1
Detection of Intelligent Mobile Target in a
Mobile Sensor Network
Jren-chit Chin
Yu Dong
Wing-Kai Hon
Abstract—We study the problem of a mobile target (the mouse)
trying to evade detection by one or more mobile sensors (we call
such a sensor a cat) in a closed network area. We view our problem
as a game between two players: the mouse, and the collection
of cats forming a single (meta-)player. The game ends when the
mouse falls within the sensing range of one or more cats. A cat
tries to determine its optimal strategy to minimize the worst case
expected detection time of the mouse. The mouse tries to determine
an optimal counter movement strategy to maximize the expected
detection time. We divide the problem into two cases based on the
relative sensing capabilities of the cats and the mouse. When the
mouse has a sensing range smaller than or equal to the cats’, we
develop a dynamic programming solution for the mouse’s optimal
strategy, assuming high level information about the cats’ movement
model. We discuss how the cats’ chosen movement model will affect
its presence matrix in the network, and hence its payoff in the
game. When the mouse has a larger sensing range than the cats, we
show how the mouse can determine its optimal movement strategy
based on local observations of the cats’ movements. We further
present a coordination protocol for the cats to collaboratively
catch the mouse by (1) forming opportunistically a cohort to
limit the mouse’s degree of freedom in escaping detection, and
(2) minimizing the overlap in the spatial coverage of the cohort’s
members. Extensive experimental results verify and illustrate the
analytical results, and evaluate the game’s payoffs as a function
of several important system parameters.
I. Introduction
Consider a strategic zone belonging to country C
and bordering with another untrusted country C ′ . For
its national safety, it is important for C to monitor the
zone for the presence of any intruder from C ′ . However,
because the strategic zone is vast and sometimes spans
difficult terrains, it is impractical for C to install an
expansive static sensor network covering the whole area,
and ensure that all the sensors are operational and
functioning according to the planning stage. Instead,
with advances in robotics and unmanned aerial vehicles
(UAVs) [11], it will be feasible for the zone to be
monitored by a balance of mobile robots, UAVs, patrol
vehicles, etc, according to the deployment conditions. It
is then important to control and coordinate the patrol
routes in order to achieve effective area monitoring.
We consider the use of a group of surveillance sensors,
under possible coordination, to secure a network area
Research was supported in part by the U.S. Oak Ridge National Lab/Office of Naval Research under grant number DE-AC0500OR22725, in part by the U.S. National Science Foundation under
grant number CNS-0305496, in part by an IBM Fellowship awarded
to Y. Dong, and in part by a Purdue Research Foundation Fellowship
awarded to J. C. Chin.
Chris Y. T. Ma
David K. Y. Yau
against one or more intruders. An intruder (e.g., an enemy vehicle) may lurk in the area to prepare for damage
activities or gather intelligence. It may also be able to
sense its environment or plan its movement to avoid
detection. Similarly, the surveillance sensors may be
mobile. For example, they are carried by robots, UAVs,
or a platoon on patrol schedules. Each sensor will have a
sensing range enabling it to detect an intruder within the
range. By moving, the sensors may be able to efficiently
cover the network area over time, although they do
not have sufficient density to ensure complete coverage
all the time. The sensors may move independently or
in a coordinated manner. The movement may also be
either deterministic or randomized. In particular, stochastic movement is effective in overcoming unforeseen or
probabilistic events in the operating environment, such as
the failure of another sensor, or the unexpected presence
of obstacles in the surveillance area.
In this paper, we model and analyze the game between
the sensors and the intruders. The sensors plan their
movement to detect the intruders as soon as possible,
in order to initiate timely response against the presence
of the intruders. The sighting of an intruder could be
reported by a wide-area cellular or 802.16 network
infrastructure, and the response action could be in the
form of raising the alert level, or sending in troops to
destroy/capture the intruders, although such actuation
issues are not explicitly considered in the paper. The
intruders, on the other hand, plan their movement to
avoid detection for as long as possible, in order to
prolong their own mission. In addition, an intruder itself
may have a sensing range allowing it to see the presence
or movement of the sensors without being detected. We
analyze the best movement strategies for both the mobile
intruders (we call such an intruder a mouse) and the
mobile surveillance sensors (we call such a sensor a cat),
under different conditions in a closed network area. We
assume that the network area is of uniform interest to a
mouse, and use the time until detection as the primary
performance metric of interest.
Our contributions in the paper are as follows:
• For a blind mouse whose sensing range is smaller
than or equal to the cats’, we develop an optimal
dynamic program for the mouse to maximize its
expected detection time, given statistical knowl-
2
•
•
•
edge about the cats’ movements in the form of
a presence matrix. We also discuss how the cats
can optimize their presence matrix to minimize
the expected detection time, assuming that each
position is equally likely to be the mouse’s starting
position. The optimal cat and mouse strategies are
in Nash equilibrium.
For a seeing mouse having a larger sensing range
than that of the cats, we show how the mouse can
use its local observations of the cats’ movements to
maximize the expected detection time.
We show how a network of cats, in playing against
the seeing mouse, can coordinate their movement
to maximize their ability to catch the mouse. First,
we show that two cats who fall within each other’s
sensing range can attempt to minimize the overlap
in spatial coverage by moving away from each
other, and that such a strategy can reduce the
detection time. Second, if the cats can additionally
communicate within a wireless range, the cats can
opportunistically form a cohort to minimize the
mouse’s degree of freedom in escaping, while maximizing the barrier coverage by the cohort members.
We show that the communication-enabled coordination protocol can perform significantly better than
the approach enabled by the sensing only.
We present extensive experimental results to evaluate how the detection time can be impacted by the
players’ strategies.
II. Problem Formulation
We study the problem of Nm mice trying to evade
detection by Nc cats in a closed network area. The
network is modeled as an X × Y rectangular region,
where X and Y are in distance units. When there are
multiple mice (i.e., Nm > 1), we assume that each
mouse will try its best to escape detection, independent
of the actions by the other mice. Hence, without loss
of generality, we will consider the case of a single
mouse. We will view our problem as a game between two
players: the mouse, and the collection of all cats forming
a single (meta-)player. The mouse has a sensing range of
Rm and a speed of Vm . For simplicity, we will assume
that each cat has a sensing range of Rc and a speed of
Vc . The assumption can be easily relaxed to include the
case when different cats have different sensing ranges
and different speeds of movement. The game ends when
the mouse falls within the sensing range of one or more
cats.
Given that the mouse is initially located at position m,
the expected detection time of the mouse is denoted by
detect
E[Tm
], which is also the mouse’s payoff in the game.
For the cats, they do not know the mouse’s initial position, so that they simply assume that the mouse’s initial
position is uniformly distributed in the network. Hence,
the cats try to minimize the “hypothetical” expected
payoff of the mouse; i.e., the game’s payoff for the cats is
detect
−Ê[E[Tm
]], where Ê denotes the expectation under
the above assumption. Notice that because the initial
position of the mouse is known to the mouse, but not
to the cats, the game is not a simple zero-sum game. A
player’s strategy in the game specifies how the player
should move in the network area. The mouse and the
cats will optimize their strategies in order to maximize
their own payoffs.1
We consider general movement strategies for the
mouse and the cats. They can be autonomous or reactive,
and they can be deterministic or probabilistic. In particular, when the mouse has a larger sensing range than
the cats (i.e., Rm > Rc ), the mouse can see some of
the cats’ movements while remaining undetected. Such
movement information can be exploited by the mouse
to avoid or delay detection. The cats, on the other
hand, can opportunistically coordinate their movement
to maximize their ability to catch the mouse. First,
when two or more cats fall within the sensing range
of each other, the cats can try to minimize the overlap
in their spatial coverage by moving away from each
other. Second, if the cats can additionally communicate
within a wireless range of Wc , they can further run a
coordination protocol to maximally detect the mouse in
a collaborative manner.
