Autonomous Graph Partitioning for Multi-Agent Patrolling Problems

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Autonomous Graph Partitioning for Multi-Agent
Patrolling Problems
Bernát Wiandt and Vilmos Simon
Budapest University of Technology and Economics
in Budapest
Magyar tudósok krt 2., Hungary
Email: [bwiandt,svilmos]@hit.bme.hu
Abstract—Patrolling algorithms are coordinating multiple
agents with the goal of visiting points of interest in a timely
manner. These algorithms play a major role in efficient use of
UAVs or other autonomous vehicles for precision agriculture,
large area monitoring or security use cases. These algorithms
are either centralized and in need of constant connection with
the agents or solve NP-hard problems to plan the routes of
individual agents on the graph. These requirements become
unfeasible when the number of agents or points of interest
grow or become dynamic. In this article we further elaborate
the performance characteristics of the Partition Based Patrolling
Strategy (PB PS) algorithm. The partitioning requires only local
interactions between agents, therefore it is scalable to a large
number of nodes and agents. On these subgraphs agents patrol
independently from each other, therefore the approach eliminates
interference between agents.
I. INTRODUCT IO N
PATROLLING is the task of visiting points of interest in
a timely manner. These points of interest can be selected
for security purposes (monitoring entry points of an area)
or points in space one would like to monitor[1] by making
photographs or measurements. Every problem that involves
points of interests distributed in space and limited possibilities
to move between them (modeled by a graph) is a patrolling
problem. When patrolling inside a building, points of interest
can be intersections or doors needed to be inspected in a
timely manner. When dealing with a large open area, like an
agricultural crop, the points of interest are locations where one
wants to make measurements, i.e. water level, infections, fruit
maturity, etc.
Points of interest are usually modeled as nodes of a graph,
where the edges represent the option to travel between them
directly and the length of the edges represent the distance or
cost of moving between them. Patrolling is usually performed
by physical agents, such as Unmanned Autonomous Vehicles
(UAVs) or robots[2], either on ground or in the air. In this
article we refer to these entities, without loss of generality, as
agents. Patrolling with a single agent is a matter of calculating
a sequence of points of interest for the agent to visit. Instead,
our focus is on multi-agent patrolling, where the task is to
efficiently coordinate an arbitrary amount of agents in order
to minimize some performance metric associated with the
patrolling task.
Coordinating multiple agents to perform patrolling can be
reduced to a multiple traveling salesman problem, therefore
it is an NP-hard problem[3]. Patrolling algorithms either plan
the routes for each agent ahead of time and feed those routes
to the agents as a priori information or try to coordinate the
agents in real time by allowing them to communicate through
some wireless medium and employing various heuristics in the
agents to decide on the next graph node to visit.
Patrolling on graphs is either implemented by allowing
every agent to visit all nodes in the graph or the graph is
partitioned into subgraphs[4], [5] and the agents work only on
their own subgraph without interfering with each other. The
first case involves interference between the agents that can
result in agents blocking each other on the way from one node
to another. When agents try to move in the same direction or
meet at an intersection, they inhibit each other from carrying
out their tasks as fast as possible. These interference events
can have a negative impact on the performance and scalability
of the patrolling algorithm. Since patrolling is usually a long
running task, these interference events will have a continuous,
most of the time unpredictable effect on the performance.
To counter this effect, researchers started investigating
partition-based patrolling. Currently partitioning is usually
done before patrolling starts, on a central entity, and the result-
ing partitions are assigned to the agents as a priori information.
This method is not always feasible as environments can be
dynamic: appearing or disappearing graph nodes and failing
or newly introduced agents can trigger the recalculation of
partitions. Therefore agents need the central entity throughout
the patrolling task and need synchronization at times of
partition changes. A self-organizing partition based strategy
is more desirable, because agents need less synchronization,
require only local information and it makes the whole system
more robust against agent failures.
