Optimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots
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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Optimal Probabilistic Ring Exploration byAsynchronous Oblivious Robots
Stéphane Devismes ◦ — Franck Petit † — Sébastien Tixeuil ⋆, ‡ ◦ VERIMAG UMR 5104, Université Joseph Fourier, Grenoble (France) † INRIA, LIP UMR 5668, Université de Lyon / ENS Lyon (France) ⋆ Université Pierre et Marie Curie - Paris 6, LIP6, France ‡ INRIA project-team Grand Large
N° 6838
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Optimal Probabilistic Ring Exploration byAsynchronous Oblivious Robots
St´ephane Devismes ◦ , Franck Petit † , S´ebastien Tixeuil ⋆, ‡ ◦ VERIMAG UMR 5104, Universit´e Joseph Fourier, Grenoble (France) † INRIA, LIP UMR 5668, Universit´e de Lyon / ENS Lyon (France) ⋆ Universit´e Pierre et Marie Curie - Paris 6, LIP6, France ‡ INRIA project-team Grand Large
Th`eme NUM — Syst`emes num´eriquesProjet Grand LargeRapport de recherche n ° Abstract:
We consider a team of k identical, oblivious, asynchronous mobile robotsthat are able to sense ( i.e. , view) their environment, yet are unable to communicate,and evolve on a constrained path. Previous results in this weak scenario showthat initial symmetry yields high lower bounds when problems are to be solved by deterministic robots.In this paper, we initiate research on probabilistic bounds and solutions in thiscontext, and focus on the exploration problem of anonymous unoriented rings ofany size. It is known that Θ(log n ) robots are necessary and sufficient to solve theproblem with k deterministic robots, provided that k and n are coprime. By contrast,we show that four identical probabilistic robots are necessary and sufficient to solvethe same problem, also removing the coprime constraint. Our positive results areconstructive. Key-words:
Robots, Anonimity, Obliviousness, Exploration xploration d’Anneau Probabiliste Optimale par desRobots Asynchrones et Amn´esiques
R´esum´e :
Nous consid´erons une ´equipe de k robots identiques, amn´esiques, asyn-chrones et mobiles qui sont capables de percevoir leur environnement mais incapablesde communiquer, et ´evoluent sur des circuits contraints. Les r´esultats pr´ec´edentsqui utilisent le mˆeme sc´enario montrent que la symm´etrie initiale potentielle induitdes bornes inf´erieures ´elev´ees d`es lors que le probl`eme doit ˆetre r´esolu par des robotsd´eterministes.Dans cet article, nous initions la recherche sur les bornes et sur les solutions pro-babilistes dans le mˆeme contexte, et nous consid´erons le probl`eme de l’explorationd’anneaux anonymes et non orient´es de taille quelconque. Il est connu que Θ(log n )robots sont n´ecessaires et suffisants dans le cas d´eterministe pour r´esoudre le probl`emeavec k robots, tant que k et n sont premiers entre eux. En contrepartie, nous mon-trons que quatre robots identiques probabilistes sont n´ecessaires et suffisants pourr´esoudre le mˆeme probl`eme, tout en supprimant la contrainte de coprimalit´e. Nosr´esultats positifs sont constructifs. Mots-cl´es :
Robots, Anonymat, Amn´esie, Exploration ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots We consider autonomous robots that are endowed with visibility sensors (but thatare otherwise unable to communicate) and motion actuators. Those robots mustcollaborate to solve a collective task, namely exploration , despite being limited withrespect to input from the environment, asymetry, memory, etc. In this context, theexploration tasks requires every possible location to be visited by at least one robot,with the additional constraint that all robots stop moving after task completion.Robots operate in cycles that comprise look , compute , and move phases. The lookphase consists in taking a snapshot of the other robots positions using its visibilitysensors. In the compute phase a robot computes a target destination based onthe previous observation. The move phase simply consists in moving toward thecomputed destination using motion actuators.The robots that we consider here have weak capacities: they are anonymous (they execute the same protocol and have no mean to distinguish themselves fromthe others), oblivious (they have no memory that is persistent between two cycles),and have no compass whatsoever (they are unable to agree on a common directionor orientation). Related works
The vast majority of literature on coordinated distributed robotsconsiders that those robots are evolving in a continuous two-dimentional Euclidianspace and use visual sensors with perfect accuracy that permit to locate other robotswith infinite precision [2, 13, 14, 10, 6, 5].Several works investigate restricting the capabilities of both visibility sensors andmotion actuators of the robots, in order to circumvent the many impossibility resultsthat appear in the general continuous model. In [1, 9], robots visibility sensors aresupposed to be accurate within a constant range, and sense nothing beyond thisrange. In [9, 4], the space allowed for the motion actuator was reduced to a one-dimentional continuous one: a ring in [9], an infinite path in [4].A recent trend was to shift from the classical continuous model to the discrete model. In the discrete model, space is partitioned into a finite number of locations.This setting is conveniently represented by a graph, where nodes represent locationsthat can be sensed, and where edges represent the possibility for a robot to movefrom one location to the other. Thus, the discrete model restricts both sensing andactuating capabilities of every robot. For each location, a robot is able to sense ifthe location is empty of if robots are positioned on it (instead of sensing the exactposition of a robot). Also, a robot is not able to move from a position to another
RR n ° S. Devismes et al.unless there is explicit indication to do so ( i.e. , the two locations are connected by anedge in the representing graph). The discrete model permits to simplify many robotprotocols by reasoning on finite structures ( i.e. , graphs) rather than on infinite ones.However, as noted in most related papers [12, 11, 7, 8], this simplicity comes withthe cost of extra symmetry possibilities, especially when the authorized paths arealso symmetric (indeed, techniques to break formation such as those of [6] cannotbe used in the discrete model).The two main problems that have been studied in the discrete robot model aregathering [12, 11] and exploration [7, 8]. For gathering, both breaking symmetry [12]and preserving symmetry are meaningful approaches. For exploration, the fact thatrobots need to stop after exploring all locations requires robots to “remember” howmuch of the graph was explored, i.e. , be able to distinguish between various stages ofthe exploration process since robots have no persistent memory. As configurationscan be distinguished only by robot positions, the main complexity measure is thenthe number of robots that are needed to explore a given graph. The vast number ofsymmetric situations induces a large number of required robots. For tree networks,[8] shows that Ω( n ) robots are necessary for most n -sized tree, and that sublinearrobot complexity is possible only if the maximum degree of the tree is 3. In uniformrings, [7] proves that the necessary and sufficient number of robots is Θ(log n ),although it is required that the number k of robots and the size n of the ring arecoprime. Note that all previous approaches in the discrete model are deterministic , i.e. , if a robot is presented twice the same situation, its behavior is the same in bothcases. Our contribution
In this paper, we initiate research on probabilistic bounds andsolutions in the discrete robot model, and focus on the exploration problem of anony-mous unoriented rings of any size. By contrast with [7] while in the same systemsetting, we show that four identical probabilistic robots are necessary and sufficientto solve the same problem, also removing the coprime constraint between the num-ber of robots and the size of the ring. Our negative result show that for any ring ofsize at least four, there cannot exist any protocol with three robots in our setting,even if they are allowed to make use of probabilistic primitives. Our positive resultsare constructive, as we present a randomized protocol with four robots for any ringof size more than eight.