III. Related Work
Meguerdichian et al. [9] derive an optimal path for
a mobile target to minimize its exposure to a set of
static sensors in getting to a given destination. They
do not assume that the target can observe the sensors,
and implicitly study the case of the blind mouse only.
Our strategic goal for the target is also different. Rather
than minimize the target’s exposure in getting to some
destination, our goal is for the target to maximize the
sensors’ expected detection time among all possible
paths.
Network coverage by mobile sensors has also been
addressed by Liu et al. [8]. Their work and ours differ
in the following aspects. First, we consider a closed
network area with explicit boundaries, whereas they
consider the network to be an unbounded infinite space.
Second, they implicitly study the case of the blind
mouse only. Third, they assume infinitely many cats
spatially distributed with a given density. We assume a
given number of cats, and consider how their movement
strategy can impact their steady state spatial distribution,
which can in turn impact their ability to detect the mouse
quickly.
1 Liu et al. [8] proposed and studied a similar game-theoretic problem
formulation, but for a different network and mobility model.
3
Our cat-and-mouse problem can be considered an
instance of pursuit-evasion games. Significant results
about various forms of the full information game (i.e.,
both pursuers and evaders know each other’s moves
completely) are discussed in [5]. For extending the
theory to games of incomplete information, they describe a princess-and-monster problem for which it is
stated that there is no known solution. The princess-andmonster is similar to our blind mouse problem, but it is
not identical. In particular, we assume that the mouse
knows the cat’s presence matrix, while the princess
has no such knowledge about the monster. In general,
instances of pursuit-evasion games and their solutions
differ fundamentally based on the amount and kind of
information available to the players.
More recently, for partial-information games, Trummel and Weisinger [12] show that the optimal solution
for a pursuer to catch a stationary target in the least time,
given a probability distribution of the target’s position,
is NP hard. Their problem uses a general discrete-time
graph model, in which any pair of vertices may be connected by an edge. The NP hardness result is in contrast
to our dynamic program solution for the blind mouse. We
are able to obtain an efficient optimal algorithm because
in our surveillance area, each cell is a neighbor of only
up to 8 geographically adjacent cells. Based on a line-ofsight capture model (i.e., an evader is captured if it is in
the line-of-sight of a pursuer), Isler et al. [6] show that
in a connected polygonal space, a pursuer is guaranteed
to capture an evader with probability approaching one,
even if the evader knows where the pursuer is all the
time. The line-of-sight model does not consider the finite
sensing range of cats as in our problem.
For multiple coordinated pursuers, Hespanha et al. [3]
study a greedy (non-optimal) policy for a swarm of
pursuers to find an evader. They show that, assuming
a similar greedy strategy by the evaders, the greedy
pursuer policy guarantees capture in finite time. Isler
et al. [7] study the question of how many pursuers
are needed to capture a limited-visibility evader with
high probability. Their result uses the same discrete-time
graph model as in [12]. Vidal et al. [13] propose localmax and global-max heuristic pursuit strategies to coordinate a group of aerial/ground pursuers in catching a
number of randomly moving evaders. They observe that
the dependency between the expected capture time and
the pursuit policy is “very complex,” which motivates
their proposal of efficient sub-optimal policies with good
practical performance. Hollinger and Singh [4] make
similar observations about the hardness of the optimal
coordinated pursuer problem, due to exponential growth
of the search space as the number of pursuers increases.
For the cat coordination problem, we similarly study the
empirical performance of heuristic strategies that can be
efficiently implemented. Part of our focus is, however,
the contrast between sensing- and communication-based
coordination (SBC and CBC) approaches. SBC coordination has been less explored in the literature, but can be
attractive because it does not assume any extra system
costs/complexities beyond the sensing resources.
IV. The Blind Mouse: Rc ≥ Rm
In this section, we focus on the case where the cats
have a longer sensing range than the mouse. In this case,
the mouse is always detected before it can see the cat;
hence, the mouse can be considered a “blind” mouse.
A. Strategy of the mouse
Since a blind mouse cannot sense a cat’s actual
movement, the mouse is assumed to have some high level
information about the cats’ movements in order to avoid
detection. Specifically, we assume that the mouse knows
the statistical movement model and the sensing range Rc
of the cats. Based on the information, we estimate the
probability of finding at least one cat for any position in
the network, and design the best strategy for the mouse
accordingly.
With a sensing range Rc , each cat controls a circular
region of area πRc2 . Roughly speaking, each cat controls
a square region—which
we call a cell—of size s × s,
√
with s = πRc . We then consider the network area
to consist of X/s by Y /s disjoint cells, with each cat
controlling the cell in which it is currently located. The
presence matrix Π is defined such that Π[x, y] denotes
the probability that at least one cat is present in cell i =
(x, y) at any instant (when the context is clear, we use πi
as a shorthand for Π[x, y]); note that Π can be obtained
since the cats’ movement model is known. For instance,
Figure 1 shows the presence matrix for a single cat (i.e.,
Nc = 1) moving under the random waypoint (RWP)
algorithm [1] in a network area of 10 × 10 cells.
0.02
0.018
0.016
0.014
0.012
Probability
0.01
0.008
0.006
0.004
10
0.002
7
0
1
2
3
X
4
4
5
6
7
8
9
Y
1
10
Fig. 1. The presence matrix with 10 × 10 cells for a single cat
moving under the RWP algorithm in the network area. The mean of
the probabilities is 0.01, while the standard deviation is 0.012.
Remark: If Nc > 1 and the cats are moving independently under the same movement model, it is easy
to obtain the presence matrix Π for Nc cats based on
4
the presence matrix of a single cat. In particular, let pi
be the presence probability of one cat in cell i. Then,
the presence probability for at least one of the Nc cats
appearing
in cell i isPπi = 1 − (1 − pi )Nc . Also note that
P
and
i πi ≤ Nc .
i pi = 1
Dynamic programming solution for the best strategy:
Given the presence matrix Π, what strategy should the
mouse use to maximize the expected detection time? One
simple greedy strategy is for the mouse to continually
move to the neighboring cell having the lowest presence
probability, and stop moving when all the current cell’s
neighbors are more dangerous (i.e., they have a higher
presence probability). The intuition is that the expected
detection time will immediately increase whenever the
mouse visits a new cell along the path. However, the
greedy strategy may not always find the best path for
the mouse, since it prevents the mouse from considering
those paths that temporarily access a more dangerous
neighboring cell but will eventually lead to a safe
network location. For example, in Figure 2, the greedy
strategy suggests that the mouse should move from cell a
to cell c along the illustrated path, and finally stop at c.
However, under most values of the mouse’s and cat’s
speeds, the optimal path for the mouse is to move from
cell a to cell b.