II. RELATED WOR KS
Patrolling strategies can be really simple heuristic based
approaches or complex learning algorithms that try to find
the optimal route for a multi-agent team. It is not always
clear whether a complex approach or a simple heuristic can
offer better performance. In [6] patrolling is cast into a multi-
agent Markov Decision Process and optimization techniques
are applied to efficiently find the optimal paths for agents
on the graph. However in the same article this solution is
compared to a simple reactive algorithm and performance
Proceedings of the Federated Conference on
Computer Science and Information Systems pp. 261–268
DOI: 10.15439/2018F213
ISSN 2300-5963 ACSIS, Vol. 15
IEEE Catalog Number: CFP1885N-ART c
2018, PTI 261

differences with regards to the instantaneous idleness metric
were negligible. This experiment and the conclusion of [7]
suggest that computationally complex approaches are not
always justified and simple reactive algorithms can match the
performance of their more elaborate counterparts.
Agents can be restricted to their own subgraph or parti-
tion and perform patrolling alone in that region. An early
theoretical study [3] gives some insight into the performance
characteristics of the partition-based and cyclic approaches
(for example the previously mentioned SingleCycle). Authors
of this article claim that the optimal patrolling strategy for a
single agent can be obtained by finding a shortest closed walk
on the graph. This can be reduced to the Traveling Salesman
Problem for which good approximations exist (for example
the Christofides algorithm with O(n3)for a 3
2approximation).
When multiple agents are considered, one strategy can be to
traverse the closed path with evenly distributed agents along
the path. The performance of such a strategy is bound by
the edge length distribution of the graph [3]. For example on
graphs with “long corridors" (edges connecting two distant
nodes) a partition-based strategy could be a better fit, because
those “long corridors" can be avoided by leaving them out of
the partitions assigned to the agents.
So far the solutions enumerated were dealing with the
multi-agent patrolling problem based on the assumption that
agents cannot interfere with each other. However, interference
can occur when two agents are too close to each other and
cannot proceed in the direction of their targets. Interference is
important when the algorithm is designed to be deployed on
physical agents and can render theoretically better algorithms
inferior in the real world. Accounting for interference results
in a much harder problem, because the theoretically best
results are no longer optimal if agents have to stop before
colliding with each other or possibly choose different targets.
Several new heuristic approaches are taking into account this
phenomena, for example in [8] authors implement Greedy
Bayesian Strategy (GBS) and State Exchange Bayesian Strat-
egy (SEBS). Both strategies require global communication
meaning that agents can communicate with all other agents
involved in the patrolling task. They employ reactive agents (so
the agents do not form plans ahead of time) and the decision
making algorithm that chooses the next graph node to visit
is based on Bayes’ rule. The latter strategy (State Exchange
Bayesian Strategy (SEBS)) aims to minimize the interference
between agents while patrolling, by communicating the agent’s
next node it is intending to visit. SEBS is able to achieve better
performance than any other heuristic studied in the article.
Interference minimalization and graphs with non uniform edge
length distributions contributed to the interest in partition
based strategies.
Numerous other approaches have been tried to address
the multi-agent patrolling problem, such as spanning tree[9],
[10] or graph partitioning based approaches[11], [12]. Others
involve task allocation based strategies[13] and evolution-
ary algorithms[14] or linear programming[15]. Auction based
strategies involve agents placing bids on nodes to patrol on the
graph. These kind of approaches implement a decentralized
task allocation, where the agents decide on the outcome and
able to reassign nodes or partitions of poorly performing
agents to others[16], [17].
III. THE MU LTI-AGE NT PATROLLI NG PROBL EM
In this article our goal is to further elaborate the perfor-
mance characteristics of our proposed PB PS algorithm [18].
In our model the only global knowledge the agents have is the
map they are patrolling on. They do not have the capability to
communicate globally with each other. Agents’ behavior will
result in solving the patrolling problem efficiently together,
without third party coordination.
In the model, the connected graph G(V, E )consists of
nodes v1, v2, ..., vn∈Vand undirected edges ei,j ∈E.
Nodes are the points of interest, they are visited by the agents
periodically. Each node has a counter that measures time
between the last visit to that node and the current time. This
counter is usually referred to as the instantaneous idleness of
the node at time tand is defined as Ivi(t) = t−I′
vi, where
I′
viis the time at the last visit to the node and vi∈V. In
some use cases patrolling is done to measure some physical
quantity at the point of interest periodically and as frequently
as possible. Let us suppose that the measurement at each point
of interest takes the same amount of time for each agent.
