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Computer Science
Theoretical Computer Science

Anonymous Meeting in Networks

Profile image of Andrzej PelcAndrzej Pelc

2016, HAL (Le Centre pour la Communication Scientifique Directe)

Abstract

A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node. Agents are anonymous (identical), execute the same deterministic algorithm and move in synchronous rounds along links of the network. An initial configuration of agents is called gatherable if there exists a deterministic algorithm (even dedicated to this particular configuration) that achieves meeting of all agents in one node. Which configurations are gatherable and how to gather all of them deterministically by the same algorithm? We give a complete solution of this gathering problem in arbitrary networks. We characterize all gatherable configurations and give two universal deterministic gathering algorithms, i.e., algorithms that gather all gatherable configurations. The first algorithm works under the assumption that a common upper bound N on the size of the network is known to all agents. In this case our algorithm guarantees gathering with detection, i.e., the existence of a round for any gatherable configuration, such that all agents are at the same node and all declare that gathering is accomplished. If no upper bound on the size of the network is known, we show that a universal algorithm for gathering with detection does not exist. Hence, for this harder scenario, we construct a second universal gathering algorithm, which guarantees that, for any gatherable configuration, all agents eventually get to one node and stop, although they cannot tell if gathering is over. The time of the first algorithm is polynomial in the upper bound N on the size of the network, and the time of the second algorithm is polynomial in the (unknown) size itself. Our results have an important consequence for the leader election problem for anonymous agents in arbitrary graphs. Leader election is a fundamental symmetry breaking problem in distributed computing. Its goal is to assign, in some common round, value 1 (leader) to one of the entities and value 0 (nonleader) to all others. For anonymous agents in graphs, leader election turns out to be equivalent to gathering with detection. Hence, as a by-product, we obtain a complete solution of the leader election problem for anonymous agents in arbitrary graphs.

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In this paper, we consider the partial gathering problem of mobile agents in arbitrary networks. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all the agents meet at the same node. The partial gathering problem requires, for a given positive integer g, that each agent should move to a node and terminate so that at least g agents should meet at each of the nodes they terminate at. The requirement for the partial gathering problem is no stronger than that for the total gathering problem, and thus, we clarify the difference on the move complexity between them. First, we show that agents require Ω(gn + m) total moves to solve the partial gathering problem, where n is the number of nodes and m is the number of communication links. Next, we propose a deterministic algorithm to solve the partial gathering problem in O(gn + m) total moves, which is asymptotically optimal in terms of total moves. Note that, it is known that agents require Ω(kn+ m) total moves to solve the total gathering problem in arbitrary networks, where k is the number of agents. Thus, our result shows that the partial gathering problem is solvable with strictly fewer total moves compared to the total gathering problem in arbitrary networks.

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Deterministic Leader Election in Anonymous Radio Networks
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Leader election is a fundamental task in distributed computing. It is a symmetry breaking problem, calling for one node of the network to become the leader, and for all other nodes to become non-leaders. We consider leader election in anonymous radio networks modeled as simple undirected connected graphs. Nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbours, or stay silent and listen. A node v hears a message from a neighbour w in a given round if v listens in this round and if w is its only neighbour transmitting in this round. If v listens in a round in which more than one neighbour transmits then v hears noise that is different from any message and different from silence. We assume that nodes are identical (anonymous) and execute the same deterministic algorithm. Under this scenario, symmetry can be broken only in one way: by different wake-up times of the nodes. In which situations is it possible to break symmetry and...

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Deterministic Leader Election in Anonymous Radio Networks 1
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Leader election is a fundamental task in distributed computing. It is a symmetry breaking problem, calling for one node of the network to become the leader , and for all other nodes to become non-leaders . We consider leader election in anonymous radio networks modeled as simple undirected connected graphs. Nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbours, or stay silent and listen. A node v hears a message from a neighbour w in a given round if v listens in this round and if w is its only neighbour transmitting in this round. If v listens in a round in which more than one neighbour transmits then v hears noise that is different from any message and different from silence. We assume that nodes are identical (anonymous) and execute the same deterministic algorithm. Under this scenario, symmetry can be broken only in one way: by different wake-up times of the nodes. In which situations is it possible to break symmetry a...

