On Swarm Optimality in Dynamic and Symmetric

Environments

⋆

Yaniv Altshuler

1

, Israel A. Wagner

1,2

and Alfred M. Bruckstein

1

1

Computer Science Department, Technion, Haifa 32000 Israel

2

IBM Haifa Labs, MATAM, Haifa 31905 Israel

Abstract. The ﬁeld of multi agents and multi robotics has become increasingly

popular during the last two decades. The motivation behind multi agents based

systems is that many tasks can be much efﬁciently completed by using multiple

simple autonomous agents (robots, software agents, etc.) instead of a single so-

phisticated one. However, when examining such systems, one may be concerned

of the price-tag attached to the decentralized nature of swarm based approaches.

Meaning, while we simplify designs and control mechanisms in order to save

costs and computation resources, how far do our systems drift from optimal-

ity ? This work examines this issue by constructing an optimal algorithm for

the Dynamic Cooperative Cleaners problem (presented and analyzed in [2]). The

performance of the SWEEP algorithm of [2] is compared to this of an optimal

algorithm. The results of this comparison show that not only that the swarm al-

gorithm produces close results to the optimal solution, but also as the problem

gets harder, the performance of the two converge. In addition, insightful results

concerning optimal swarms in symmetric environments are presented.

1 Introduction

The ﬁelds of multi agents and multi robotics have become increasingly popular during

the last two decades. A multi agents system, or a swarm, can generally be deﬁned as a

decentralized group of multiple autonomous units, either homogenous or heterogenous,

such that those units are simple and possess limited capabilities. Many research efforts

have been invested examining distributed systems models inspired by biology (behav-

ior based control model — [16,13], ﬂocking and dispersing models — [22,18,19] and

predator-prey approach — [14,21]), physics [12], and economics [7–11]. The motiva-

tion behind multi agents and multi robotics systems is that by using multiple simple

autonomous agents instead of a single sophisticated one, many tasks can be performed

cheaply and easily, while the performance of those system remains satisfactory. In addi-

tion, such systems have several other advantages such as high scalability and adaptivity.

Application for which multi robotics systems ﬁt successfully include covering, explo-

ration and patrolling [1], construction of complex structures and self-assembly [17, 15,

26], mapping and localizing [20, 27] and transportation [23,25, 24].

⋆

This research was partly supported by the Ministry of Science Infrastructural Grant No. 3-942

Altshuler Y., A. Wagner I. and M. Bruckstein A. (2005).

On Swarm Optimality in Dynamic and Symmetr ic Environments.

In Proceedings of the 1st International Workshop on Multi-Agent Robotic Systems, pages 64-71

DOI: 10.5220/0001196900640071

Copyright

c

SciTePress

However, when designing such systems, one must take into account the decrease in

performance which is inherent in such approaches, in comparison to an optimal central-

ized system (albeit much more complex and expensive).

In this work we examine this issue by constructing a centralized optimal algorithm

for the Dynamic Cooperative Cleaners problem (presented and analyzed in [2]. Similar

works appear in [4–6]). This problem assumes a grid, part of which is “dirty”, when the

“dirty” part is a connected region of the grid. On this dirty region several agents move,

each having the ability to “clean” the place it is located in. A deterministic evolution of

the environment is assumed, simulating a spreading contamination, or ﬁre.

A cleaning algorithm designed to be used by a swarm of cleaning agents is pre-

sented and discussed in [2]. In addition, a lower bound over the cleaning time of agents

employing any algorithm (i.e. a lower bound over an optimal algorithm for the prob-

lem) is presented. The performance of the optimal algorithm described in section 4 is

compared to those of the sub-optimal SWEEP algorithm (described in section 3) and

the generic lower bound for the problem of [2].

The results of this comparison surprisingly show that although the SWEEP algo-

rithm is fully decentralized and extremely simple, assuming completely autonomous

agents and using no explicit form of communication, the improvements which can be

achieved by using an optimal algorithm is in no way signiﬁcant enough to justify the

immense costs and complexity of such an algorithm.

In addition, several insightful and counter-intuitive results concerning optimal clean

strategies for symmetric environments are presented in section 6.

2 The Dynamic Cooperative Cleaners Problem

Let us assume that the time is discrete. Let G be a two dimensional grid, whose vertices

have a binary property of ‘contamination’. Let cont

t

(v) state the contamination state

of the vertex v in time t, taking either the value “on” or “off”. Let F

t

be the dirty sub-

graph of G at time t, i.e. F

t

= {v ∈ G | cont

t

(v) = on}. We assume that F

0

is a single

connected component. Let a group of k agents that can move across the grid G (moving

from a vertex to its neighbor in one time step) be placed in time t

0

on F

0

.

