Look, Compute, Move Model

We considerrobotsmodelled aspoints in theirenvironment thatarecontinuously ob-taininginformation fromtheirsurroundings, and moveeitherin synchronous fashion, or hâvetheirwalkcontrolledby anadversary, whichmeansthat thoughthe paththey would take is determined by the algorithm and could not be steered from, the speed they useto walk on it is controlled by an adversary (which could use négative speeds as well) at ailtime, providedthat they coveredtheentirety ofthepath. Thefollowing papers consider a different model: in this model, robots will in turn obtain informa­

tion about their surroundings, often obtaining information with perfect visibility of the entirety of the environment (LOOK), use a deterministic algorithm to décidé of theirnext destination (COMPUTE), and fînally, move in a given direction (MOVE).

The notion of synchronism is different here as well: for robots to be asynchronous means that they execute their LCM cycles independently from one another. If they are synchronous, they start each LCM cycle exactly at the same time. This model also allows for the introduction ofa new synchronism pattern: the semi-synchronous model.

The problem of searching the plane by a mobile agent has first been identified as a field of study in [11]. The model used can be considered a precursor to the Look-Compute-Move (LCM) model that will be discussed later on. In this paper, a mobilerobotistaskedwith asearchproblem inanunknown,unboundedenvironment, and itsperformances are evaluated by consideringthe totaldistance covered between every probe position taken. Various scénarios are explored, with varying level of a priori information given to the robot, such as the distance to the target and the

general direction ofthe target. The paper also examines the possibiiity oferrors in the interprétation of the environment by the robot.

ThegatheringproblemforasynchronousrobotsoperatinginLCMcyclesisstudied in [72]. Robots are equipped with multiplicity détection: during their Look phase, they see if any node is occupied by zéro robots, one robot, or two or more robots (without knowing the spécifie amount). This paper identifies the initial configura­

tions in which the gathering problem can be solved, and provides an algorithm that generates a solution whenever possible.

The graph exploration problem by a swarm ofrobots operating in asynchronous LCM cycles is studied in [115]. In a given laps of time, the swarm of robots rnust explore ail nodes, then put an end to their activities (quiescent State). The efficiency ofthe algorithm is measured by the numberofrobots required to complététheexplo­

ration in thegiven time. Itisshown theeven in n-node treeswith amaximum degree of4, fi(n) robots are necessary to complété the exploration. Ifthe maximum degree ) robots are sufficient, and this solution is asymptotically

logre loglogn

of the tree is 3, 0(

optimal.

[191] studies rendezvous oftwo asynchronous oblivious anonymous robots in the two-dimensional Euclidean spacein theabsence ofacommon coordinate System. An algorithm that allows forrendezvous in a finitenumberofcycle isprovided under the assumption that the compassésdiffer from at much 7r/4.

Basedonthefactthatrendezvousbetween twosemi-synchronousrobotsistrivially solvable when their coordinate Systems are consistent, [145] explores the magnitude of consistency between coordinate Systems required robots to ensure rendezvous in the semi-synchronous and asynchronous model. In the paper, robots hâve unreliable compassés: their bearings may deviate from an absolute reference direction. From there, two scénarios are considered: in the first one, the compass is static, and the

déviation remains the samethroughout the executionofthe algorithm. In the second the compass is dynarnic, and the déviation may vary at the start of every new LCM cycle. For robots with static compass in both the semi-synchronous and asyn-chronous setting, a déviation of $ < 7r/2 allows for a rendezvous; a déviation of at most <F < 7r/4is requiredforsemi-synchronouswithdynarnic compassésto meet; and adéviation ofat most $ < 7r/6 is required for asynchronouswith dynarniccompassés to meet.

one

In a similar fashion, [144] considers gathering of more than two robots with un-reliable compassés, both for the case of static and dynarnic compassés, under the semi-synchronoussetting. An algorithmallowinggatheringis provided foracompass that accepts déviations ofup to 7t/2 — e, with e > 0. This algorithm is proven to be optimal.Forany déviationgreaterthan 7t/2, it is proven that noalgorithm canensure either rendezvous or gathering.

[200] considers anonymous, oblivious robots operating in LCM pattern with an interesting added feature: both robots carry coloured lights that are visible to both robots. They are placed either in the plane or on a line, and their purpose is to reduce (or increase) the distance between them bv a constant factor without using distance information. A solution is provided with spécifications on the number of colours required to ensure rendezvous. Every synchronism model is explored. In the asynchronous model, three colours are enough to ensure rendezvous for any initial configuration.

In [123], anonymous robots hâve to meet in the plane under the asynchronous setting. Twosettingsarestudied: inthefirstone,robots areincapableofremembering the layout of the environment in their previous cycle, but can communicate with constant-size messages (finite-communication model), and in the second, robots can remember the layout of the environment in their previous cycle using constant-size

memory, but are unable to communicate{finite-state model). Those settings can be modelled as robots carrying lights: in the finite-communication model, robots can only see the other robot’s light, whereas in the finite-state model, robots can only see their own. It is shown that finite-communication model allows rendezvous for asynchronous robots, and finite-state model allows rendezvous for semi-synchronous robots.

The relationship between synchronism models is studied in [77], under the as-sumption that robots are equipped with lightsof differentcolours that are persistent - that is, the light is not automatically reset at the end of each LCM cycle. It is shown that asynchronous robots with a constant number ofcolours are more pow-erful than semi-synchronous robots without colours. Furthermore, it is shown that there is no différence between asynchronous and semi-synchronous robotswhen they are equippedwith visiblelights. Asynchronous robotswithvisiblelightsarealsomore powerful than synchronised robots under the assumption that they hâve the ability to remember a single snapshot of the graph.