Outline
The remaining of the paper is divided as follows. Section 2 presents thesystem model that we use throughout the paper. Section 3 provides evidence that
INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots
Distributed System
We consider systems of autonomous mobile entities called agents or robots evolving into a graph . We assume that the graph is a ring of n nodes, u ,. . . , u n − , i.e. , u i is connected to both u i − and u i +1 — every computation overindices is assumed to be modulus n . The indices are used for notation purposes only:the nodes are anonymous and the ring is unoriented , i.e. , given two neighboringnodes u , v , there is no kind of explicit or implicit labelling allowing to determinewhether u is on the right or on the left of v . Operating in the ring are k ≤ n anonymous robots.A protocol is a collection of k programs , one operating on each robot. Theprogram of a robot consists in executing Look-Compute-Move cycles infinitely manytimes. That is, the robot first observes its environment (Look phase). Based on itsobservation, a robot then (probabilistically or deterministically) decides — accordingto its program — to move or stay idle (Compute phase). When an robot decides amove, it moves to its destination during the Move phase.The robots do not communicate in an explicit way; however they see the positionof the other robots and can acquire knowledge from this information. We assumethat the robots cannot remember any previous observation nor computation per-formed in any previous step. Such robots are said to be oblivious (or memoryless ).The robots are also uniform and anonymous , i.e, they all have the same programusing no local parameter (such that an identity) allowing to differentiate any ofthem.
Computations
Time is represented by an infinite sequence of instants 0, 1, 2, . . .At every instant t ≥
0, a non-empty subset of robots is activated to execute a cycle.The execution of each cycle is assumed to be atomic : every robot that is activatedat instant t instantaneously executes a full cycle between t and t + 1. Atomicityguarantees that at any instant the robots are on some nodes of the ring but not onedges. Hence, during a Look phase, a robot sees no robot on edges.We assume that during the Look phase, every robot can perceive whether severalrobots are located on the same node or not. This ability is called MultiplicityDetection . We shall indicate by d i ( t ) the multiplicity of robots present in node u i at instant t . More precisely d i ( t ) = j indicates that there is j robots in node u i at RR n ° S. Devismes et al.instant t . If d i ( t ) ≥
2, then we say that there is a tower in u i at instant t (or simplythere is a tower in u i when it is clear from the context). We say a node u i is free atinstant t (or simply free when it is clear from the context) if d i ( t ) = 0. Conversely,we say that u i is occupied at instant t (or simply occupied when it is clear from thecontext) if d i ( t ) = 0.Given an arbitrary orientation of the ring and a node u i , γ + i ( t ) (respectively, γ − i ( t )) denotes the sequence h d i ( t ) d i +1 ( t ) . . . d i + n − ( t ) i (resp., h d i ( t ) d i − ( t ) . . . d i − ( n − ( t ) i ).The sequence γ − i ( t ) is called mirror of γ + i ( t ) and conversely. Since the ring is unori-ented, agreement on only one of the two sequences γ + i ( t ) and γ − i ( t )) is impossible.The (unordered) pair { γ + i ( t ) , γ − i ( t ) } is called the view of node u i at instant t (weomit “at instant t ” when it clear from the context). The view of u i is said to be symmetric if and only if γ + i ( t ) = γ − i ( t ). Otherwise, the view of u i is said to be asymmetric .By convention, we state that the configuration of the system at instant t is γ +0 ( t ). Any configuration from which there is a probability 0 that a robot movesis said terminal . Let γ = h x x . . . x n − i be a configuration. The configuration h x i x i +1 . . . x i + n − i is obtained by rotating γ of i ∈ [0 . . . n − γ and γ ′ are said undistinguable if and only if γ ′ can be obtained by rotating γ orits mirror. Two configurations that are not undistinguable are said distinguable .We designate by initial configurations the configurations from which the system canstart at instant 0.During the Look phase of some cycle, it may happen that both edges incidentto a node v currently occupied by the robot look identical in the snapshot, i.e. , v lies on a symmetric axis of the configuration. In this case, if the robot decides tomove, it may traverse any of the two edges. We assume the worst case decision insuch cases, i.e. , that the decision to traverse one of these two edges is taken by anadversary.We call computation any infinite sequence of configurations γ , . . . , γ t , γ t +1 , . . . such that (1) γ is a possible initial configuration and (2) for every instant t ≥ γ t +1 is obtained from γ t after some robots (at least one) execute a cycle. Any transition γ t , γ t +1 is called a step of the computation. A computation c terminates if c containsa terminal configuration.A scheduler is a predicate on computations, that is, a scheduler define a set of admissible computations, such that every computation in this set satisfies the sched-uler predicate. Here we assume a distributed fair scheduler. Distributed means that,at every instant, any non-empty subset of robots can be activated. Fair means thatevery robot is activated infinitively often during a computation. A particular case of INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots sequential fair scheduler: at every instant, one robotis activated and every robot is activated infinitively often during a computation.In the following, we call sequential computation any computation that satisfies thesequential fair scheduler predicate.