0.0005
0.0005
0.0005
0.0075
0.0075
0.0075
0.0075
0.0005
0.0075
0.03
0.03
0.03
0.03
0.03
0.03
0.0075
0.0075
0.03
0.021
0.023
0.025
0.03
0.0075
0.02
ALGO()
Initialize E[Tidetect ] of each cell i with E[Tistay ]
For each cell i, mark i as its next step
Put each cell i into a max-heap using E[Tidetect ] as key
while heap is not empty
Extract cell i from heap with largest key
for each neighbor cell k of i
move ]
Calculate E[Tk,i
move ] > E[T detect ]
if E[Tk,i
k
move ]
Update cell k’s key E[Tkdetect ] to E[Tk,i
Re-insert cell k into the max-heap
Mark i as k’s next step
end if
end for
end while
Fig. 3. DP
ALGO :
A dynamic program for the mouse’s best strategy.
In the algorithm, we initialize E[Tidetect ] to E[Tistay ]
for each cell i in Line 1, and insert the cell into a heap
with key value E[Tidetect ], sorted in decreasing order
(Line 3). For each iteration (Lines 4-14), we first extract
the cell i with the largest key value from the heap, so that
the value E[Tidetect ] will not be further updated. Then,
for each neighboring cell k of i, we update E[Tkdetect ]
move
to E[Tk,i
] if the latter value is larger, meaning that
the mouse starting at cell k can benefit from moving to
cell i. If there was an update, we reinsert cell k into the
max-heap with the updated key value (Lines 6-9). Also,
we mark cell i to be the next cell to move to when the
mouse is in cell k (Line 10).
b
a
0.0075
0.03
0.018
0.03
0.03
0.03
0.03
0.0075
0.0075
0.03
0.03
0.016
0.014
0.01
0.03
0.0075
Theorem 1. DP ALGO is an optimal algorithm for the
mouse, against any presence matrix Π.
"hill" of high probability
optimal escape path
local greedy escape path
c
0.0075
0.03
0.03
0.03
0.03
0.03
0.03
0.0075
0.0075
0.0075
0.0075
0.0075
0.0075
0.0075
0.0075
0.0075
Fig. 2.
DP
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A greedy movement strategy may not be optimal.
To avoid missing the optimal path, we apply the
dynamic program DP ALGO as shown in Figure 3.
In the figure, there are three groups of variables,
move
namely E[Tidetect ], E[Tistay ], and E[Tk,i
]. The value
detect
E[Ti
] represents the expected detection time when
the mouse starts at cell i and uses the best path among
the paths considered so far. This value will be updated
as the algorithm proceeds, and will eventually hold the
desired expected detection time when the mouse uses the
best path among all possible paths. The value E[Tistay ]
denotes the expected detection time when the mouse
starts at cell i and stays there forever. Finally, the value
move
E[Tk,i
] denotes the expected detection time when the
mouse starts at cell k, moves to a neighbor cell i, and
follows the best strategy once it reaches cell i.
Proof: In general, a mouse strategy specifies what
the mouse should do when it is in a certain cell (whether
to stay or move to a particular adjacent cell). We shall
argue that the action determined by DP ALGO at each
cell is optimal. In particular, we shall prove by induction
the following: the jth best cell is extracted in Line 5 of
the jth iteration, whose optimal action is to move to its
next step.2 In addition, upon the extraction of any cell i,
E[Tidetect ] will be correctly set to the expected detection
time under any optimal mouse strategy.
In the first iteration, the cell i1 with the largest
E[T detect ] is extracted. This cell must be the one with
the largest E[T stay ], so that it must be the best cell. Here,
we determine that the mouse should “stay” (since its next
step is itself), and such an action is clearly optimal.
Next, suppose that the j − 1 best cells are extracted in
the first j − 1 iterations. For the jth best cell ij , we can
see that its optimal action is either to stay, or to move
to the best adjacent cell (if it yields a longer expected
detection time). Note that if the latter case is true, such
an adjacent cell, say w, must be one of the best j − 1
2 Here, the jth best cell is the cell which, when chosen as the initial
position, has the jth longest expected detection time, under an optimal
mouse strategy.
5
cells, so that E[T detect ] of ij is set to the optimal value
upon w’s extraction (Lines 6-13). Then in both cases, the
jth best cell must have the largest E[T detect ] among all
the remaining cells at the jth iteration, so that it will
be extracted. Furthermore, its optimal action and the
corresponding E[T detect ] are set correctly as required.
This completes the induction step.
It remains to show how to compute E[Tistay ] and
move
E[Tk,i
]. To do so, we use a concept called the cell
sojourn time, which is the length of time that a cat stays
in the current cell before it moves to a neighboring cell.
The expected sojourn time, denoted by E[T s ], can be
calculated since the cats’ movement model is known.
For the purpose of estimation, we may assume that the
status of whether any cat is present in a certain cell
is unchanged during the time interval [ℓt, (ℓ + 1)t), for
t = E[T s ] and any non-negative integer ℓ. Then, we can
estimate E[Tistay ] by
E[Tistay ] ≈
∞
X
ℓ=0
(ℓE[T s ])πi (1 − πi )ℓ = E[T s ]/πi . (1)
move
To calculate E[Tk,i
] based on the optimal
detect
], we let tk,i be the time taken by the mouse
E[Ti
to move from (the center of) cell k to (the boundary of)
cell i. Then, the mouse will either be caught in cell k, or
it can reach cell i so that it is expected to be caught after
another time of E[Tidetect ]. The probability that it can
reach cell i without being caught can be estimated by
s
(1 − πi )tk,i /E[T ] . Using an approach similar to the one
move
used to estimate E[Tistay ], we can estimate E[Tk,i
]
by
place for the mouse to stay, which reduces the (worstcase) expected detection time of the mouse. In fact, the
uniform presence matrix is an optimal strategy for the
cats, as shown in the following theorem:
Theorem 2. When the mouse applies DP ALGO, an
optimal strategy for the cat is to set its presence matrix
uniform.
Proof: Let M denote an arbitrary presence matrix,
and U denote the uniform presence matrix. When the
mouse applies DP ALGO, we let tM and tU denote the
expected time for the cat to catch the mouse when using
M and U , respectively. We shall show that tU ≤ tM ,
thus proving the theorem.
To link the two quantities, we consider a further
expected time t′M , which corresponds to the case that
the cat uses presence matrix M , and the mouse simply
never moves. It is easy to see that t′M ≤ tM , since if the
mouse never moves, it will never improve (i.e., lengthen)
the detection time. On the other hand, when the network
has x × y cells,
X 1
1
t′M =
×
(2)
xy
M
[i,
j]
i,j
X 1
1
≥
×
(3)
xy
U [i, j]
i,j
= tU ,
(4)
so that tU ≤ t′M ≤ tM . Here, Eq. 3 follows from the fact
that for any a, b > 0, (1/a) + (1/b) ≥ 1/((a + b)/2) +
1/((a + b)/2).
The presence matrix can be generalized for Nc cats, in
which case each cell is associated with a probability that
Z tk,i
at least one cat is present in the cell. The sum of these
tk,i
t
πi (1 − πi ) E[T s ] tdt + (1 − πi ) E[T s ] (E[Tidetect ] + tk,i ) probabilities can be as large as Nc , which is achievable
0
when the cats are always in disjoint cells. When the
It can be shown that the dynamic program has time network has x × y cells, and each cell is associated
complexity O(n log n), where n = XY /s2 . We omit the with a probability Nc /(xy), we call this presence matrix
analysis due to limited space.
maximum-uniform.
Repeating the arguments in Theorem 2, one can easily
show the following result:
B. Strategy of the cats
Based on the strategy in the previous section, the
mouse may eventually move to a safer position that has
a lower cat presence probability than its current position,
thereby maximizing the expected detection time. Hence,
the best strategy for the cats seems to maximize the
minimum presence probability among all the cells in Π.
