Node Labels in Local Decision

The role of unique node identifiers in network computing is well understood as far as symmetry breaking is concerned. However, the unique identifiers also leak information about the computing environment - in particular, they provide some nodes with information related to the size of the network. It was recently proved that in the context of local decision, there are some decision problems such that (1) they cannot be solved without unique identifiers, and (2) unique node identifiers leak a sufficient amount of information such that the problem becomes solvable (PODC 2013). In this work we give study what is the minimal amount of information that we need to leak from the environment to the nodes in order to solve local decision problems. Our key results are related to scalar oracles $f$ that, for any given $n$, provide a multiset $f(n)$ of $n$ labels; then the adversary assigns the labels to the $n$ nodes in the network. This is a direct generalisation of the usual assumption of unique node identifiers. We give a complete characterisation of the weakest oracle that leaks at least as much information as the unique identifiers. Our main result is the following dichotomy: we classify scalar oracles as large and small, depending on their asymptotic behaviour, and show that (1) any large oracle is at least as powerful as the unique identifiers in the context of local decision problems, while (2) for any small oracle there are local decision problems that still benefit from unique identifiers.Comment: Conference version to appear in the proceedings of SIROCCO 201

Topology recognition with advice

In topology recognition, each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes. We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size $n$ and diameter $D\le \alpha n$, for any constant $\alpha< 1$. In most cases, our bounds are asymptotically tight

Leader Election for Anonymous Asynchronous Agents in Arbitrary Networks

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

Optimal Computation in Leaderless and Multi-Leader Disconnected Anonymous Dynamic Networks

We give a simple characterization of which functions can be computed deterministically by anonymous processes in disconnected dynamic networks, depending on the number of leaders in the network. In addition, we provide efficient distributed algorithms for computing all such functions assuming minimal or no knowledge about the network. Each of our algorithms comes in two versions: one that terminates with the correct output and a faster one that stabilizes on the correct output without explicit termination. Notably, these are the first deterministic algorithms whose running times scale linearly with both the number of processes and a parameter of the network which we call "dynamic disconnectivity". We also provide matching lower bounds, showing that all our algorithms are asymptotically optimal for any fixed number of leaders. While most of the existing literature on anonymous dynamic networks relies on classical mass-distribution techniques, our work makes use of a recently introduced combinatorial structure called "history tree", also developing its theory in new directions. Among other contributions, our results make definitive progress on two popular fundamental problems for anonymous dynamic networks: leaderless Average Consensus (i.e., computing the mean value of input numbers distributed among the processes) and multi-leader Counting (i.e., determining the exact number of processes in the network). In fact, our approach unifies and improves upon several independent lines of research on anonymous networks, including Nedic et al., IEEE Trans. Automat. Contr. 2009; Olshevsky, SIAM J. Control Optim. 2017; Kowalski-Mosteiro, ICALP 2019, SPAA 2021; Di Luna-Viglietta, FOCS 2022.Comment: 35 pages, 1 figure. arXiv admin note: text overlap with arXiv:2204.0212

Non Trivial Computations in Anonymous Dynamic Networks

In this paper we consider a static set of anonymous processes, i.e., they do not have distinguished IDs, that communicate with neighbors using a local broadcast primitive. The communication graph changes at each computational round with the restriction of being always connected, i.e., the network topology guarantees 1-interval connectivity. In such setting non trivial computations, i.e., answering to a predicate like "there exists at least one process with initial input a?", are impossible. In a recent work, it has been conjectured that the impossibility holds even if a distinguished leader process is available within the computation. In this paper we prove that the conjecture is false. We show this result by implementing a deterministic leader-based terminating counting algorithm. In order to build our counting algorithm we first develop a counting technique that is time optimal on a family of dynamic graphs where each process has a fixed distance h from the leader and such distance does not change along rounds. Using this technique we build an algorithm that counts in anonymous 1-interval connected networks

Reliable broadcast in anonymous distributed systems with fair lossy channels

Reliable Broadcast (RB) is a basic abstraction in distributed systems, because it allows processes to communicate consistently and reliably to each other. It guarantees that all correct process reliable deliver the same set of messages. This abstraction has been extensively investigated in distributed systems where all processes have different identifiers, and the communication channels are reliable. However, more and more anonymous systems appear due to the motivation of privacy. It is significant to extend RB into anonymous system model where each process has no identifier. In another hand, the requirement of reliable communication channels is not always satisfied in real systems. Hence, this paper is aimed to study RB abstraction in anonymous distributed systems with fair lossy communication channels. In distributed systems, symmetry always mean that two systems should be considered symmetric if they behave identically, and two components of a system should be considered symmetric if they are indistinguishable. Hence, the anonymous distributed systems is symmetry. The design difficulty of RB algorithm lies in how to break the symmetry of the system. In this paper, we propose to use a random function to break it. Firstly, a non-quiescent RB algorithm tolerating an arbitrary number of crashed processes is given. Then, we introduce an anonymous perfect failure detector AP?. Finally, we propose an extended and quiescent RB algorithm using AP?

Deterministic Symmetry Breaking in Ring Networks

We study a distributed coordination mechanism for uniform agents located on a circle. The agents perform their actions in synchronised rounds. At the beginning of each round an agent chooses the direction of its movement from clockwise, anticlockwise, or idle, and moves at unit speed during this round. Agents are not allowed to overpass, i.e., when an agent collides with another it instantly starts moving with the same speed in the opposite direction (without exchanging any information with the other agent). However, at the end of each round each agent has access to limited information regarding its trajectory of movement during this round. We assume that $n$ mobile agents are initially located on a circle unit circumference at arbitrary but distinct positions unknown to other agents. The agents are equipped with unique identifiers from a fixed range. The {\em location discovery} task to be performed by each agent is to determine the initial position of every other agent. Our main result states that, if the only available information about movement in a round is limited to %information about distance between the initial and the final position, then there is a superlinear lower bound on time needed to solve the location discovery problem. Interestingly, this result corresponds to a combinatorial symmetry breaking problem, which might be of independent interest. If, on the other hand, an agent has access to the distance to its first collision with another agent in a round, we design an asymptotically efficient and close to optimal solution for the location discovery problem.Comment: Conference version accepted to ICDCS 201

Self-stabilizing network orientation algorithms in arbitrary rooted networks

Network orientation is the problem of assigning different labels to the edges at each processor, in a globally consistent manner. A self-stabilizing protocol guarantees that the system will arrive at a legitimate state in finite time, irrespective of the initial state of the system. Two deterministic distributed network orientation protocols on arbitrary rooted, asynchronous networks are proposed in this work. Both protocols set up a chordal sense of direction in the network. The protocols are self-stabilizing, meaning that starting from an arbitrary state, the protocols are guaranteed to reach a state in which every processor has a valid node label and every link has a valid edge label. The first protocol assumes an underlying depth-first token circulation protocol; it orients the network as the token is passed among the nodes and stabilizes in O(n) steps after the token circulation stabilizes, where n is the number of processors in the network. The second protocol is designed on an underlying spanning tree protocol and stabilizes in O(h) time, after the spanning tree is constructed, where h is the height of the spanning tree. Although the second protocol assumes the existence of a spanning tree of the rooted network, it orients all edges--both tree and non-tree edges--of the network