Problem to be solved
We consider the exploration problem, where k robotscollectively explore a n -sized ring before stopping moving forever. More formally, aprotocol P deterministically (resp. probabilistically ) solves the exploration problemif and only if every computation c of P starting from a towerless configuration satisfies:1. c terminates in finite time (resp. with expected finite time ).2. Every node is visited by at least one robot during c .The previous definition implies that every initial configuration of the system inthe problem we consider is towerless .Using probabilistic solutions, termination is not certain, however the overallprobability of non-terminating computations is 0. In this section, we show that the exploration problem is impossible to solve in oursettings ( i.e. , oblivious robots, anonymous ring, distributed scheduler, . . . ) if thereis less than four robots, even in a probabilistic manner (Corollary 2). The proof ismade in two steps: ˆ The first step is based on the fact that obliviousness constraints any explorationprotocol to construct an implicit memory using the configurations. We showthat if the scheduler behaves sequentially, then in any case except one, it isnot possible to particularize enough configurations to memorize which nodeshave been visited (Theorem 1 and Lemma 5). ˆ The second step consists in excluding the last case (Theorem 2).Lemmas 1 to 4 proven below are technical results that lead to Corollary 1. Thelatter exhibits the minimal size of a subset of particular configurations required tosolve the exploration problem.
RR n ° S. Devismes et al.
Definition 1 (MRP)
Let s be a sequence of configurations. The minimal rele-vant prefix of s , noted MRP ( s ) , is the maximal subsequence of s where no twoconsecutive configurations are identical. Lemma 1
Let P be any (probabilistic or deterministic) exploration protocol for k robots in a ring of n nodes. For every sequential computation c of P that terminates,we have |MRP ( c ) | ≥ n − k + 1 . Proof.
Let c be a sequential computation that terminates. In the initialconfiguration of c exactly k nodes are already visited because there is at most onerobot in each node. So, n − k nodes are dynamically visited before c terminates. Asthe computation is sequential, the computation contains at least n − k + 1 differentconfigurations: the initial one plus one configuration per node to be dynamicallyvisited. Hence, |MRP ( c ) | ≥ n − k + 1. ✷ Lemma 2
Let P be any (probabilistic or deterministic) exploration protocol for k robots in a ring of n > k nodes. For every sequential computation c of P thatterminates, MRP ( c ) has at least n − k + 1 configurations containing a tower. Proof.
Assume, by the contradiction, that there is a sequential computation c of P that terminates and such that MRP ( c ) has less than n − k + 1 configurationscontaining a tower.There exists a suffix c ′ of c starting from a configuration α without tower followeda suffix s that only contains configurations with a tower. As α is a configurationwithout tower, c ′ is an admissible sequential computation of P . Moreover, as c terminates, c ′ terminates too. Hence, |MRP ( c ′ ) | = n − k + 1 by Lemma 1 and allrobots must be visited before c ′ reaches its terminal configuration. As a consequence, c ′ contains exactly n − k steps of the form ββ ′ with β = β ′ . Now, the first of thesesteps in c ′ is a step where one robot moves to a node already occupied by anotherrobot (remember that the computation is sequential and the first step in MRP ( c ′ )is a step from a configuration without tower to a configuration with a tower). Hence, c ′ contains at most n − k − c ′ terminates beforeall robots are visited, a contradiction. ✷ Lemma 3
Let P be any (probabilistic or deterministic) exploration protocol for k robots in a ring of n > k nodes. For every sequential computation c of P thatterminates, MRP ( c ) has at least n − k + 1 configurations containing a tower of lessthan k robots. INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots Proof.
Assume, by the contradiction, that there is a sequential computation c of P that terminates and such that MRP ( c ) has less than n − k + 1 configurationscontaining a tower of less than k robots.There exists a suffix c ′ of c starting from a configuration α without tower followeda suffix s that only contains configurations with a tower. As α is a configurationwithout tower, c ′ is an admissible sequential computation of P . Moreover, as c terminates, c ′ terminates too. Hence, MRP ( c ′ ) is constituted of a configurationwith no tower followed by at least n − k + 1 configurations containing a tower byLemma 2 and all robots must be visited before c ′ reaches its terminal configuration.As the first configuration of c ′ is without tower, for every configuration α of MRP ( c ′ ) with a tower there exists a unique step in MRP ( c ′ ) of the form α ′ α with α ′ = α . Now, as c ′ is sequential, for each of these steps, if α contains a tower of k robots, then no new node is visited during α ′ = α . By contradiction assumption,there is less than n − k + 1 of steps β ′ β such that β contains a tower of less than k robots. Moreover, no node is visited during the first of these steps (rememberthat the computation is sequential and the first of these steps is a step from aconfiguration without tower to a configuration with a tower). Hence, less that n + k steps allow to dynamically visit new nodes in c ′ and, as c ′ is sequential, c ′ terminatesbefore all robots are visited, a contradiction. ✷ Lemma 4
Let P be any (probabilistic or deterministic) exploration protocol for k robots in a ring of n > k nodes. For every sequential computation c of P thatterminates, MRP ( c ) has at least n − k + 1 configurations containing a tower of lessthan k robots and any two of them are distinguable. Proof.