The maximin strategy implies that when the cats are
moving independently under the same movement model,
the best choice for each cat is to move in a way such
that the presence probability is the same in each cell. (We
call the resulting presence matrix in which all the entries
have equal values a uniform presence matrix.) With a
uniform presence matrix, there is no particularly safe
Corollary 1. When the mouse applies DP ALGO, an
optimal strategy for Nc cats is to set the presence matrix
maximum-uniform.
Theorem 3. Nash equilibrium is achieved when the
mouse applies DP ALGO and the cats apply the
maximum-uniform presence matrix.
Proof: A direct consequence of Theorem 1 and
Corollary 1.
To yield a uniform presence matrix in the single-cat
case, one simple example movement is to sequentially
and circularly scan all the network cells. However, the
deterministic nature of the cat’s movement may allow
6
the mouse to accurately predict where the cat will be
and therefore easily avoid it. Hence, we will present in
Section V-B a probabilistic movement strategy that can
achieve a presence matrix close to being uniform.
As mentioned, when there are Nc cats, it is better
for them to move in disjoint areas of the network, as the
sum of πi in the presence matrix is equal to Nc , which is
greater than the sum of πi if the cats move independently.
This suggests that if the cats are allowed to move in
a coordinated way, we should always assign them to
monitor disjoint parts of the network. Accordingly, the
best strategy for the cats is to divide the network into Nc
equally sized partitions, in which one cat moves within
one partition optimally to yield a maximum-uniform
presence matrix.
V. The Seeing Mouse (Rc < Rm ), Independent Cats
In this section, we discuss the case when the mouse
has a larger sensing range than the cats. In this case,
once a cat enters the mouse’s sensing range, the mouse
can know the cat’s movement in advance without being
detected. We focus on the case that the mouse’s speed
is less than the cats’ (i.e., Vm < Vc ); we believe that
the other case is not as interesting since intuitively, the
faster mouse can always avoid being detected. In the
following, we first propose a strategy for the mouse to
escape from the cats based on such advance knowledge.
We then discuss some possible strategies that the cat may
use to reduce the detection time, knowing that the mouse
may run away when it sees the cat. We assume in this
section that the cats act independently.
A. Strategy of the mouse
We first study the special case in which there is only
one cat in the network. Then, we generalize the strategy
to the case of multiple cats.
Avoiding a single cat: When the mouse tries to escape
from a cat, it is better if it can move in a direction β such
that the minimum distance between the mouse and the
cat (assuming that the cat does not change its speed and
moving direction) in the future is as large as possible.
Let C and M be the current positions of the cat and
the mouse, respectively. Let α be the moving direction
of the cat. Then, the position of the cat at time t, denoted
by C(t), can be expressed as
C + (Vc cos α · ı̂ + Vc sin α · ̂) t.
(5)
Similarly, if the mouse chooses to move in a direction β at a speed of Vm , the position of the mouse at
time t, denoted by M (β, t), can be expressed as
M + (Vm cos β · ı̂ + Vm sin β · ̂) t.
(6)
γ is maximized when this angle is π/2
Mouse
V
Vm -Vc
to maximize γ
Mouse
Vm
γ
-Vc
β *=α+arccosVm
Vc
Vc
Vc
α
Cat
α
Cat
~ = V~m − V~c
(a) V
Fig. 4.
γ
β*
(b) To maximize γ.
The mouse’s choice of β ∗ to maximize γ.
Then, the distance between the cat and the mouse
at time t is equal to d(β, t) = kC(t) − M (β, t)k. By
differentiating d(β, t) with respect to t, we can find the
minimum distance between the mouse and the cat for
t ≥ 0. We denote such a distance by d∗ (β).
In other words, in order to maximize the distance from
the cat in the future, the mouse should find a direction β ∗
such that
β ∗ = argmax d∗ (β).
(7)
β
Remark: In fact, by a simple geometric argument, the
optimal β ∗ for the single-cat case can be easily obtained.
Essentially, the cat is moving in a vector V~c = Vc cos α ·
ı̂ + Vc sin α · ̂, while the mouse choosing a direction β
is moving in a vector V~m = Vm cos β · ı̂ + Vm sin β · ̂.
Equivalently, we may assume that the cat is stationary,
~ = V~m − V~c
while the mouse is moving in a direction V
towards the cat. To maximize the distance between the
cat and the mouse, we want (the absolute value of) the
angle γ between V~ and the vector from the mouse to the
~ −M
~ ) to be as large as possible. As shown
cat (i.e., C
in Fig. 4, we have
Vm
β ∗ = α + arccos
.
(8)
Vc
Avoiding multiple cats: We have shown how the mouse
can choose the optimal direction β ∗ when there is only
one cat. In the case of multiple cats, we need to find a β ∗
that maximizes the minimum distance to all the cats in
the future. Using similar notations, we let d∗i (β) denote
the minimum distance between the mouse and the ith
cat in the future, assuming that the mouse is moving in
the direction β. In other words, if there are j cats within
the sensing range of the mouse, we have
n
o
β ∗ = argmax min d∗1 (β), d∗2 (β), . . . , d∗j (β) . (9)
β
Notice that in the case of multiple cats, a direction β
that maximizes the minimum distance for one cat does
not necessarily yield a short minimum distance for
another cat. Finding the optimal β ∗ may require us to
consider all the intersections between any two of the j
7
curves d∗i (β), so as to decide the best direction; in the
worst-case, there can be Ω(j 2 ) such intersections. An
alternative way is to obtain a close estimation of β ∗ , by
choosing a small angle δ and computing all d∗i (β) values
for β = 0, δ, 2δ, 3δ, . . . , ⌊2π/δ⌋δ, and then finding β ∗
based on only these values. The drawback of the above
estimation is that we may miss the optimal β ∗ when it
is not a multiple of δ. However, there are only O(j/δ)
values to compute, and the simplicity of the method will
usually allow us to obtain a good enough β ∗ efficiently
in practice.
Degree of freedom and a revised strategy: In general,
it is not necessary for the mouse to choose the optimal
direction β ∗ to avoid being caught. It is because when
choosing a direction β, a mouse will not be detected
as long as its minimum distance to all the cats in the
future is larger than the cats’ sensing range Rc . Hence,
the mouse may choose a direction from a feasible set B
such that
∗
(10)
B = β min {di (β)} > Rc .
1≤i≤j
Thus, the larger the size of B is, the more directions
the mouse can choose from, so that it is more likely for a
mouse to escape successfully. Note that the mouse may
have fewer choices when it is located at the boundaries
or the corners of the network area. In fact, if the cats
move under the RWP algorithm and the mouse uses the
above strategy to escape (so that it chooses to move
at an angle of β ∗ when one or more cats are around,
and stops moving when no cat is around), the results in
Fig. 5 show that the mouse will likely be “pushed” to the
corners/boundaries of the network, and then be caught
there.
may be offset by the predominant presence of the cats
near the center.
B. Strategy of the cats
Since a cat knows that the mouse may see it before it
sees the mouse, the cat may assume that the mouse will
try to run away from it in advance. One logical choice of
strategy for the cats is then to visit more frequently the
center region of the network, where the mouse has more
freedom to move and escape. As shown in Figure 1, a cat
can achieve the goal of visiting the center region more
often by playing the RWP strategy. On the other hand,
the cats may also benefit from a movement strategy that
will yield a uniform presence matrix, so that they will
eliminate safe havens for the mouse to hide in. As both
kinds of strategy have their merits, we will compare their
empirical performance in Section VII.