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Symmetry Breaking In Anonymous Networks: Characterizations
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We address the enumeration and the leader election problems over partially anonymous and multi-hop broadcast networks. We consider an asynchronous communication model where each process broadcasts a message and all its neighbours receive this message after arbitrary and unpredictable time. In this paper, we present necessary conditions that must be satisfied by any graph to solve these problems and we show that these conditions are sufficient by providing an enumeration algorithm on the one hand and a leader election algorithm on the other hand. From the complexity viewpoint, our algorithms offer a polynomial complexity (memory, number of messages and size of messages).

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We present distributed protocols for electing a leader among k mobile agents that are dispersed among the n nodes of a graph. While previous solutions for the agent election problem were restricted to specific topologies or under specific conditions, the protocols presented in this paper face the problem in the most general case, i.e. for an arbitrary topology where the nodes of the graph may not be distinctly labelled and the agents might be all identical (and thus indistinguishable from each other). In such cases, the agent election problem is often difficult, and sometimes impossible to solve using deterministic means. We have designed protocols for solving the problem that-unlike previous solutions-are effective, meaning that they always succeed in electing a leader under any given setting if at all it is possible, and otherwise detect the fact that election is impossible in that setting. We present several election protocols, all effective. Starting with the straightforward solution, that requires an exponential amount of edge-traversals by the agents, we describe significantly more efficient algorithms; in the latter the total number of edge-traversals made by the agents is always polynomial, their difference is in the amount of bits of storage they required at the nodes.

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Communication Complexity of Consensus in Anonymous Message Passing Systems
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Lecture Notes in Computer Science, 2011

We consider the message complexity of achieving consensus in synchronous anonymous message passing systems. Unlabeled processors (nodes) communicate through links of a network. In each round every processor can exchange messages with all neighbors and the duration of each transmission is one round. An adversary wakes up some subset of processors at possibly different times and assigns them arbitrary numerical input values. All other processors are dormant and do not have input values. Any message wakes up a dormant processor. The goal of consensus is to wake up all processors and have them agree on one of the input values. We seek deterministic consensus algorithms using as few messages as possible. As opposed to most of the literature on consensus, the difficulty of our scenario are not faults (we assume that the network is fault-free) but the arbitrary network topology combined with the anonymity of nodes. For unknown n-node networks we show a consensus algorithm using O(n 2 ) messages; this complexity is optimal for this class. We show that if the network is known, then the complexity of consensus decreases significantly. Our main contribution is an algorithm that uses O(n 3/2 log 2 n) messages on any n-node network and we show that some networks require Ω(n log n) messages to achieve consensus. We also observe that availability of distinct labels of nodes helps to improve complexity of consensus for known networks but has no effect for the class of unknown networks. Indeed, even with labeled nodes, Ω(n 2 ) messages are sometimes necessary if the network is unknown but for known labeled networks consensus can be always achieved with O(n) messages.

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Gathering asynchronous oblivious agents with local vision in regular bipartite graphs
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We consider the problem of gathering identical, memoryless, mobile agents in one node of an anonymous graph. Agents start from different nodes of the graph. They operate in Look-Compute-Move cycles and have to end up in the same node. In one cycle, an agent takes a snapshot of its immediate neighborhood (Look), makes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case makes an instantaneous move to this neighbor (Move). Cycles are performed asynchronously for each agent. The novelty of our model with respect to the existing literature on gathering asynchronous oblivious agents in graphs is that the agents have very restricted perception capabilities: they can only see their immediate neighborhood. An initial configuration of agents is called gatherable if there exists an algorithm that gathers all the agents of the configuration in one node and keeps them idle from then on, regardless of the actions of the asynchronous adversary. (The algorithm can be even tailored to gather this specific configuration.) The gathering problem is to determine which configurations are gatherable and find a (universal) algorithm which gathers all gatherable configurations. We give a complete solution of the gathering problem for regular bipartite graphs. Our main contribution is the proof that the class of gatherable initial configurations is very small: it consists only of ''stars'' (an agent A with all other agents adjacent to it) of size at least 3. On the positive side we give an algorithm accomplishing gathering for every gatherable configuration.