Consider any sequential computation c of P that terminates.By Lemma 3, MRP ( c ) has x configurations containing a tower of less than k robots where x ≥ n − k + 1.We first show that (**) if c contains at least two different configurations thatare undistinguable, then there exists a sequential computation c ′ that terminates andsuch that MRP ( c ′ ) has x ′ configurations containing a tower of less than k robotswhere x ′ < x . Assume that there two different undistinguable configurations γ and γ ′ in c having a tower of less than k robots. Without loss of generality, assume that γ occurs at time t in c and γ ′ occurs at time t ′ > t in c . Consider the two followingcase:1. γ ′ can be obtained by applying a rotation of i to γ . Let p be theprefix of c from instant 0 to instant t . Let s be the suffix of c starting at RR n ° S. Devismes et al.instant t ′ + 1. Let s ′ be the sequence obtained by applying a rotation of − i tothe configurations of s . As the ring and the robots are anonymous, ps ′ is anadmissible sequential computation that terminates. Moreover, by construction MRP ( ps ′ ) has x ′ configurations containing a tower of less than k robots where x ′ < x . Hence (**) is verified in this case.2. γ ′ can be obtained by applying a rotation of i to the mirror of γ . Wecan prove (**) in this case by slightly modifying the proof of the previous case:we have just to apply the rotation of − i to the mirrors of the configurationsof s .By (**), if MRP ( c ) contains less than n − k + 1 distinguable configurations witha tower of less than k robots, it is possible to (recursively) construct an admissiblecomputation c ′ of P such that MRP ( c ′ ) has less than n − k + 1 configurationscontaining a tower of less than k robots, a contradiction to Lemma 3. Hence, thelemma holds. ✷ From Lemma 4, we can deduce the following corollary:
Corollary 1
Considering any (probabilistic or deterministic) exploration protocolfor k robots in a ring of n > k nodes, there exists a subset S of at least n − k + 1 configurations such that:1. Any two different configurations in S are distinguable, and2. In every configuration in S , there is a tower of less than k robots. Theorem 1 ∀ k, ≤ k < , ∀ n > k , there is no exploration protocol (even proba-bilistic) of a n -size ring with k robots. Proof.
First, for k = 0, the theorem is trivially verified. Consider then the case k = 1 and k = 2: with one robot it is impossible to construct a configuration withone tower; with two robots it is impossible to construct a configuration with onetower of less than k robots ( k = 2). Hence, for k = 1 and k = 2, the theorem is adirect consequence of Corollary 1. ✷ Lemma 5 ∀ n > , there is no exploration protocol (even probabilistic) of a n -sizering with three robots. INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots Proof.
With three robots, the size of the maximal set of distinguable configura-tions containing a tower of less than three robots is ⌊ n/ ⌋ . By Corollary 1, we havethen the following inequality: ⌊ n/ ⌋ ≥ n − k + 1From this inequality, we can deduce that n must be less of equal than four and weare done. ✷ From this point on, we know that, assuming k <
4, Corollary 1 prevents the existenceof any exploration protocol in any case except one: k = 3 and n = 4 (Theorem 1and Lemma 5). Actually, assuming that the scheduler is sequential is no sufficientto show the impossibility in this latter case: Indeed, there an exploration protocolfor k = 3 and n = 4 if we assume a sequential scheduler. The protocol works asshown in Figure 1.Figure 1: Protocol for n = 4 and k = 3. (The arrows show the destinations of therobots if they are activated.)We now show the impossibility in this latter using a (non-sequential) distributedscheduler. This proof is established by enumerating and testing all possible protocolsfor k = 3 and n = 4. Theorem 2
There is no exploration protocol (even probabilistic) of a n -size ringwith three robots for every n > . Proof.