We use the following simple random algorithm, which
we call the bouncing strategy, for the cats to approximate
closely the uniform presence matrix. In the algorithm, a
cat moves in a straight line until it hits the boundary
or a corner of the network area. Whenever it reaches
the boundary/corner, it selects a direction randomly and
uniformly from all the feasible directions and proceeds
to move in the selected direction. Figure 6 shows the
presence matrix for a cat moving under the bouncing
model. Alternative algorithms are known that can provably achieve uniform presence for the cats [2], [10].
0.018
0.016
0.014
0.012
0.01
Probability
0.008
0.006
0.004
9
0.002
7
5
0
1
2
3
Y
3
4
5
X
6
7
8
9
1
10
Fig. 6. The presence matrix with 10×10 cells for a single cat moving
under the bouncing strategy in the network area. The matrix is close
to the uniform presence matrix.
Fig. 5. The black dots show the positions where the mouse is detected
in 50,000 simulation runs. The mouse is caught at the the network
corners and boundaries in most cases.
This leads to a revised strategy for the mouse to
always move towards the center region of the network
whenever there is no cat detected within its sensing
range. We call this the centric strategy. Our results in
Section VII show that the centric strategy works well
among other strategies evaluated, when the cats have
a uniform presence matrix. When the cats use RWP,
however, the mouse’s degree of freedom at the center
Remark. In the seeing mouse case, the cats have
no information about the mouse during the game. An
optimal cat strategy can be defined in the minimax sense,
i.e., the optimal strategy minimizes the hypothetical expected detection time over all possible mouse strategies.
However, given that there are infinitely many possible
mouse strategies, and that the cats have no information
about what the mouse might choose, we conjecture that
finding the optimal cat strategy is undecidable. Hence,
we propose only heuristic cat strategies in the above
discussion.
8
VI. The Seeing Mouse and Coordinated Cats
In this section, we consider the seeing mouse, and a
network of cats who may coordinate their movement to
maximize their ability to detect the mouse. We consider
two coordination approaches. The first one is enabled by
the cats’ ability to see each other within their sensing
range. The second one is enabled by the cats’ additional
ability to communicate over a wireless range of Wc .
A. Sensing-based coordination (SBC)
When a cat moves close to other cats, their sensing
regions overlap, which results in inefficient use of the
sensing resources. Such an overlap can be detected
partially by the cats involved. Specifically, they can see
each other when they fall within the sensing range of
each other. When that happens, the cats can try to move
away from each other in order to reduce the overlap in
spatial coverage. We call such a coordination approach
sensing-based coordination (SBC).
In SBC, a cat moves according to the usual algorithm
(e.g., bouncing or RWP) until it detects one or more other
cats within its sensing range. When that happens, the
first cat will try to maximally avoid the other cat(s), i.e.,
to minimize the expected future overlap in coverage by
the cats. This can be done by using the same algorithm
that a seeing mouse uses to avoid a detected cat, namely
the geometry-based algorithm presented in Section V-A.
According to the algorithm, the cat computes the optimal
β ∗ in order to optimally move away from the other cat(s).
B. Communication-based coordination (CBC)
In SBC, the cats coordinate by observing each other’s
movements only. If the cats can also communicate over
a wireless channel of range Wc , they can further coordinate their strategies to reduce the expected detection time
of the mouse. First, if Wc > Rc , the coordination can
occur sooner and therefore be more proactive. Second,
the cats can actively exchange information about their
strategies (e.g., the planned future movements), instead
of using passive observations only. We call such a coordination approach communication-based coordination
(CBC).
The CBC algorithm we use (Fig. 7) is motivated by
the observation in Section V-A about the importance of
the mouse’s degree of freedom in enabling its escape
strategy. Specifically, we aim to allow the cats to team
up and form a cohort in searching for the mouse. The
concerted efforts of the cohort members will then form a
larger barrier that limits the mouse’s degree of freedom
in passing the barrier. The algorithm in Fig. 7 allows
cohorts to form in an opportunistic manner. We assume
that each cat’s plan of movement consists of a sequence
of trips, and the destination of one trip is the starting
point of the next trip, and so on. We call the destination
of a trip a waypoint. Cohorts are then formed as follows.
A cat, say C, that is acting independently (i.e., not in any
cohort currently) will continuously attempt to establish
wireless communication with another cat, which happens
when the other cat, say D, comes within a distance of
Wc of C. With a probability of 1 − pf , C will ignore
the communication and repeat looking for a new cat for
communication. Otherwise (i.e., with a probability of
pf ), C asks D about its current destination waypoint.
C will then abandon its current waypoint and join D by
adopting a new waypoint that is “close to” D’s waypoint.
(The “close to” notion will be made precise in the
following.) Once C decides to join D and is on its way
to the new waypoint, it considers itself committed and
will not attempt to establish communication with other
cats. Once C has reached the new waypoint, however, it
will repeat the procedure of entering another cohort.
Notice that while a more effective barrier is formed
by a larger cohort, a larger cohort size will result in
a fewer number of cohorts since the number of cats is
fixed. A small number of cohorts limits the ability to
spread out the cats over different parts of the surveillance
area where the mouse may be found. Hence, a tradeoff
between the cohort size and the number of cohorts is
useful. The pf parameter aims to provide this tradeoff,
in which a larger pf value is more likely to produce
larger cohorts.
Notice also that, while the teaming of cats forms a
useful barrier against the escape of the mouse, it is also
important to avoid the inefficient coverage overlap that
can happen with uncontrolled teaming. This is the reason
when C decides to follow D in the above example, it
does not try to go to the same waypoint as D, but selects
a new waypoint close to D’s. Specifically, the new
waypoint is computed as follows: C selects a waypoint
that is 2Rc away from D’s and whose direction from D’s
is perpendicular to D’s movement vector. The waypoint
selection strategy is illustrated in Figure 8. Notice that
two cats having the same speed are likely to arrive at the
new waypoints at about the same time, and will form a
maximum barrier against the mouse’s escape.
In general, we have conjectured about the hardness
of the optimal independent cat strategy against the seeing mouse. The problem becomes more difficult when
coordinated cats are considered. A number of related
heuristic algorithms for coordination are known [3], [7],
[13]. Our goal in this paper is not to extensively cover
the design space of CBC, but to contrast between the
SBC and CBC approaches, and evaluate how the cats
may use coordination to improve against effective mouse
strategies identified in our previous discussions.
9
Start
Establish
communication with
another cat
No
strategy, in three sets of runs. From the table, notice
that the mouse can achieve a significantly higher average
detection time by playing the dynamic programming
strategy, as the analysis in Section IV-A shows.
Following
another cat?
Yes
Is this cat in set
G?
Yes
Communication
established?
No
Reached
waypoint?
No
Yes
No
Yes
Follow?
Change waypoint
Select a new
waypoint and clear
set G
No
Move
Add cat to set G
End
Fig. 7.
The communication-based coordination (CBC) approach.
Fig. 8. Coordinated waypoint selection to minimize coverage overlap
in CBC.
VII. Experimental Results
A. The blind mouse: Rc ≥ Rm
This section evaluates the case of a blind mouse.
Unless otherwise specified, we report average results
over 100 simulation runs, each lasting 200,000 seconds.
We omit the standard deviations, because they are very
small compared with the means. In the discussion, when
a node deterministically cycles through all the cells in
the network area, we will refer to the movement as the
scan strategy.