Agents’ movement is restricted to the edges between con-
nected nodes. Agents have a priori knowledge of the graph
nodes and edges, and are capable of moving between nodes
along the edges with constant and uniform speed. Agents
are able to communicate with each other, therefore influence
each other’s decisions by broadcasting messages via a wireless
medium.
Patrolling algorithms have very different characteristics with
respect to the routes they take, the order in which they visit
certain nodes, etc. To be able to compare their performances,
we have to define metrics that represent the goodness of an
algorithm from a certain point of view. The metric used for
this task is usually the average idleness or the worst idleness.
Both of these metrics are based on the instantaneous idleness
defined before as Ivi(t). This basic metric can be averaged in
a window or throughout the simulation to obtain the average
idleness, defined as:
Ivi(t) = Ivi(tvi)·Ci+Ivi(t)
Ci+ 1 (1)
where Ciis the number of visits to vi. The average graph
idleness is the average of average idleness values, such that:
IG(t) = 1
|V|
|V|
X
i=1
Ivi(t)(2)
A problem with the average graph idleness metric is that
there can be two patrolling algorithms with the same average
graph idleness but significantly different behaviors. Average
graph idleness masks important characteristics of patrolling
algorithms and as a consequence, its value can be misleading
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when comparing different algorithms. The idleness times be-
tween visits are samples from an unknown distribution and to
reason about that unknown distribution based on the mean of a
limited amount of its samples can be misleading. Therefore in
this article we use the distribution of the worst idleness values
to compare performances. The worst idleness associated with
a node between times taand tbis defined as the maximum
instantaneous idleness value:
W Ivi=max{Ivi(ta), ..., Ivi(tb)}(3)
The worst idleness times are the worst case scenario, there-
fore by gathering enough samples, one can be reasonably
sure of an upper bound of the idleness times that can be
observed during patrolling. Patrolling use cases like reading
measurements from sensors or taking photographs of certain
physical locations usually benefit from a known maximum
time between measurements. The worst idleness distribution
makes it possible to evaluate patrolling algorithms based on
this criteria. It also shows the fairness of the algorithm: the
difference between the minimum and maximum worst idleness
values on the graph over time. Moreover the distribution of
worst idleness can be easily visualized using box plots.
IV. PARTITIO N BAS ED PATROLLI NG ST RATE GY (PBP S)
The patrolling strategy we use in this article follows the
partition based approach. Based on [3], [19], the performance
of such a patrolling algorithm is in theory inferior compared
to a cycle-based strategy on graphs without “long corridors".
However, if the model accounts for the effects of interference
between agents, this may no longer be true. Partitioning the
graph between the agents is not a new idea, authors in [8] and
[5] improved their patrolling algorithms by limiting the amount
of interference between agents. Partition based strategies take
this idea to the extreme as there is no interference between
agents, they patrol their own partitions independently.
PBPS was published in [18] along with the first results
obtained on well-known graphs. Here we give only a short in-
troduction to the algorithm. PB PS has two important assump-
tions about the environment: agents can localize themselves
and they know the structure of the graph a priori. It should be
noted, that these assumptions are not typical in self-organized
systems, because they require agents with greater knowledge
than usual. However the information requirement of PBPS
is the same as other patrolling algorithms, such as SE BS or
Greedy Bayesian Strategy (GBS). The problem of discovering
the environment and forming consensus about it among the
agents is not discussed in this article.