Lemma 5 excludes the existence of any exploration protocol for threerobots in a ring of n >
RR n ° S. Devismes et al.Assume, by the contradiction, that there exists an exploration protocol P forthree robots in a ring of four nodes. Then, any possible initial configuration isundistinguable with the configuration presented in Figure 2. Moreover, any possibleterminal configuration contains a tower and so is undistinguable with one of thethree configurations presented in Figure 3.Figure 2: Initial configuration for n = 4 and k = 3. (The indices are used fornotation purposes only.)Figure 3: Terminal configurations for n = 4 and k = 3. (The indices are used fornotation purposes only.)Consider that the system is initially in the configuration of Figure 2. Three casesare possible at instant 0 using P : INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots ˆ There is a strictly positive probability that robot R a (resp. robot R c ) moves tonode u if activated by the scheduler. In this case, assume that the scheduleractivates R a until it moves. The probability that R a eventually moves is 1(resp. R a moves in one step if P is determistic). Once R a has moved, R b has astrictly positive probability to move to node u if activated. Assume then thatthe scheduler activates R b until it moves. The probability that R b eventuallymoves is 1. Repeating this scheme for R c and so on, it is possible to constructa distributed fair computation that does not terminate in finite expected time(resp. in finite time, if if P is determistic), a contradiction. ˆ There is a strictly positive probability that robot R a (resp. robot R c ) moves tonode u if activated by the scheduler. In this case, there an admissible com-putation where R a and R c moves to node u in the first step. At instant 1,the system is in a configuration that is undistinguable with configuration (i)of Figure 3. As node u is still not visited in this case, any configuration thatis undistinguable with configuration (i) cannot be terminal. There is also anadmissible computation where only R a moves to node u in the first step. Atinstant 1, the system is in a configuration that is undistinguable with config-uration (ii) of Figure 3. As node u is still not visited in this case, any con-figuration that is undistinguable with configuration (ii) cannot be terminal.Moreover, assuming that the system reaches a configuration undistinguablefrom configuration (i) of Figure 3 at instant 1, there is a strictly positive prob-ability that the three robots moves (the configuration is not terminal and allrobots have the same view). If they move, the adversary can choose whichincident edge they traverse because the configuration is symmetric. Hence, wecan obtain a configuration undistinguable with configuration (iii) of Figure 3and where node u is still not visited. Thus, any configuration that is undis-tinguable with configuration (iii) cannot be terminal. Hence, no configurationcan be terminal, a contradiction. ˆ There is a strictly positive probability that robot R b moves if activated by thescheduler. Assume that the scheduler activates R b until it moves. Then,the probability that R b eventually moves is 1. Once R b decide to move, theadversary can choose the edge that R b traverses because the view from R b is symmetric. Hence, the system can reache the configuration γ : R a is innode u , R b and R c and in node u . This configuration is undistinguable with If P is deterministic the probability is 1 and if activated, R a moves in one step.RR n ° S. Devismes et al.configuration (iii) in Figure 3 and node u is still not visited. Consider thetwo following cases: – The probability that R a moves, if activated, is 0. Then, there is a strictlypositive probability that R c (resp. R b ) moves if activated. Assume thatthe scheduler activates R a and then R c until R c moves. The probabilitythat R c eventually moves is 1 and as the the view from R c is symmetric,the adversary can decide which edge R c will traverse. Assume that theadversary forces R c to go to node u , the system reaches a configurationundistinguable with the initial configuration. Repeating the same schemeinfinitively often, we obtain a distributed fair computation that does notterminate in finite expected time, a contradiction. – The probability that R a moves ,if activated, is strictly positive. Assumethat the scheduler activates R a until it moves. Then, the probabilitythat R a eventually moves is 1 and as the the view from R a is symmetric,the adversary can decide which edge R a will traverse. Assume that R a moves to node u , the system reaches the following configuration: R a isin node u , R b and R c are in node u , and node u is still not visited.This configuration is undistinguable with configuration (ii) in Figure 3.Consider the two following cases: * The probability that R c (resp. R b ) moves, if activated, is strictlypositive. · Assume that the destination of R c , if R c , is node u . Then,the system reaches a configuration undistinguable from initialconfiguration. Repeating same the scheme infinitively often, weobtain a distributed fair computation that does not terminate infinite expected time, a contradiction. · Assume that the destination of R c , if R c moves, is node u . Then,the destination of R b , if R b moves, is node u too. Hence, thereis an admissible computation where R b and R c move to node u . In this case, the system reaches a configuration that is notdistinguable from configuration (i) in Figure 3 while node u isstill not visited. In this case, no configuration can be terminal, acontradiction. * The probability that R b (resp. R c ) moves, if activated, is 0. Then,the probability that R a moves is strictly positive. Consider the twofollowing cases: INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots · Assume that the destination of R a , if R a , is node u . In this case,there is an admissible computation where R a move to node u :the system reaches a configuration that is not distinguable fromconfiguration (i) in Figure 3 while node u is still not visited. Inthis case, no configuration can be terminal, a contradiction. · Assume that the destination of R a , if R a , is node u . Assume thatthe scheduler activates R b , R c , and then R a until R a moves. Theprobability that R a eventually moves is 1 and we retreive a con-figuration that is undistinguable with configuration γ . Repeatingthe same scheme infinitively often, we obtain a fair distributedcomputation that does not terminate in finite expected time, acontradiction.In all cases, we obtain a contradiction: there no exploration protocol for three robotsin a ring of n > ✷ From Theorems 1 and 2, we can deduce the following corollary:
Corollary 2 ∀ k, ≤ k < , ∀ n > k , there is no exploration protocol (even proba-bilistic) of a n -size ring with k robots. In this section, we propose a probabilistic exploration protocol for k = 4 robots ina ring of n > nodes. We first define some useful terms in Subsection 4.1. Wethen give the general principle of the protocol in Subsection 4.2. Finally, we fullydescribe and prove the protocol in Subsection 4.3. Below, we define some terms that characterize the configurations.We call segment any maximal non-empty elementary path of occupied nodes.The length of a segment is the number of nodes that compose it. We call x -segment any segment of length x . An isolated node is a node belonging to a 1-segment.We call hole any maximal non-empty elementary path of free nodes. The length ofa hole is the number of nodes that compose it. We call x -hole any hole of length x . Inthe hole h = u i , . . . , u k ( k ≥ i ) the nodes u i and u k are terms as the extremities of h .We call neighbor of an hole any node that does not belong to the hole but is neighbor RR n ° S. Devismes et al.of one of its extremities. In this case, we also say that the hole is a neighboring hole of the node. By extension, any robot that is located at a neighboring node of a holeis also referred to as a neighbor of the hole.We call arrow a maximal elementary path u i , . . . , u k of length at least four suchthat ( i ) u i and u k are occupied by one robot, ( ii ) ∀ j ∈ [ i + 1 . . . k − u j is free,and ( iii ) there is a tower of two robots in u k − . The node u i is called the arrow tail and the node u k is called the arrow head . The size of an arrow is the number of freenodes that compose it, i.e. , its the length of the arrow path minus 3. Note that theminimal size of an arrow is 1 and the maximal size is n −