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On the complexity of universal leader election
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Proceedings of the 2013 ACM symposium on Principles of distributed computing, 2013

Electing a leader is a fundamental task in distributed computing. In its implicit version, only the leader must know who is the elected leader. This paper focuses on studying the message and time complexity of randomized implicit leader election in synchronous distributed networks. Surprisingly, the most "obvious" complexity bounds have not been proven for randomized algorithms. In particular, the seemingly obvious lower bounds of Ω(m) messages, where m is the number of edges in the network, and Ω(D) time, where D is the network diameter, are non-trivial to show for randomized (Monte Carlo) algorithms. (Recent results, showing that even Ω(n), where n is the number of nodes in the network, is not a lower bound on the messages in complete networks, make the above bounds somewhat less obvious). To the best of our knowledge, these basic lower bounds have not been established even for deterministic algorithms, except for the restricted case of comparison algorithms, where it was also required that nodes may not wake up spontaneously and that D and n were not known. We establish these fundamental lower bounds in this paper for the general case, even for randomized Monte Carlo algorithms. Our lower bounds are universal in the sense that they hold for all universal algorithms (namely, algorithms that work for all graphs), apply to every D, m, and n, and hold even if D, m, and n are known, all the nodes wake up simultaneously, and the algorithms can make any use of node's identities. To show that these bounds are tight, we present an O(m) messages algorithm. An O(D) time leader election algorithm is known. A slight adaptation of * A preliminary version of this article appeared in the proceedings of the 32nd ACM Symposium on Principles of Distributed Computing (PODC) 2013, pages 100-109.

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Deterministic gathering of anonymous agents in arbitrary networks
Andrzej Pelc

A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node. Agents are anonymous (identical), execute the same deterministic algorithm and move in synchronous rounds along links of the network. An initial configuration of agents is called gatherable if there exists a deterministic algorithm (even dedicated to this particular configuration) that achieves meeting of all agents in one node. Which configurations are gatherable and how to gather all of them deterministically by the same algorithm? We give a complete solution of this gathering problem in arbitrary networks. We characterize all gatherable configurations and give two universal deterministic gathering algorithms, i.e., algorithms that gather all gatherable configurations. The first algorithm works under the assumption that an upper bound n on the size of the network is known. In this case our algorithm guarantees gathering with detection, i.e., the existence of a round for any gatherable configuration, such that all agents are at the same node and all declare that gathering is accomplished. If no upper bound on the size of the network is known, we show that a universal algorithm for gathering with detection does not exist. Hence, for this harder scenario, we construct a second universal gathering algorithm, which guarantees that, for any gatherable configuration, all agents eventually get to one node and stop, although they cannot tell if gathering is over. The time of the first algorithm is polynomial in the upper bound n on the size of the network, and the time of the second algorithm is polynomial in the (unknown) size itself. Our results have an important consequence for the leader election problem for anonymous agents in arbitrary graphs. Leader election is a fundamental symmetry breaking problem in distributed computing. Its goal is to assign, in some common round, value 1 (leader) to one of the entities and value 0 (nonleader) to all others. For anonymous agents in graphs, leader election turns out to be equivalent to gathering with detection. Hence, as a by-product, we obtain a complete solution of the leader election problem for anonymous agents in arbitrary graphs.