The goal of the algorithm is to partition the graph into non-
overlapping connected subgraphs and in the process create a
global partitioning that enables the efficient patrolling of the
graph. All agent’s partitions are empty in the beginning. A
partitioning of Gis defined as P={P1...Pk}, such that P1∪
... ∪Pk=Vand Pi∩Pj=∅.{G1...Gk}refer to subgraphs
induced by the partitioning, thus Gi= (V∩Pi, E ∩(Pi×
Pi)). The partition growing process is self-organized, such that
there is no global coordination between agents and no global
T0
T1
T2
agent1agent2agent3
Fig. 1: Claiming of nodes between agents in order to achieve
a balanced partitioning.
state shared among them. Agents grow their own partitions by
claiming nodes from each other’s partitions and broadcasting
their new partition after claiming a new node. In the process
(see for example Figure 1) of growing partitions, there can
be local violations to the strict non-overlapping partitioning
introduced above, but these are temporary and will resolve
over time. The algorithm always reaches a state, where the
partitions are stable, i.e. no more changes made by the agents.
When agents arrive at a stable configuration, each partition
will belong to one and only one agent and each partition will
be a connected subgraph with no overlapping nodes with other
partitions. The resulting partitioning assigns a subgraph to each
agent to patrol with roughly equal number of nodes.
V. SIMU LATI ON E NVIRONME NT
We have conducted numerous simulations to investigate the
performance characteristics of PB PS and compared patrolling
on the formed partitions to well-known patrolling algorithms.
First we describe the environment in which the experiments
were carried out as it was designed to resemble possible real-
world usage scenarios. We used the osmnx python library[20]
to download maps of two Hungarian cities: Budapest and
Komárom. We chose Budapest, since it is the capital of Hun-
gary and like large cities it has a lot of intersections and shorter
edges. Komárom on the other hand tends to have differently
distributed edge lengths and node degrees (as can be seen
on Fig. 2 and 3) as it is a smaller city on the countryside.
Node degrees tend to be lower for Komárom but edge lengths
are significantly longer than the ones for Budapest. We chose
two different real maps to model possible usage scenarios
and also to have environments with different characteristics.
Our partitioning strategy is based on the assumption that the
edge length distribution of the graph is more or less even.
The map excerpt from Komárom has a longer tail in its edge
length distribution, therefore more variety than the map excerpt
from Budapest. The downloaded maps were simplified as the
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0.0
0.2
0.4
0.6
0.8
0 5 10 15 20 25
Node degree
density
Degree distribution for "Budapest"
0.0
0.2
0.4
0.6
0 5 10 15 20 25
Node degree
density
Degree distribution for "Komarom"
0.00
0.05
0.10
0.15
0.20
0 5 10 15 20 25
Node degree
density
Degree distribution for "Small world"
Fig. 2: Degree distributions of the maps used for the experi-
ments.
0.000
0.005
0.010
0.015
0.020
0 50 100 150 200
Edge length
density
Edge length distribution for "Budapest"
0.0000
0.0025
0.0050
0.0075
0.0100
0 50 100 150 200
Edge length
density
Edge length distribution for "Komarom"
0.00
0.01
0.02
0.03
0.04
0 50 100 150 200
Edge length
density
Edge length distribution for "Small world"
Fig. 3: Edge length distributions of the maps used for the
experiments.
original Open Street Map data contains a lot of short edges and
unnecessary nodes. The simplification step was done using a
built-in functionality in osmnx. A third map was introduced,
called Small World, which is a generated small world type
graph (Barabási-Albert graph) with a completely different
node degree distribution than the other two maps. This graph
has 50 nodes and was grown with 3edges preferentially
attached to existing nodes using the networkx python library’s
built in graph generator. This third environment has some
really high degree nodes and claiming those nodes in the
partitioning phase can effect other partitions, as these high
degree nodes act as access points to the rest of the graph.
We carried out experiments in these graphs with three
different patrolling algorithms with parameters given in Table
I. First, Conscientious Reactive[21] was used as a baseline as
it is a greedy heuristic with no communication requirements
among the agents. The second algorithm tested was State
Exchange Bayesian Strategy[8], which is based on the idea that
agents can perform better if they exchange their intentions and
Parameter Value
Number of agents 1,2,4,...,32
Double number of agents every 200000s
Number of iterations 10
Agent speed 1m/s
TABLE I: Simulation parameters for the local vs. global metric
experiment.