4. Note also that whenthere is an arrow in a configuration, the arrow is unique. An arrow is said primary if its size is 1. An arrow is said final if its size is n − i ) the arrow is formedby the path u , u , u , u ; the arrow is primary; the node u is the tail and the node u is the head. In Configuration ( ii ), there is a final arrow (the path u , u , u , u , u , u ). Finally, the size of the arrow in Configuration ( iii ) (the path u , u , u , u , u ) is 2. Our protocol (Algorithm 1) proceeds in three distinct phases: ˆ Phase I : Starting from a configuration without tower, the robots move alongthe ring in such a way that ( i ) they never form any tower and (2) form aunique segment (a 4-segment) in finite expected time. INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots ˆ Phase II : Starting from a configuration with a unique segment, the four robotsform an primary arrow in finite expected time. The 4-segment is maintaineduntil the primary arrow is formed. ˆ Phase
III : Starting from a configuration where the four robots form a pri-mary arrow, the arrow tail moves toward the arrow head in such way thatthe existence of an arrow is always maintained. The protocol terminates whenrobots form a final arrow. At the termination, all nodes have been visited.Note that the protocol we propose is probabilistic. As a matter of fact, as most aspossible the robots move deterministically. However, we use randomization to breakthe symmetry in some cases: When the system is in a symmetric configuration,the scheduler may choose synchronously to activated some processes in such waythat the system stays in a symmetric configuration. To break the symmetry despitethe choice of the scheduler, we proceed as follows: The activated nodes toss a coin(with a uniform probability) during their Compute phase. If they win the toss, theydecide to move, otherwise they decide to stay idle. In this case, we say that therobots try to move . Conversely, when a process deterministically decides to movein its Compute phase, we simply say that the process moves . Algorithm 1
The protocol. if the four robots do not form a final arrow then if the configuration contains neither an arrow nor a 4-segment then Execute Procedure
P hase I; else if the configuration contains a 4-segment then Execute Procedure
P hase
II; else / ∗ the configuration contains an arrow ∗ / Execute Procedure
P hase
III; end if end if end if IPhase I is described in Algorithm 2. The aim of this phase is to eventually forma 4-segment without creating any tower during the process. Roughly speaking, inasymmetric configurations, robots moves determiniscally (Lines 3, 8, 22, 26). Bycontrast, in symmetric configurations, robots moves probabilistically using
Try to
RR n ° S. Devismes et al. move (Lines 13 and 18). Note that in all case, we prevent the tower formation byapplying the following constraint: a robot can move through a neighboring hole H only if its length is at least 2 or if the other neighboring robot can move through H . Algorithm 2
Procedure
P hase I. if the configuration contains a 3-segment then if I am the isolated robot then Move toward the 3-segment through the shortest hole; end if else if the configuration contains a unique 2-segment then / ∗ Two robots are isolated ∗ / if I am at the closest distance from the 2-segment then Move toward the 2-segment through the hole having me and an extremity of the 2-segmentas neighbors; end if else if the configuration contains (exactly) two 2-segments then if I am a neighbor of a longuest hole then
Try to move toward the other 2-segment through my neighboring hole; end if else / ∗ the four robots are isolated ∗ / Let l max be the length of the longuest hole; if every robot is neighbor of a l max -hole then Try to move through a neighboring l max -hole; else if l max -hole then if I am neighbor of only one l max -hole then Move toward the robot that is neighbor of no l max -hole through my shortestneighboring hole; end if else / ∗ l max -hole ∗ / if I am neighbor of the unique l max -hole then Move through my shortest neighboring hole; end if end if end if end if end if end if
The following lemma (Lemma 6) shows that no tower can by creating duringPhase I. The next one (Lemma 7) shows that executing Algorithm 2, a 4-segmentis eventually created.
Lemma 6
If the configuration at instant t contains neither a -segment nor a tower,then the configuration at instant t + 1 contains no tower. INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots Proof.
Let γ be the configuration at instant t . First, note that the robotsexecutes P hase
I (Algorithm 2) in γ . Note also that γ satisfies one of the followingcases: ˆ γ contains a -segment . In this case, only the (unique) isolated robot can moveand, if it does, it moves to a free node (see Line 3). Hence, no tower is createdat instant t + 1. ˆ γ contains a unique -segment . Two cases are possible: – There is a unique isolated robot R at the closest distance from the -segment .In this case, only R can move and, if it does, it moves to free node (seeLine 8), so no tower is created at instant t + 1. – The two isolated robots are at the same distance from the -segment .In this case, the two isolated robots can move but as they follow theirshortest path to the 2-segment (see Line 8) and there is no tower in γ ,they follow distinct paths and no tower is created at instant t + 1.Hence, in the two subcases no tower is created at instant t + 1. ˆ γ contains two -segments . In this case, as there is four robots and the sizeof the ring is greater than 8, the size of the longuest hole is at least three.In such a configuration, the only possible moves are the moves where robotsmove through one of their neighboring holes of length at least two (see Line13). Hence, all moving robots move to a different free node: no tower is createdat instant t + 1. ˆ γ contains four isolated robots . Let l max be the length of the longuest hole in γ . In this case, as there is four robots and the size of the ring n is greater than8, l max ≥
2. Consider then the following three subcases: – Every robot is neighbor of a l max -hole . In this case, the configuration issymmetric. Every robot can move in the next step but to a neighboringhole of size at least two (see Line 18). So, all moving robots move to adifferent free node. Hence, no tower is created at instant t + 1. – Three robots are neighbors of a l max -hole . Let R be the robot that isnot neighbor of any l max -hole. In this case, the robots that may move(at most two) move through their neighboring hole having R as otherneighbor (see Line 22). As R cannot move, no tower is created at instant t + 1. RR n ° S. Devismes et al. – Two robots, say R and R , are neighbors of the unique l max -hole . Inthis case, only R and R can move. If R (resp. R ) moves, then R (resp. R ) through its neighboring hole having not R (resp. R ) asother neighbor (see Line 26). So, all moving robots move to a differentfree node. As a consequence, no tower is created at instant t + 1.In all cases, the configuration obtained at instant t + 1 contains no tower and thelemma holds. ✷ Lemma 7
Starting from any initial configuration, the system reaches in finite ex-pected time a configuration containing a -segment. Proof.
