on all internal walls.
Figure 4: A 5-Lego gallery
We represent the N-Lego gallery as a non-directed finite graph G =
(V, E) with no loops nor multiple edges and with edge connectivity
e(G). Denote a(G) as the second smallest eigenvalue of the corre-
sponding Laplacian matrix. From Fiedler(1973), [2]
2e(G)(1 − cos(π/N)) ≤ a(G) ≤ N. (6)
The second largest eigenvalue for transition matrix P is
λ2 = 1 − q · a(G), (7)
where q ∈ (0, 1). Therefore, from equation (6) we know that
1
qN
≤ trel ≤
1
a e(G)(1 − cos(π/N))
≤
2N2
q π2
, (8)
that is, tmix for an N-Lego gallery is bounded between O(1/N ) and
O(N2
/ ).
General Gallery
For any well-defined gallery, we design an algorithm of dissecting the
gallery and transform it into a Lego gallery to give an upper bound of
tmix. We find that for any gallery with s sides, its tmix = O(s2
/ ).
References
[1] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov Chains and Mixing Time,
American Mathematical Soc, 2009.
[2] Miroslav Fiedler, Algebraic Connectivity of Graphs, Czechoslovak Math. J. 23(98): 298 - 305, 1973.
[3] P. J. Davis, Circulant Matrices, Wiley-Interscience, NY, 1979.
Acknowledgements
This project is the result of a 2015 summer research conducted at ICERM (Institute for Computational and
Experimental Research in Mathematics)/ Brown University under the direction of Professor Yuliy Barysh-
nikov and Professor Maxim Arnold. All remaining errors are our own. Please contact chang.he@centre.edu
and yangf2@carleton.edu for any questions.
![Mixing Time in Robotic Explorations
Chang He 1
and Shun Yang 2
1
Department of Mathematics, Centre College 2
Department of Mathematics, Carleton College
Introduction
We use a Markov model to evaluate the mixing time of a point robot
R that navigates autonomously by randomly moving in parallel to
the coordinate axes in an open domain B, which is a collection of
rectangular rooms having internal walls restricting the motion of the
robot, as shown in Figure 1. We call such domain a gallery. For a
given gallery with s sides and a bottleneck ratio of , we find that the
mixing time for the robot to explore the entire gallery has an upper
bound of O(s2
/ ). We also give exact bounds of the mixing time in
some special gallery configurations.
Model and Definitions
Figure 1: A possible path of a robot in a well-defined gallery
Motion of the Robot: Each step the robot takes consists of a pair
of ordered moves (hi, vi). Define the horizontal move hi as a mapping
from point p → p , where p is a random point on the largest
continuous horizontal segment in the gallery containing the starting
point p. The vertical move vi is determined in the same manner. We
use a finite Markov chain to model this process with the gallery as the
state space and each rectangular room as a state. We denote the
corresponding transition matrix by P.
Mixing Time tmix: Mixing time is the time it takes for a Markov
Chain to get close to its stationary distribution. Let
d(t) := max
x∈Ω
||Pt
(x, ·) − π||TV , where · TV is the total variation
distance between the long-term distribution of P and its stationary
distribution π. The mixing time is defined as
tmix(δ) := min{t : d(t) ≤ δ}, where δ is a substantially small value.[1]
Coupling Time tcoup: The coupling of two Markov chains Xt, Yt
with X0 = x and Y0 = y but the same transition matrix P is defined
as the process (Xt, Yt)∞
t=0. We define coupling time,
tcoup := min{t : Xt = Yt}, as the first time the two chains meet, and
after which the two chains have exactly the same trajectories.[1]
Relaxation Time trel: Let λ2 be the second largest eigenvalue of P,
i.e. λ2 = max{|λ| : λ = 1}, then relaxation time trel := 1
1−λ2
. We will
use the fact that tmix is bounded from above by trel.[1]
Comb Gallery and Snake Gallery
ε
Figure 2: A 6-comb gallery
ε
Figure 3: A 6-snake gallery
We consider galleries with parallel internal walls that have same-size
bottleneck openings. The comb gallery has all openings on internal
walls aligned with no overlapping in height; the snake gallery has no
two adjacent openings aligned. We make the additional assumption
that the first and last room of a snake gallery are directly connected
by an external bottleneck for it to be called ouroboric.
Comb Gallery: The transition matrix P for the comb gallery can
be written as
P = (1 − ) · I + /N · J, (1)
where I is the identity matrix and J is the matrix of all ones. The
second largest eigenvalues for P is 1 − , meaning that
trel = O(1/ ), and so is tmix.
Ouroboric Snake: The block transition matrix Pn for the nth
room is in the form of
Pn =
n − 1 n n + 1
n /2 1 − /2
. (2)
Since P is circulant, the kth
eigenvector rk has the form [3]
rk = c · (1, e−2πik/N
, . . . , e−2πik(N−1)/N
) , (3)
where k = 0, 1, . . . , N − 1 and c is a constant. Then
trel = N2
/8π2
= O(N2
/ ), and so is tmix.
Non-ouroboric Snake: We introduce a random point Y into
the model that evolves through the same Markov chain as the robot
does and design a coupling algorithm that matches the height of Y
and R. The coupling time can be approximated through the
recurrence relation
t(n + 1) − 2t(n) + t(n − 1) + 1/ = 0, (4)
with boundary conditions
t(0) = t(1) + 2/ , t(N) = 0. (5)
Therefore, tcoup = O(N2
/ ), and so is tmix.
Lego Gallery
We also consider an N-Lego gallery that has N connected rectangular
rooms with the same size bottleneck openings on all internal walls.
Figure 4: A 5-Lego gallery
We represent the N-Lego gallery as a non-directed finite graph G =
(V, E) with no loops nor multiple edges and with edge connectivity
e(G). Denote a(G) as the second smallest eigenvalue of the corre-
sponding Laplacian matrix. From Fiedler(1973), [2]
2e(G)(1 − cos(π/N)) ≤ a(G) ≤ N. (6)
The second largest eigenvalue for transition matrix P is
λ2 = 1 − q · a(G), (7)
where q ∈ (0, 1). Therefore, from equation (6) we know that
1
qN
≤ trel ≤
1
a e(G)(1 − cos(π/N))
≤
2N2
q π2
, (8)
that is, tmix for an N-Lego gallery is bounded between O(1/N ) and
O(N2
/ ).
General Gallery
For any well-defined gallery, we design an algorithm of dissecting the
gallery and transform it into a Lego gallery to give an upper bound of
tmix. We find that for any gallery with s sides, its tmix = O(s2
/ ).
References
[1] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov Chains and Mixing Time,
American Mathematical Soc, 2009.
[2] Miroslav Fiedler, Algebraic Connectivity of Graphs, Czechoslovak Math. J. 23(98): 298 - 305, 1973.
[3] P. J. Davis, Circulant Matrices, Wiley-Interscience, NY, 1979.
Acknowledgements
This project is the result of a 2015 summer research conducted at ICERM (Institute for Computational and
Experimental Research in Mathematics)/ Brown University under the direction of Professor Yuliy Barysh-
nikov and Professor Maxim Arnold. All remaining errors are our own. Please contact chang.he@centre.edu
and yangf2@carleton.edu for any questions.](https://image.slidesharecdn.com/cf7d561d-6eea-4d60-a6c1-2c6fdcba1fb9-160424035111/85/mixingtimeposter-1-320.jpg)