0
50
100
150
1 2 4 8 16 32
Number of agents
Size of partitions
Partition sizes for "Budapest"
Fig. 4: Partition sizes for Budapest
factor the intentions of other agents in when making a decision
which node they visit next. The third was PBPS, which forms
partitions on the map in a self-organized manner and agents
perform patrolling on their own partition only.
VI. PARTI TIONI NG A LG ORITH M RE SULTS
The goal of this experiment was to gain insight into the
performance characteristics of the partition forming part of the
PBPS algorithm. To this end we have disabled the mobility of
the agents and tracked the partitioning events throughout the
simulation. Partitioning events are either node free or nonfree
node claims, or partition resets. In the beginning of every
simulation run, only one agent is on the map at a randomly
chosen node and it starts forming its partition. After this initial
agent claims the whole map as its partition there are no more
partitioning events occurring. The simulator waits until there
are no more partitioning events for 5000 time steps and then
doubles the number of agents on the map. The agents already
on the map retain their partitions and the new agents are
assigned a random starting node as their initial partition. This
mechanism for adding agents sometimes cause overlaps in the
partitioning, because the previous generation of agents claimed
the whole map together (every node belongs to one and only
one agent). However this is not a problem as the first time a
new agent broadcasts its initial partition, the overlaps will be
resolved by the old agent releasing that node from its partition.
The number of agents simulated on each map are exponen-
tially increasing (1,2,...,16). The exponential increment in
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0
50
100
150
1 2 4 8 16
Number of agents
Size of partitions
Partition sizes for "Komarom"
Fig. 5: Partition sizes for Komárom
0
50
100
150
1 2 4 8 16
Number of agents
Size of partitions
Partition sizes for "Small world"
Fig. 6: Partition sizes for Small World
the number of agents can reveal the scalability characteristics
of the partition forming part of PB PS with respect to the
different maps. As a performance metric, the time to arrive
at a stable partitioning was measured for different number of
agents (1,2,4,...,16) and at the same time the number of
nodes in each partition were tracked.
This was done in order to confirm whether a stable parti-
tioning is also a balanced one. A balanced partitioning in this
terminology is where the number of nodes in each agent’s
partition is close to the number_of_nodes
number_of _agents number. In other
words, a balanced partitioning is where each partition has a
roughly equal amount of nodes. Depending on the structure of
the map this balanced property might not be achievable, since
the tested maps are not full graphs with an edge connecting
every node. Figures 4, 5 and 6 report the distribution of
Num. of agents 1 2 4 8 16 32
Budapest belváros
Mean 146.0 73.0 36.5 18.25 9.125 4.5625
Var 0.0 0.0 0.2564 0.3670 0.6383 0.3534
IDI 0.0 0.0 0.0070 0.0201 0.0699 0.0774
Komárom
Mean 59.0 29.5 14.75 7.3750 3.6875 N/A
Var 0.0 0.2631 0.5000 1.1487 0.3671 N/A
IDI 0.0 0.0089 0.0338 0.1557 0.0995 N/A
Small world
Mean 50 25 12.5 6.25 3.125 N/A
Var 0.0 0.0 0.2564 0.1898 0.1729 N/A
IDI 0.0 0.0 0.0205 0.0303 0.0553 N/A
TABLE II: Mean, variance and index of dispersion values of
resulting partition sizes for all three tested maps and different
number of agents.
Fig. 7: Times of last partitioning events for versus number of
agents for Budapest
partition sizes versus the number of agents. These are box
plots[22] where the box contains 75% of the observed values.
The blue curve shows the balanced partition sizes for differ-
ent number of agents, its value is number_of _nodes
number_of _agents , where
number_of _nodes is constant and a property of the map. The
results here show that in all simulated experiments the parti-
tioning algorithm was able to arrive at solutions close to the
optimal. Some exceptions show up as outliers on the box plots
and indicate that the initial conditions have some influence
on the final partitioning. All simulations were done 10 times
with different starting positions for the agents, therefore each
box represents the distribution of 10 ∗number_of _agents
different resulting partition sizes.
Additionally the mean, variance and index of dispersion[23]
metrics were calculated for the three different maps and for
the different agent numbers. These results are summarized in
Table II.