Any initial configuration contains no tower. If the initial configurationcontains a 4-segment, the lemma trivially holds. Consider now the case where theinitial configuration contains neither a 4-segment nor a tower.By Lemma 6, while the system does not reaches a configuration containing a4-segment, the system remains in configurations containing no tower. For a given n -size ring network, the number of such configuration is finite . So, to prove the lemma,we have just to show that from any configuration containing neither a 4-segmentnor a tower, there is always a strictly positive probability that the system eventuallyreaches a configuration containing a 4-segment (despite the choices of the scheduler).To see this, consider a configuration γ containing neither a 4-segment nor a towerand split the study into the following cases:1. γ contains a -segment . In this case, only the unique isolated robot can moveand by the fairness property, it eventually does: it moves toward the 3-segmentthrough the shortest hole (see Line 3). So, until the system reaches a config-uration containing a 4-segment, only the isolated robot moves and at eachmove the length of the shortest hole decreases. Hence, the system reaches aconfiguration containing a 4-segment in finite time .2. γ contains a unique -segment . Following the same scheme as in the previouscase, we can see that the system reaches a configuration containing a 4-segment in finite time .3. γ contains two -segments . In this case, the robots that are neighbors of alonguest hole (at least two) can try to move (see Line 13). So, by fairnessproperty, a non-empty set of these robots, say S , is eventually activated bythe scheduler. Now, every robot in S decides with a uniform probability to INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots S decides to move. In this case, we retreive the previous case and we are done.4. γ contains four isolated nodes . Let l max be the length of the longuest hole in γ . Let study the following subcases:(a) Only two robots are neighbors of a l max -hole . In this case, the two robotsthat are neighbors of the unique l max -hole can move. So, by fairness prop-erty, either one or both of them eventually move through their shortestneighboring hole (see Line 26). After such moves, either ( i ) the sys-tem is still in a configuration containing four isolated nodes and wheretwo robots are neighbors of a unique longuest hole but the size of thelonguest hole increased, or ( ii ) the system is in a configuration contain-ing a unique 2-segment, or ( iii ) the system is in a configuration containingtwo 2-segments. Hence, the system reaches in finite time a configurationsatisfying ( ii ) or ( iii ), i.e. , we eventually retreive the cases 2 or 3, and weare done.(b) Exactly three robots are neighbors of a l max -hole . Let R be the robotthat is not neighbor of a l max -hole. Let R and R be the two robotsthat are neighbor of exactly one l max -hole. In this case, only R and R can move (see Line 22) and by fairness property at least one of themeventually does. If only one of them moves, then we retreive Subcase4.(a) or Case 2, and we are done. If both R and R move, then thesystem reaches ( i ) either a configuration where exactly three robots areneighbors of a longuest hole of length l max + 1 or ( ii ) a configurationcontaining a 3-segment. In Case ( i ), if we repeat the argument, we cansee that we eventually retreive Subcase 4.(a), Case 1, or Case 2, and weare done. In Case ( ii ), we directly retreive Case 1 and we are done.(c) The four robots are neighbors of a l max -hole . In this case, the configura-tion is symmetric and all robots try move (see Line 26). Now, despite thechoice of the scheduler, there is a strictly positive probability that onlyone robot probabilistically decides to move. In this case, the robot movesthrough one of its neighboring l max -hole of size at least two (to providethe tower creation). As a consequence, we retreive Subcases 4.(a) or 4.(b)and we are done.Hence, in all cases there is a strictly positive probability that the system eventuallyreaches a configuration containing a 4-segment from γ and the lemma holds. ✷ RR n ° S. Devismes et al.
IIPhase II is described in Algorithm 3: Starting from a configuration where there is a 4-segment on nodes u i , u i +1 , u i +2 , u i +3 , the system eventually reaches a configurationwhere a primary arrow is formed on nodes u i , u i +1 , u i +2 , u i +3 . To that goal, weproceed as follows: Let R and R be the robots located at the nodes u i +1 and u i +2 of the 4-segment. R and R try to move to u i +2 and u i +1 , respectively. Eventuallyonly one of these robots moves and we are done, as proven in the two next lemmas. Algorithm 3
Procedure
P hase
II. if I am not located at an extremity of the 4-segment then Try to move toward my neighboring node that is not an extremity of the 4-segment; end if Lemma 8
Let γ be a configuration containing a -segment u i , u i +1 , u i +2 , u i +3 . If γ is the configuration at instant t , then the configuration at instant t + 1 is eitheridentical to γ or the configuration containing the primary arrow u i , u i +1 , u i +2 , u i +3 . Proof.
Let R (resp. R ) be the robot located at node u i +1 (resp. u i +2 ) in γ . In γ , all robots executes Algorithm 3 (see Algorithm 1). So, from γ , only R and R can move: R can move to node u i +2 and R can move to node u i +1 (see Algorithm3). When one or both of these robots, we obtain a configuration containing either a4-segment or a primary arrow in u i , u i +1 , u i +2 , u i +3 and the lemma holds. ✷ Lemma 9
From a configuration containing a -segment, the system reaches a con-figuration containing a primary arrow in finite expected time. Proof.