1) Benefits of dynamic programming solution for
the mouse: We compare the performance of different
movement strategies for the mouse in Table I. In the
table, the column “DP” refers to the case when the mouse
determines its movement by the dynamic programming
strategy in Section IV-A. The mouse is initially located
at the center of the network area. The column labeled
“Stay” in Table I is used for baseline comparison, and
refers the to case when the mouse simply stays at its
starting position (i.e., it does not move at all afterward).
The experiments use one cat. Its movement strategy is
shown as the rows of Table I, and is chosen to be
(1) the uniform scan strategy, (2) the bouncing strategy
in Section V-B, and (3) the random waypoint (RWP)
Mc \ Mm
uniform scan
Bouncing
RWP
DP
1083.26
628.66
2823.26
RWP
415.31
442.23
271.73
Stay
511.50
305.03
226.13
2) Uniform presence matrix benefits the cat: The
“DP” column in Table I also shows how a uniform
presence matrix may benefit the cat. From the results,
notice that the simple scan strategy can greatly reduce
the detection time compared with the non-uniform RWP
strategy. However, as discussed in Section IV-B, the
deterministic nature of the scan strategy may allow the
mouse to predict where the cat will be in advance and
thus effectively avoid the cat. To avoid the problem, the
bouncing strategy can closely approximate the uniform
presence matrix without being deterministic.
3) Effects of the cat’s sensing range: In this experiment, there are one cat and one mouse moving in a
500 m × 500 m network area. The mouse plays the
dynamic programming strategy. The cat uses (1) the
RWP strategy and (2) the bouncing strategy, in two
different runs. We measure the average detection time of
the mouse for different sensing ranges, Rc , of the cat.
Fig. 9 shows that as Rc increases, the average detection
time decreases as an inversely proportional function of
Rc , showing that the cat’s ability to detect the mouse
increases proportionally to its sensing capability.
60000
Average detection time (s)
Yes
TABLE I
AVERAGE DETECTION TIME ( IN S ) FOR DIFFERENT CAT AND MOUSE
MOVEMENT STRATEGIES IN A 500 M BY 500 M NETWORK .
Vc = Vm = 10 M / S , Rc = 25 M , AND THE MOUSE IS INITIALLY
LOCATED AT THE CENTER OF THE NETWORK .
RWP
Bouncing
50000
40000
30000
20000
10000
0
0
10
20
30
Rc (m)
40
50
60
Fig. 9. Average detection time (in s) of the mouse versus the cat’s
sensing range Rc (in m); Vc = Vm = 10 m/s.
4) Number of cats: In this experiment, we measure
the average detection times when the number of cats
is varied. The network area is 500 m × 500 m, and the
sensing range of each cat is 25 m. Similar to the previous
experiment, the mouse plays the dynamic programming
strategy, while the cats independently play either the
RWP or the bouncing strategy. Fig. 10 shows that as
the number of cats increases, the average detection time
10
decreases, approximately like inversely proportional to
Nck , where k is a constant slightly larger than one.
Average detection time (s)
3000
RWP
Bouncing
2500
2000
1500
1000
500
0
0
10
20
30
40
50
Nc
Fig. 10. Average detection time (in s) versus the number of cats Nc
in a 500 m by 500 m network area; Vc = Vm = 10 m/s.
5) Effects of Vc and Vm on dynamic programming
solution: In this experiment, there are one cat and one
mouse. We illustrate how the expected detection time of
the mouse varies with different speeds Vc and Vm of the
cat and the mouse, respectively. The mouse plays the
dynamic programming strategy. Fig. 11 shows that the
average detection time is reduced when the cat moves
faster (i.e., Vc is higher). This is because a faster cat
can cover a larger area in the same amount of time.
Notice also that the detection time increases when the
mouse moves at a higher speed. This is because a faster
mouse can move from its current position to a safe
position more quickly, and benefit from staying in the
safe position longer. We also find that the mouse can
benefit more from moving at a higher speed, if the cat
itself is moving at a higher speed. This shows that a fast
cat will force the mouse to be fast to avoid detection.
Average detection time (s)
3000
2500
2000
1500
1000
500
0
0
0
20
40
50
60
80
100
100
Vm (m/s)
Vc (m/s)
Fig. 11. Average detection time (in s) versus the speeds (in m/s) of
the cat and the mouse in a 500 m by 500 m network. Rc of the cat is
15 m.
We further show that the path of movement computed
by the dynamic program in Figure 3 is dependent on
the speed of the mouse. In Figures 12(a) and 12(b), the
gray level represents the cat’s presence probability (the
darker an area, the lower the cat’s presence probability
in the area). The cat has a speed of 10 m/s. The arrows
in the figure show the escape paths of the mouse in a
500 m × 500 m network area divided into 10 × 10 cells.
The paths when the mouse has a speed of 10 m/s are
shown in Figure 12(a). The paths when the mouse has
a higher speed of 15 m/s are shown in Figure 12(b). In
the figure, cells j and k are cell i’s neighbor. Cell k is
safer than j, and both of them are safer than i. Assume
that the mouse is currently in cell i. Notice that when
the mouse has the lower speed, it will move to the less
safe neighbor j because j is closer in distance than k.
Choosing the closer neighbor allows the mouse to leave
the more dangerous cell i soon (considering that the
mouse moves slowly). When the mouse has the higher
speed, it can afford to move a longer distance before
leaving cell i. Hence, it will choose to move directly to
the safer neighbor k although k is farther away.
i j
k
0.0030
0.0115
i j
k
0.0200
(a) Vc = Vm = 10 m/s.
0.0030
0.0115
0.0200
(b) Vc = 10 m/s, Vm = 15 m/s.
Fig. 12.
Escape paths calculated by the algorithm
MaximizeDetectTime() for a 500 m by 500 m network
area divided into 10 by 10 grids, for different speeds of the mouse.
B. The seeing mouse (Rm > Rc ), independent cats
We now evaluate the case of a seeing mouse, when
the cats do not coordinate. In the experiments, unless
otherwise specified, the cats play the bouncing strategy
and Rc = 5 m and Rm = 10 m. We report maximum,
minimum, and average results over 5,000 simulation runs
within a 100 m by 100 m area.
1) Movement direction β for the mouse: In this
experiment, we report the optimal direction β ∗ calculated
by Eqn. 9 to maximize the minimum distance between
the cats and the mouse. We use Vc = Vm = 5 m/s.
There are six cats and one mouse. Figure 13, shows
the minimum distance of the mouse from each cat, as
a function of the mouse’s direction of movement β. In
the figure, the thick line shows the minimum distance of
the mouse from any of the cats. Of the thick line, the
dark segment shows the range of the movement angles
over which the minimum distance (from any of the cats)
is maximized, and thus gives the range of the optimal
maximin solutions computed. Any angle that falls within
the range can be used by the mouse as its optimal
movement direction.
2) Effects of Rc and Nc : In this experiment, we
use ten cats, and show how Rc can affect the range of
maximin solutions available to the mouse as it tries to
find an optimal angle of movement. In Figure 14, we
report the fraction of the movement angles that are in
11
Minimum distance between cat and mouse (m)
90
Cat 1
Cat 2
Cat 3
Cat 4
Cat 5
Cat 6
Minimum
Maximin
80
70
60
50
40
Range of maximin solutions
30
Minimum distance to all the cats
20
10
0
0
50
100
150
200
Beta (degree)
250
300
350
Fig. 13. The minimum distance d∗ (β) calculated by Equation 9
between six cats and one mouse as a function of the mouse’s movement
direction β. The cats and the mouse are initially randomly placed in
a 100 m by 100 m network.
the feasible set B (defined by Equation 10) as a function
of Rc . The results show that the fraction of feasible
solutions decreases like exponentially as Rc increases
(up to Rm ), showing that the mouse’s degree of freedom
in choosing a solution is severely restricted as the cat’s
sensing capability increases.