The next set of results discuss the time requirements of
the partitioning algorithm, paying attention to its scaling
properties. The question here is how does adding more agents
to a map influence the time needed to arrive at a stable
configuration? A stable configuration is one, where no more
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Fig. 8: Times of last partitioning events for versus number of
agents for Komárom
Fig. 9: Times of last partitioning events for versus number of
agents for Small World
partitioning events occur. One of the advantages of using a
self-organized approach is that in theory it is scalable and
can enable multi-agent patrolling on larger graphs for a larger
number of agents, than centralized algorithms. Figures 7, 8 and
9 show the distribution of time of the last partitioning event for
each different simulated agent population versus the number of
agents across all 10 iterations. Each group of values consists of
10 measurements and a curve is fitted to show the observed
general trend of the values. It is important to note that the
5000 time step period between agent additions is subtracted
from the values on the figures to show an ideal situation where
one could determine the exact moment when the partitioning
stops. The figures show that although the number of agents
increase exponentially, the time needed to partition the graph
increases only linearly. This indicates, that for the tested
maps, the algorithm behaves well with a large number of
agents and might be suitable to handle very large number
of agents partitioning the same graph simultaneously. The
variance of the values increase for larger number of devices.
This indicates that the intermediate resulting partitionings and
randomly assigned starting nodes for added agents (initial
condition for each addition step) has an increasingly large
effect when the number of agents are high. It is possible
that by carefully choosing the insertion point for new agents,
the observed variance might be decreased. However in this
analysis we were interested in the possible outcomes and was
aiming of getting a full picture of the dynamic behavior PBPS,
rather than optimizing its parameters for ideal performance.
VII. PATROLLI NG R ESULTS
After investigating the different aspects of the partitioning,
we compared the PB PS algorithm with two other algorithms:
Conscientious Reactive (CR) and State Exchange Bayesian
Strategy (SEBS). Conscientious Reactive (CR) is a simple
greedy reactive algorithm, where the agent makes decisions
based only on the visits made by itself. Other agents are not
taken into account, therefore no communication is needed by
this algorithm. Upon arriving to a node it resets the node’s
idle timer in its own local data structure and evaluates the
node’s neighbors in the graph. For the next node to visit the
agent chooses the one with the largest instantaneous idle time
value. CR is a trivial algorithm that requires only an agent with
the capability to move and store the graph and timers in its
internal memory, therefore the algorithm is an ideal baseline
to compare other, more sophisticated algorithms to.
The State Exchange Bayesian Strategy (SE BS) strategy
takes into account the interference caused by other agents in
the same area. Agents can inhibit each others’ movements
by standing in the way or forcing other agents to move
slower to avoid collision. This effect can cause strategies
like CR and G BS to perform poorly in situations, where
the number of agents in an area becomes high. GB S and
SEBS builds on the idea of Bayesian decisions, where the
a priori probability to visit a neighbor node is offset by the
instantaneous idleness of the node known to the agent. This
way GB S can take into account priorities input by the designer
and the actual state of the system (instantaneous idleness of the
nodes). SE BS extends GBS by having agents broadcast their
intentions (the node they are intend to visit next) for their
immediate neighbors. This information is used in all agents
when evaluating the agent’s next node to visit. If one or more
agents intend to visit the same node, it is not beneficial to let
another one go in that same direction, even if the instantaneous
idleness value of the node is high. SEBS aims to lower the
amount of time an agent loses by executing collision avoidance
behaviors, causing them to slow down or chose a different
target node altogether.
In these simulation runs, each group size (number of agents)
ran for 200000 time steps, and after the number of agents were
doubled. This process was repeated until there were 16 agents
for Komárom and Small World, and 32 agents for Budapest on
the map. We ran 10 iterations for each algorithm-map pair with
random starting positions for every agent for every iteration.