By Lemma 8, we know that starting from a configuration γ containinga 4-segment, the system either remains in the same configuration or reaches a con-figuration containing a primary arrow. Let R and R be the robots that are notlocated at the extremity of the 4-segment in γ . Only R and R can (probabilisti-cally) decide to move in γ . Also, by the fairness property, eventually one or bothof them are activated. Now, despite the choice of the scheduler, there is a strictlypositive probability that only one of them probabilistically decide to move: in thiscase, the system reaches a configuration containing a primary arrow (see Algorithm3) and we are done. ✷ INRIA ptimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots IIIPhase III is described in Algorithm 4. This phase is fully deterministic: Let H bethe hole between the tail and the head of arrow. The robot located at the arrow tailtraverses H . When it is done, the system is in a terminal configuration containinga final arrow: all nodes have been visited as shown is the theorem below. Algorithm 4
Procedure
P hase
III. if I am the arrow tail then Move toward the arrow head through the hole having me and the arrow head as neighbor; end if Theorem 3
Algorithm 1 is a probabilistic exploration protocol for robots in a ringof n > nodes. Proof.
The proof of the theorem is based on the two following claims:1.
Any configuration containing a final arrow is terminal . Proof:
Immediate, see Line 1 of Algorithm 1.2.
From a configuration containing a non-final arrow of length x , the systemeventually reaches a configuration containing a x + 1 -arrow. Proof:
In such a configuration, only the arrow tail can move. By the fairnessproperty, the robot located at the arrow tail moves in finite time : it movesthrough its neighboring hole having the arrow head as other neighbor (seeAlgorithm 4). As a consequence, the size of the arrow is incremented to x + 1and we are done.Using the two previous claims, we now prove the lemma in two step: ˆ Termination.
Any computation of Algorithm 1 terminates in finite expectedtime.
Proof:
Immediate from Lemmas 7, 9, Claims 1 and 2. ˆ Partial Correctness.
When a computation of Algorithm 1 terminates, anynode has been visited.
Proof:
By Lemma 7, starting from any initial configuration, the systemreaches in finite expected time a configuration containing a 4-segment say
RR n ° S. Devismes et al. u i , u i +1 , u i +2 , u i +3 . By Lemmas 8 and 9, from this configuration the sys-tem reaches in finite expected time a configuration containing an arrow on u i , u i +1 , u i +2 , u i +3 . Hence, when the phase III starts, nodes u i , u i +1 , u i +2 , and u i +3 are already visited. By Claim 2, the robots executes then Algorithm 4until the computation terminates. Let P be the path u i − , . . . , u i − n +4 . ByClaim 2, until the computation terminated, only the robot located at the ar-row tail can move and it it move following P . Hence, when the computationterminates all nodes of P have been visited ( i.e. , nodes u i − , . . . , u i − n +4 ) and,as nodes u i , u i +1 , u i +2 , u i +3 have also been visited, we are done. ✷ We provided evidence that for the exploration problem in uniform rings, random-ization could shift complexity from Θ(log n ) to Θ(1). While applying randomizationto other problem instances is an interesting topic for further research, we would liketo point out immediate open questions raised by our work:1. Though we were able to provide a general algorithm for any n (strictly) greaterthan eight, it seems that ad hoc solutions have to be designed when n isbetween five and eight (included).2. Our protocol is optimal with respect to the number of robots. However, theefficiency (in terms of exploring time) is only proved to be finite. Actuallycomputing the convergence time from our proof argument is feasible, but itwould be more interesting to study how the number of robots relates to the timecomplexity of exploration, as it seems natural that more robots will explorethe ring faster. References [1] H. Ando, Y. Oasa, I. Suzuki, and M. Yamashita. Distributed memoryless pointconvergence algorithm for mobilerobots with limited visibility.
IEEE Transac-tions on Robotics and Automation , 1999.[2] Yuichi Asahiro, Satoshi Fujita, Ichiro Suzuki, and Masafumi Yamashita. A self-stabilizing marching algorithm for a group of oblivious robots. In Baker et al.[3], pages 125–144.
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Principles of Dis-tributed Systems, 12th International Conference, OPODIS 2008, Luxor, Egypt,December 15-18, 2008. Proceedings , volume 5401 of
Lecture Notes in ComputerScience . Springer, 2008.[4] Zohir Bouzid, Maria Gradinariu Potop-Butucaru, and S´ebastien Tixeuil.Byzantine-resilient convergence in oblivious robot networks. In
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Parallel ProcessingLetters , 19(1):175–184, 2009.[6] Yoann Dieudonn´e, Ouiddad Labbani-Igbida, and Franck Petit. Circle formationof weak mobile robots.
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Lecture Notes in Computer Science , pages 105–118. Springer,2007.[8] Paola Flocchini, David Ilcinkas, Andrzej Pelc, and Nicola Santoro. Remember-ing without memory: Tree exploration by asynchronous oblivious robots. InAlexander A. Shvartsman and Pascal Felber, editors,
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Lecture Notes in Computer Science , pages 33–47. Springer, 2008.[9] Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer.Gathering of asynchronous robots with limited visibility.
Theor. Comput. Sci. ,337(1-3):147–168, 2005.[10] Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer.Arbitrary pattern formation by asynchronous, anonymous, oblivious robots.
Theor. Comput. Sci. , 407(1-3):412–447, 2008.[11] Ralf Klasing, Adrian Kosowski, and Alfredo Navarra. Taking advantage ofsymmetries: Gathering of asynchronous oblivious robots on a ring. In Bakeret al. [3], pages 446–462.[12] Ralf Klasing, Euripides Markou, and Andrzej Pelc. Gathering asynchronousoblivious mobile robots in a ring.
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