1
Fraction of feasible solutions
0.9
0.8
0.7
Maximum
0.6
Median
0.5
0.4
Maximum
0.3
Mean
Mean
0.2
Median
0
Minimum
Minimum
0.1
0
10
20
30
Sensing range (m)
40
50
Fig. 14. The fraction of feasible solutions calculated by Equation 10
as a function of the cat’s sensing range Rc . Ten cats are randomly
placed in a 100 m by 100 m network. Rm = ∞.
We next consider the effects of Nc , the number of cats.
In Figures 15 and 16(a), we show that as Nc increases,
both the optimal minimum distance (of the mouse) from
any of the cats and the average detection time decrease.
140
Maximum
Mean
Median
Minimum
Maximin distance to any cats (m)
120
100
80
60
40
Minimum
20
0
0
20
40
60
Number of cats
80
100
Fig. 15. Statistics of the maximin distance of the mouse from any of
the cats, as a function of the number of cats Nc . Both the cats and the
mouse move according to the bouncing strategy in a 100 m by 100 m
network area.
3) Effects of Rm and Vm : We show the impact of
the mouse’s sensing range Rm on the detection time in
Figure 16(b). As Rm increases, the mouse can detect
the cats earlier and are better able to make the necessary
moves to avoid the cats. Hence, the detection time
increases significantly at first. As Rm increases beyond
9 m, however, the detection time does not change much.
This shows that information about the cats that are far
away (as captured by a large sensing range Rm ) may not
be that useful to the mouse in determining its immediate
course of action.
The results in Figure 16(c) show that the detection
time generally increases as the mouse’s speed increases.
This is because a faster mouse can avoid the cats more
quickly and allow itself more choices of the optimal
movement direction β ∗ .
4) Comparison of different strategies: We discussed the bouncing and centric strategies for the seeing
mouse in Section V-A. These strategies define what the
mouse should do when it sees no cat within its sensing
range. An alternative strategy in such a situation is for
the mouse to simply stay where it is, presumably to
conserve energy, and we call it the static strategy. In
this experiment, we compare the performance of these
strategies for the mouse, while ten cats in the network
play either the bouncing or the RWP strategy.
The average detection time results are shown in
Fig. 18. As a baseline case for comparison, we also show
a “stay” strategy in which the mouse simply stays at
its starting position; i.e., the mouse never moves (i.e.,
Vm = 0) and does not apply Equation 9 to determine its
angle of movement. From Fig. 18, notice that bouncing
is the best strategy for the cats, where their presence
matrix is approximately uniform. For the mouse, its best
strategy depends on the cats’ strategy. When the cats
play the RWP strategy, the best strategy for the mouse
is bouncing. When the cats play the bouncing strategy,
the best strategy for the mouse is centric. We further
quantify how deviations from the center position by the
mouse will impact the average detection times in this
case. We use an off-centric mouse that will move to the
closet point (from the mouse’s current location) that is
k distance away from the center, and vary k in a set of
experiments. (When k = 0, we have a centric mouse.)
The results in Figure 17 verify that a mouse moving
closer to the center can increase the average detection
time when the cats have a uniform presence matrix.
The above results show an interesting interplay between the cats’ and the mouse’s movements. When the
cats use the bouncing strategy, the mouse benefits by
moving toward the center of the network, where it has a
higher degree of freedom as discussed in Section V-A.
However, this higher degree of freedom is offset, in the
case of this experiment, when the cats use the RWP
12
4500
2000
Mean
Median
Minimum
Maximum
3000
Detection time (s)
Detection time (s)
3500
2500
2000
1500
Maximum
Mean
Mean
1600
Median
1400
Minimum
2000
Median
Minimum
1200
1000
800
1500
1000
600
1000
400
500
Minimum
500
0
2500
Maximum
1800
Detection time (s)
4000
Minimum
Minimum
200
0
5
10
Number of cats
15
0
20
0
2
4
6
8
10
12
Mouse sensing range (m)
(a)
14
16
18
0
0
5
10
Mouse speed (m/s)
(b)
15
20
(c)
400
1800
350
Average detection time (s)
Average Detection Time (s)
Fig. 16. Statistics of the detection times as a function of (a) Nc , (b) Rm , and (c) Vm . The cats randomly move in a 100 m by 100 m network.
Both the cats and the mouse use the bouncing strategy.
300
250
200
150
100
50
0
0
10
20
30
40
50
Distance away from the center (m)
1600
1400
RWP Cat
RWP/SBC Cat
Bouncing Cat
Bouncing/CBC Cat
RWP/CBC Cat
1200
1000
800
600
400
200
0
Bouncing
Centric
RWP
Seeing Mouse Strategies
Fig. 19. Average detection time comparison of the SBC and CBC
coordination approaches, for different combinations of the mouse and
basic cat movement algorithms.
strategy to achieve a higher presence in the center area.
The static strategy performs the worst in all of the cases
measured.
coordination, Wc and pf are set to be 10 m and 0.7,
respectively, unless otherwise stated.
RWP
Mc
Bouncing
Stay
Static
Centric
1600
1400
1200
1000
800
600
400
200
0
Bouncing
Average detection
time (s)
Fig. 17. Average detection times of off-centric mouse strategy for
different values of k in a 100 m × 100 m surveillance area. Vc =
Vm = 10 m/s, Rc = 5 m, Rm = 10 m, Nc = 10.
Mm
Fig. 18. Average of the mouse detection time (in s) under different
strategies for the cats and the mouse in a 100 m by 100 m network.
V c = 5 m/s, V m = 10 m/s, Rc = 5 m, and Rm = 10 m.
C. Seeing mouse and coordinated cats
We evaluate the performance of the coordination approaches in Section VI for the cats playing against a
seeing mouse. We use 10 cats in a surveillance area of
100 m by 100 m; the cats and the mouse all have the
same speed of 10 m/s; the cat and mouse sensing ranges
are 5 m and 10 m, respectively. When there is no cat
within the sensing range of a seeing mouse, the mouse
plays the bouncing, centric, or RWP strategy, and we
correspondingly call it the bouncing, centric, or RWP
mouse. For each set of parameters, we report the average
detection times of 1000 simulation runs in Figure 19. The
error bars show the 95% confidence interval. For CBC
The results in Figure 19 show that the coordination
approaches are highly useful for the RWP cats, but have
relatively small effects for the bouncing cats. This is
because the bouncing cats are evenly spread out in the
surveillance area as evidenced by their uniform presence
matrix. For SBC, they get few chances to meet in the
first place, and therefore do not need help from SBC to
move away from each other. For CBC, the cats similarly
have few chances to get close and attempt the opportunistic clustering if the communication range is small.
If the communication range becomes larger, the pairing
opportunities increase. However, the cats who decide to
cluster may arrive at the coordinated waypoints far apart
from each other, which reduces the effectiveness of the
barrier formed. For the RWP cats, SBC coordination
reduces the average detection time by at least 161%
(against the centric mouse), by decreasing the overlap
in spatial coverage relative to the uncoordinated cats.