The structure of the graphs and the starting position of the
agents can influence their measured performance, therefore in
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0
20000
40000
60000
80000
1 2 4 8 16
Number of agents
Worst idleness
Strategies
CR
SEBS
PBPS
Fig. 10: Worst idleness time distributions for different strate-
gies versus number of agents for Budapest
0
10000
20000
30000
40000
50000
1 2 4 8 16
Number of agents
Worst idleness
Strategies
CR
SEBS
PBPS
Fig. 11: Worst idleness time distributions for different strate-
gies versus number of agents for Komárom
0
1000
2000
3000
4000
1 2 4 8 16
Number of agents
Worst idleness
Strategies
CR
SEBS
PBPS
Fig. 12: Worst idleness time distributions for different strate-
gies versus number of agents for Small World
order to gain as much information about the performances
of different algorithms as possible, random non-overlapping
starting points were chosen for the agents when they were
added to the team. In the case of CR and SE BS strategies
this means a random node from the graph and in the case of
PBPS, this results in a random node from the agent’s partition.
Partitioning strategies have a tendency to perform worse
when they start on a graph, because they base their decisions
on idleness times (and other agents’ intentions in the case
of SE BS), but in the beginning this data is not available for
them. Therefore the first 100000 time steps for each different
configuration were dropped in order to let the agents “warm
up" and the worst idleness times for each node were collected
for the last 100000 of each configuration. Figures 10, 11 and
12 report the worst idleness time distribution measured on the
whole map for different strategies and different number of
agents.
For one agent, CR and PBPS are almost identical, this is
caused by the fact that PBPS employs CR as the patrolling
strategy in the agents’ partitions. In all tested configurations
the performance of PB PS were better than the other tested
algorithms. Comparing worst idleness distributions between
SEBS and PBPS reveals that PBPS requires on average half
the amount of agents to achieve the same performance as
SEBS. In the case of map Komárom this effect is even more
pronounced as the required number of agents for PB PS is one
fourth of that for SE BS.
Another interesting facet is that SE BS tends to have a
distribution with a longer tail than CR and P BPS. This means
that the majority of values are larger than the mean for maps
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Budapest and Komárom (see in Figures 10 and 11), which
indicates that judging the performance of SE BS by the mean
of its worst idleness times can be misleading, by indicating
that the average time between visits to nodes in the map is
smaller than what can be interpreted from the distribution
of the values. This property means that performance metrics
reported in articles like [8] can be only part of the whole
picture as one cannot judge every aspect of the performance
of the algorithms accurately from the average idleness values.
VIII. CON CL US ION
We have elaborated on the performance characteristics of
the PB PS algorithm first published in [18]. The result of the
partitioning algorithm is a set of subgraphs formed by the
individual agents cooperating with each other that is used by
patrolling agents as their patrolling tasks. These subgraphs
are formed cooperatively by the agents on the graph without
any central help or control. Using subgraphs of the original
graph allows to completely avoid interference between agents,
therefore have them performing in a parallel and deterministic
manner. The cost function used in this article was the differ-
ence in number of nodes between partitions, but changing this
cost function opens up possibilities for designers of patrolling
systems to tailor the resulting partitions to their use case.
Simulations were performed on three different maps, rep-
resenting different environments: the inner city of Budapest
with shorter edges and more neighbors, part of the small
city Komárom with longer edges and fewer neighbors and
a random graph, which is a generated small world type
graph (Barabási Albert graph). PBPS arrived at a balanced
partitioning in every simulation run with low variance in the
number of nodes assigned to an individual agent. Moreover
PBPS scales well with the number of agents, meaning that
the time needed to finish partitioning increased sublinearly
with the number of agents tested. This property is in contrast
with planning approaches, that usually have worse scaling
properties, making it viable to employ this strategy on larger
graphs and with a larger amount of agents.
We compared the performance of agents using different
patrolling strategies, such as CR, SEBS and PBPS. Patrolling
performance was judged by the distribution of worse idleness
times for all nodes in the graph, represented as box plots. In
every tested case PBPS came out ahead, and on two maps the
necessary number of agents for PB PS were half of what was
required by CR and SEBS for the same performance. This
results in a tradeoff decided by the designer or implementor
later: either employ the same amount of agents and get lower
worst idleness times throughout the graph, or in a resource
constrained scenario, change the patrolling strategy and get
the same performance level with only half the agents required
by other strategies.
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