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Leader election for anonymous asynchronous agents in arbitrary networks
Andrzej Pelc

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We study the problem of leader election among mobile agents operating in an arbitrary network modeled as an undirected graph. Nodes of the network are unlabeled and all agents are identical. Hence the only way to elect a leader among agents is by exploiting asymmetries in their initial positions in the graph. Agents do not know the graph or their positions in it, hence they must gain this knowledge by navigating in the graph and share it with other agents to accomplish leader election. This can be done using meetings of agents, which is difficult because of their asynchronous nature: an adversary has total control over the speed of agents. When can a leader be elected in this adversarial scenario and how to do it? We give a complete answer to this question by characterizing all initial configurations for which leader election is possible and by constructing an algorithm that accomplishes leader election for all configurations for which this can be done.

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Electing a Leader Among Anonymous Mobile Agents in Anonymous Networks With Sense-of-Direction
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We consider a collection of r anonymous asynchronous mobile agents dispersed on an arbitrary anonymous network of size n. Neither r nor n are known a priori by the agents. We examine the problem of electing a leader among those agents and study the conditions for its solvability. We show that, without sense of direction, the problem is unsolvable, even if restricted to instances for which gcd(r, n) = 1. We also show that, with sense of direction, the problem remains unsolvable, but it becomes solvable if restricted to instances for which gcd(r, n) = 1. Since sense of direction can be given to any m-edge graph in O(m) time, our result shows that one can easily label the edges of an anonymous network in order to significantly improve its computational power.

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Latecomers Help to Meet: Deterministic Anonymous Gathering in the Plane
Andrzej Pelc

arXiv (Cornell University), 2018

A team of anonymous mobile agents represented by points freely moving in the plane have to gather at a single point and stop. Agents start at different points of the plane and at possibly different times chosen by the adversary. They are equipped with compasses, a common unit of distance and clocks. They execute the same deterministic algorithm. When moving, agents travel at the same speed normalized to 1. When agents are at distance at most , for some positive constant unknown to them, they see each other and can exchange all information known to date. Due to the anonymity of the agents and the symmetry of the plane, gathering is impossible, e.g., if agents start simultaneously at distances larger than . However, if some agents start with a delay with respect to others, gathering may become possible. In which situations such latecomers can enable gathering? To answer this question we consider initial configurations formalized as sets of pairs {(p 1 , t 1 ), (p 2 , t 2 ), . . . , (p n , t n )}, for n ≥ 2 where p i is the starting point of the i-th agent and t i is its starting time. An initial configuration is gatherable if agents starting at it can be gathered by some algorithm, even dedicated to this particular configuration. Our first result is a characterization of all gatherable initial configurations. It is then natural to ask if there is a universal deterministic algorithm that can gather all gatherable configurations of a given size. It turns out that the answer to this question is negative. Indeed, we show that all gatherable configurations can be partitioned into two sets: bad configurations and good configurations. We show that bad gatherable configurations (even of size 2) cannot be gathered by a common gathering algorithm. On the other hand, we prove that there is a universal algorithm that gathers all good configurations of a given size. Then we ask the question of whether the exact knowledge of the number of agents is necessary to gather all good configurations. It turns out that the answer is no, and we prove a necessary and sufficient condition on the knowledge concerning the number of agents that an algorithm gathering all good configurations must have.

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Four Shades of Deterministic Leader Election in Anonymous Networks
Andrzej Pelc

Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures, 2021

Leader election is one of the fundamental problems in distributed computing: a single node, called the leader, must be specified. This task can be formulated either in a weak way, where one node outputs 'leader' and all other nodes output 'non-leader', or in a strong way, where all nodes must also learn which node is the leader. If the nodes have distinct identifiers, then such an agreement means that all nodes have to output the identifier of the elected leader. For anonymous networks, the strong version of leader election requires that all nodes must be able to find a path to the leader, as this is the only way to identify it. In this paper, we study variants of deterministic leader election in arbitrary anonymous networks. Leader election is impossible in some anonymous networks, regardless of the allocated amount of time, even if nodes know the entire map of the network. This is due to possible symmetries in the network. However, even in networks in which it is p...

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