Census of the capability of robots operating in LCM cycles under synchronous, semi-synchronous and asynchronoussettingsare presented in [184] and [182], and the problems ofrendezvous, gathering and pattern formation are discussed.

Gathering in the ring environment is discussed in [152], under the asynchronous setting. It is shown that for an odd number of robots, gathering is only possible if their initial configuration is not periodic, and an algorithm is provided to solve the gathering problem. For even number of robots, the feasibility is decided except for one type of symmetrical initial configurations, and algorithms that ensure gathering are provided.

The impact of svmmetries on gathering possibilities for asynchronous robots in the graph are studied in [151]. On an undirected ring with at least 18 robots, it is

proven that an approach that focuses on preserving symmetries solvesthe gathering problem for ail starting positions when the gathering is feasible.

[46| studiesthegatheringproblemforobliviousasynchronousdisoriented (no com-mon compass) robots operating in LCM cycles in the two-dimensional Euclidean space. The paper solves the gathering problem for n > 2 robots for any initial con­

figuration, even under such weak assumptions.

In [183], the minimal assumptions required for anonymous oblivious robots with-out communication to gather in the two-dimensional Euclidean space are studied.

It is shown that in the synchronous case, robots require either multiplicity détec­

tion or infinité time, and in the asynchronous case, robots require either multiplicity détection, acommon compass, unboundedmemory or infinité time in order to meet.

The assumption ofunlimited visibility is dropped in [121]. The paper considère the gathering problem for asynchronous oblivious robots operating in LCM cycles.

The paperprovides an algorithm that allowsgathering in a finite amount oftime foi-robots withlimitedvisibility, providedthattheyshareacommoncompass. Fromthis resuit, the authors show that orientation is as powerful as instantaneous movement with respect to gathering.

Similarly, in [80], the gathering problem with synchronous robots with limited visibility in the two-dimensional Euclidean space is considered. The robots rnust, whenchoosing their next move, ensure that the unit disk graph defined bythe view-ing range of the robots remains connected at ail time. An algorithm that ensures gathering in time 0(n2) is provided for the gathering.

In [74] and[73], obliviousrobotsoperatinginLCMcyclesareplacedonthediscrète ring. They are asynchronous, and during their Move phase, they can décidé either to stay in place or to move to an adjacent node. Two problems are considered:

the gathering problem, and the exclusive searching problem, in which ail edges are

either traversée!, or both nodes adjacent to an edge areoccupied, without two robots ever occupving the same node. A description of the initial configurations that allow resolution ofthosetwoproblems is provided, and algorithmssolvingthe problems are provided.

The notion of rendez-vous with détection is explored in [102], where two agents hâve to meet in a graph and then become aware of this meeting. They must then déclaré sirnultaneously that the meeting took place, then stop. This paper uses the synchronousmodel. Two variations are considered: the local beeping model, and the globalbeepingmodel. Feasibilityisdiscussedunder variousprémisses,andalgorithms are provided when possible.

2.4.3 Deployment

Rendezvousand gathering problems canbe seen as a spécifie subset ofa larger prob-lem: the pattern formation problem. In the pattern formation problem, a group of k > 2 robots must form a spécifie pattern. This includes rendezvous, but also the formation of a regular k-gon, a formation in which no more than two robots form a line, a formation in which robots are équidistant, etc.

In [89], robots operating in LCM cycles are located in the two-dimensional Eu-clidean spaee. Two robotscanonlysee oneanotherifthereis no otherrobotbetween theni. Their purpose is to develop a formation in which ail robots can see ail other robots, in afinite laps oftime. This problem is called the Mutual Visibility Problem.

Each robot is equipped with visible lights that can take up to c different colours.

Various scénarios are considered: robots with prior knowledge oftheir environment, number ofavailable colours, and synchronism ofthe robots. It is proven that in the semi-synchronouscase, the Mutual Visibility Problemcan be solved withoutcollision

with c = 2, and with c = 3 ifthe robots are asynchronous.

[101] considers the even deployment ofoblivious agents with no prior knowledge of the environment on the discrète ring. The size of the swarm is unknown as weli.

Two variations ofthis problem: the first one, called dynamical uniform deployment, requiresforthe agentstospreadevenly, then keepmovingas theyholdthis formation.

The second variation, called quiescent spread, requires forthe agentsto stop moving once they are evenlyspread. For the first variation, it is shown that the problem can only be resolved if either the ring is oriented, or the agents hâve a visibility range of at least [n/k\. An optimal algorithm is proposed for an oriented ring and agents withavisibilityrangeofatleast \n/k\. For thesecond variation, the problem cannot be solved ifthe agent can only measure the distance to its two neighbours.

In [120], asetofn^4 asynchronousanonymousobliviousrobotswithnocommon compass mustarrangethemselvesto form aregular n-gon. This problemis called the uniform circleformation problem. The paperproves that this problem can be solved without any further assumption for any initial configuration.

Arbitrary pattern formation is studied in [122], where asynchronous oblivious robotsworkinginLCM cyclesoperatinginthe two-dimensionalEuclideanspace must form a pattern that is given in advance. It is show that without common compass, this task is impossible. With a compass, an odd number of robots can solve this problem, but not an even number ofrobots. With two different compassés (say, one pointing North and one pointingEast), then an even number ofrobots can solve the problem.

In [205], robots operating in LCM cycles must form a predetermined pattern. It is shown that oblivious robots can form any pattern non-oblivious robots can, with the exceptionoftworobotstrying toformapoint, which isonlypossible ifthe robots are not oblivious. The paper therefore proves that memory is not useful in the task

ofpattern formation, apart from the aforementioned exception.

Varying Environments