Compared with SBC, CBC coordination reduces the
average detection time by at least 384% (against the
bouncing mouse). This shows the additional benefits of
teaming up the cats to reduce the mouse’s degree of
freedom in evading the cats. The benefits are particularly
obvious for the centric mouse and the RWP mouse,
as the two kinds of mouse do tend to concentrate in
the center and can maximally benefit from the high
freedom of movement there if not countered by the cats’
13
coordination strategy.
Figure 20 shows the impact of Wc on the average
detection time of CBC, for the mouse strategies of
bouncing, centric, and RWP, respectively, when pf is
fixed at 0.7. We omit the standard deviations as they
are small compared with the averages. As previously
explained, CBC coordination is highly useful for the
RWP cats, but has less effect for the bouncing cats.
For the RWP cats, the performance of CBC is highly
dependent on the communication range. In general, it
increases quickly as Wc increases, up to a value of 2Rc ,
after which further increases in Wc have little impact.
Notice also that the RWP/CBC cats are most effective
against the centric mouse. This is because whereas the
centric mouse aims to achieve a high degree of freedom
to escape at the center, the CBC approach is specifically
designed to counter that freedom. The design is so
effective that whereas the RWP cats without coordination
have far worse performance than the bouncing cats, the
RWP cats can outperform the bouncing cats (with or
without coordination) when CBC is enabled. For the
centric mouse, when the communication range is large
enough, RWP/CBC detects the mouse twice as fast as
uncoordinated bouncing, and about 1.5 times faster than
bouncing/CBC.
The 3D plots in Figure 21 systematically explore the
effects of Wc and pf on the average detection time,
for 10 bouncing/CBC cats. The corresponding plots for
10 RWP/CBC cats are shown in Fig. 22. The evaluated
mouse strategies are bouncing, centric, and RWP. Notice
from the figures that a higher pf generally improves the
performance of the coordination, over the entire range of
Wc evaluated. This suggests the need to maximize the
chance for the cats to work together in our experimental
setting. For Wc , we point out that further to the observation we made about Fig. 8, the lowest detection time
is achieved, across the space of the measurements, when
Wc is about equal to 2Rc . Increasing Wc beyond 2Rc
generally has little effect on the performance. The near
optimality of Wc = 2Rc appears quite robust across a
large number of experiments that we have run that are
not reported in this paper.
expected time to detect the mouse. For a seeing mouse,
we presented an efficient optimal algorithm based on
geometric arguments that allows the mouse to maximally
delay detection by reacting to the cats’ movements
within its sensing range. We then discussed possible
counter strategies by the cats to reduce the expected
detected time. Furthermore, we discussed and evaluated
SBC and CBC coordination approaches for the cats.
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VIII. Conclusions
We have considered the cat-and-mouse game between
an intelligent mobile target trying to evade detection and
a group of mobile sensors trying to detect the target as
quickly as possible. For a blind mouse, we presented
and analyzed a dynamic program for the mouse to
maximize its expected detection time, assuming high
level knowledge about the cats’ movements. We further
argued that the cats’ choice of a maximum-uniform
presence matrix can minimize the “hypothetical” (assuming mouse’s starting positions are uniformly distributed)
Jren-Chit Chin is a Ph.D. candidate in the
Department of Computer Science at Purdue
University. He received his B. Sc. in Computer Engineering from Iowa State University
in 2005. His current area of research includes
target localization and sensor coverage in
mobile sensor network.
14
RWP/CBC Cat
Bouncing/CBC Cat
Bouncing Cat
1000
800
600
400
200
0
0
5
10
15
Communication range (m)
1200
800
600
400
200
0
0
20
(a) Bouncing mouse algorithm
1200
RWP/CBC Cat
Bouncing/CBC Cat
Bouncing Cat
1000
Average detection time (s)
Average detection time (s)
Average detection time (s)
1200
5
10
15
Communication range (m)
1000
RWP/CBC Cat
Bouncing/CBC Cat
Bouncing Cat
800
600
400
200
0
0
20
5
10
15
20
Communication range (m)
(b) Centric mouse algorithm
(c) RWP mouse algorithm
240
220
200
0
180
160
20
0.5
15
10
5
0
1
500
400
300
200
20
0
0.5
15
10
5
pf
Communication range (m)
0
1
Average detection time (s)
260
Average detection time (s)
Average detection time (s)
Fig. 20. Effects of communication range on average detection time. Network size is 100 m × 100 m, Vc = Vm = 10 m/s, Rc = 5 m,
Rm = 10 m, Nc = 10, pf = 0.7.
400
350
300
0
250
0.5
200
20
15
(a) Bouncing mouse
10
pf
Communication range (m)
5
0
1
pf
Communication range (m)
(b) Centric mouse
(c) RWP mouse
1200
1000
800
600
0
400
200
20
0.5
15
10
5
0
Communication range (m)
(a) Bouncing mouse
Fig. 22.
1
1200
1000
800
600
400
0
200
0
20
0.5
15
10
5
pf
Communication range (m)
(b) Centric mouse
0
1
Average detection time (s)
1400
Average detection time (s)
Average detection time (s)
Fig. 21. Average detection time as a function of Wc and pf for 10 bouncing/CBC cats and various mouse strategies. Network size is 100 m
× 100 m, Vc = Vm = 10 m/s, Rc = 5 m, and Rm = 10 m.
1500
1000
500
0
0
20
0.5
15
10
pf
5
0
1
pf
Communication range (m)
(c) RWP mouse
Average detection time as a function of Wc and pf for 10 RWP/CBC cats and various mouse mobility strategies.
Yu Dong received his B.E. in Industry Automation Control from University of Science and Technology Beijing (USTB), China;
M.E. in Information System Engineering
from Osaka University, Japan; M.S. and Ph.
D. in Computer Science from Purdue University, West Lafayette, IN, USA. He was the
recipient of an IBM Ph.D. fellowship and is
now at IBM Silicon Valley Laboratory in San
Jose, CA. His research interests are networking, databases, and multimedia systems.
Wing-Kai Hon is an assistant professor in
Department of Computer Science at National
Tsing Hua University, Taiwan. He received
his Ph.D. degree from University of Hong
Kong in 2005 and has visited Purdue University in 2004–2006. His research interests
include data compression, indexing, and algorithm design.
Chris Y. T. Ma is a Ph.D. student in the
Department of Computer Science at Purdue
University. He received his B.Eng. in Computer Engineering from the Chinese University of Hong Kong in 2004, and his M.Phil. in
Computer Science and Engineering from the
Chinese University of Hong Kong in 2006.
His research interests include performance
study of wireless networks and mobile sensor
networks.
David K. Y. Yau received the B.Sc. degree
from the Chinese University of Hong Kong,
and the M.S. and Ph.D degrees from the
University of Texas at Austin, all in computer
science. From 1989 to 1990, he was with the
Systems and Technology group of Citibank,
NA. He is currently an associate professor
of computer science at Purdue University,
West Lafayette, IN, USA. Dr. Yau was the
recipient of an NSF CAREER award for
research in quality of service provisioning.
His other research interests are in protocol design/implementation,
wireless/sensor networks, and network security. Dr. Yau serves on the
editorial board of IEEE/ACM Trans. Networking. He was TPC co-chair
(2006) and Steering Committee member (2007–09) of IEEE IWQoS,
and Vice General Chair (2006) and TPC co-chair (2007) of IEEE ICNP.