Each agent is equipped with a sensor capable of telling the condition of the square it

is currently located in, as well as the condition of the squares in the 8−Neighbors group

of the this square. An agent is also aware of other agents which are located in its square,

and all the agents agree on a common “north”. Each square can contain any number

of agents simultaneously. When an agent moves to a vertex v, it has the possibility

of causing cont(v) to become off. The agents do not have any prior knowledge of the

shape or size of the sub-graph F

0

except that it is a single connected component. Every

d time steps the contamination spreads. That is, if t = nd for some positive integer n,

then (∀v ∈ F

t

, ∀u ∈ 4−N eighbors(v) : cont

t+1

(u) = on). The agents’ goal is to

clean G by eliminating the contamination entirely, so that (∃t

success

: F

t

success

= ∅).

In addition, it is desired that this t

success

will be minimal.

3 The SWEEP Cleaning Algorithm

For solving the Dynamic Cooperative Cleaners problem the SWEEP algorithm was

suggested in [2]. The algorithm can be described as follows. Generalizing an idea from

computer graphics (which is presented in [3]), the connectivity of the contaminated re-

gion is preserved by preventing the agents from cleaning what is called critical points

— points which disconnect the graph of contaminated grid points. This ensures that the

agents stop only upon completing their mission. At each time step, each agent cleans

its current location (assuming this is not a critical point), and moves to its rightmost

neighbor — a local movement rule, creating the effect of a clockwise traversal of the

contaminated shape. As a result, the agents “peel” layers from the shape, while preserv-

ing its connectivity, until the shape is cleaned entirely.

4 An Optimal Cleaning Algorithm

In order to ﬁnd the minimal cleaning time possible for a given contaminated shape F

0

,

we have designed a system in which all the cleaning agents are controlled by a central

unit (referred to as the queen). Upon initialization, the queen is given the complete

information regarding the contaminated shape. While the agents are traveling along the

grid, the queen is immediately aware of any new information discovered by the agents.

The queen’s orders as to the next desired movements of the agents are also immediately

transferred to the agents, which carry them out automatically.

Since the agents are completely bound to the desires of the queen, lacking even

the slightest amount of autonomy, this mechanism can be thought of as one big robot,

equipped with numerous long “cleaning hands”. Implementing such mechanism will

of course be very complicated and there is no doubt that such a robot will be very

expensive (remembering that the “cleaning hands” are capable of moving to very large

distances from one another). However, since we are interested in the optimality issue,

costs and complexity aspects are not of interest to us, at the moment.

For calculating the optimal solution to the problem, the queen uses the well known

A

∗

algorithm [28]. Upon initialization, the queen exhaustively searches for the shortest

path within the states space, where the initial state contains F

0

and the initial locations

of the agents. Every state can be developed into 2

k

states (since every agents can make

a single move clockwise or counterclockwise). Since the queen possesses complete

information regarding F

0

and knows exactly what the agents are doing, there is no

need for the agents to not clean certain contaminated squares (note that in the SWEEP

algorithm for example, sometimes the agents do not clean contaminated squares in order

to preserve the connectivity of the contaminated shape). Every d transitions between

states, the queen simulates a contamination spread. The success state is a state in which

there are no longer contaminated squares.

Once ﬁnding the shortest path within the states space from the initial state to a

success state, the queen starts moving the agents according to this path.

The optimality of this algorithm is immediately derived from the optimality of the

A

∗

algorithm in ﬁnding shortest paths in states spaces.

5 Experimental Results

A computer simulation of both the optimal algorithm described in section 4 and of

the SWEEP algorithm of [2] was constructed. Note that the size of the states spaces

covered by the optimal algorithm was at least :

2

k·f (F

0

)

(1)

where k is the number of agents, F

0

is the initial contaminated shape, and f() is the

lower bound over the cleaning time of [2], deﬁned as :

f(F

0

) ≡ min{t|t ∈ N ∧ S

t

≤ 0} (2)

where :

S

t+d

≥ S

t

− d · k + (

p

8 · (S

t

− d · k) − 4 + 2) (3)

d is the number of time steps between contamination spreads and S

0

is the initial

area of F

0

, namely — S

0

= |F

0

|.

Three types of shapes were examined, varying in their convexity (i.e. their

S

0

c

0

value,

where c

0

is the initial circumference of the shape). Shapes whose

S

0

c

0

value is high are

referred to as convex shapes. Similarly, semi-convex and concave shapes are deﬁned.

For each type of shape, three values for S

0

were chosen. For each value of S

0

,

three values for k were selected. For each set of values, six random simulations were

performed, in order to achieve statistical signiﬁcance.

Figure 1 presents the results of the simulations, including the predictions of the

lower bound presented in equation 2. Note that although the sub-optimal cleaning al-

gorithm achieves results which are roughly 400% slower than the estimations of the

lower bound, the optimal algorithm achieves results which are also relatively far from

the lower bound’s estimations. This demonstrates the fact that the sub-optimal algo-

rithm’s performance are in fact, much closer to the optimal performances than expected

initially in [2].

After analyzing the results and trying to ﬁnd a correlation between the number of

agents to the

perf ormance

optimal

perf ormance

sub−optimal

ratio, it seems that such correlation does not exist.

However, when focusing on the largest number of agents examined (8 agents), it can

be seen that as the size of F

0

gets bigger, the

perf ormance

optimal

perf ormance

sub−optimal

ratio gets smaller.

This holds for all three types of shapes. The meaning of this observation is that as the

problem is getting harder (larger initial shapes and more agents), the beneﬁt from using

an optimal algorithm reduces (meaning, the sub-optimal algorithm achieves better and

better results). This is demonstrated in ﬁgure 2.

6 Symmetric Environments

In this section we shall present several comparisons of the SWEEP sub-optimal al-

gorithm and our optimal algorithm, in geometrical symmetric environments. Figure 3

presents the symmetric initial shapes chosen for this comparison. Four agents were

placed in symmetric places along the shapes, and both algorithms were tested.

30 40 50

10

15

20

25

30

35

40

Size of contaminated shape

Time steps

4 agents

6 agents

8 agents

Convex Shapes

30 40 50

10

15

20

25

30

35

40

Size of contaminated shape

Time steps

4 agents

6 agents

8 agents

Semi−Convex Shapes

30 40 50

20

30

40

50

60

70

80

Size of contaminated shape

Time steps

4 agents

6 agents

8 agents

Concave Shapes

30 40 50

0

5

10

15

20

25

30

35

40

45

50

Size of contaminated shape

Time steps

4 agents

6 agents

8 agents

Average

Fig.1. Comparison of sub optimal and optimal algorithms. The lower and thicker three lines rep-

resent the cleaning time of the optimal algorithm whereas the upper lines represent the SWEEP

cleaning algorithm’s performance. In the right chart on the bottom, the lower three lines represent

the lower bound of the optimal solution, as appears in equation 2.

30 40 50

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

Size of contaminated shape

Optimal / Sub−Optimal

Convex Shapes

Concave Shapes

Semi Convex Shapes

Optimality Ratio

(8 Agents)

Fig.2. Comparison of sub optimal and optimal algorithms. The Y -axes represents the

perf ormance

optimal

perf ormance

sub−optimal

ratio for various sizes. Note that as the problem gets bigger, the sub-

optimal performance of the algorithm gets closer to those of the optimal algorithm.

Fig.3. Symmetric initial shapes referred to as Shape-I, Shape-II and Shape-III respectively (left

shape is Shape-I).

Shape−I Shape−II Shape−III

0

50

100

150

200

250

300

350

400

450

Cleaning Time

Fig.4. Comparison of sub optimal and optimal algorithms. As can be seen once again, the per-

formance of the sub-optimal algorithm are only roughly 60% worse than those of the optimal

algorithm.

The comparison between the cleaning time of the optimal and sub-optimal algo-

rithms are presented in ﬁgure 4.

It is interesting to state that although the SWEEP algorithm, according to its deﬁn-

ition, generates a symmetric behavior of the agents (meaning, the agents are traversing

the shape in the same manner, creating symmetric effects over it) this is not the case for

the optimal algorithm. Counter-intuitively, the optimal algorithm generates a remark-

ably non-symmetric behavior. The agents are divided into non-identical and non-stable

groups, while the effects the activities of the agents have on the shape transform it

quickly into a truly non-symmetric shape. Interestingly enough, this behavior occurs in

all three shapes (as well as in other symmetric shapes).

This insightful observation may lead to a change in the concepts which shape the

design of swarm algorithms in the future. Hitherto, it seems that most such works tend

to be biased by the belief that symmetric behavior generates high quality results while

being implemented into swarms’ behavior. Nevertheless, by ignoring this miss-belief,

an improvement in the performance of such algorithms may be achieved.

The reason for this may rely in low robustness of symmetric behaviors, which are

prone to “error resonance”, meaning — signiﬁcantly increasing the effect of a small

error embedded within the swarm’s algorithm. By intentionally avoiding a symmetric

behavior, such traps may also be avoided, increasing the swarm’s robustness and sub-

sequently, its performance.

A demonstration of the above appears in ﬁgure 5.

Fig.5. The upper drawings demonstrate the symmetric nature of the SWEEP algorithm, while

the lower drawings demonstrate the non-symmetric behavior that was shown to be optimal for

Shape-II.

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