The Dancing problem requires a swarm of n autonomous mobile robots to form a sequence of patterns, aka perform a choreography. Existing work has proven that some crucial restrictions on choreographies and initial configurations (e.g., on repetitions of patterns, periodicity, symmetries, contractions/expansions) must hold so that the Dancing problem can be solved under certain robot models. Here, we prove that these necessary constraints can be dropped by considering the LUMI model (i.e., where robots are endowed with a light whose color can be chosen from a constant-size palette) under the quite unexplored sequential scheduler. We formalize the class of Universal Dancing problems which require a swarm of n robots starting from any initial configuration to perform a (periodic or finite) sequence of arbitrary patterns, only provided that each pattern consists of n vertices (including multiplicities). However, we prove that, to be solvable under LUMI, the length of the feasible choreographies is bounded by the compositions of n into the number of colors available to the robots. We provide an algorithm solving the Universal Dancing problem by exploiting the peculiar capability of sequential robots to implement a distributed counter mechanism. Even assuming non-rigid movements, our algorithm ensures spatial homogeneity of the performed choreography.
In this paper, we resolve a long-standing open problem in self-stabilization asking whether it is possible to construct a spanning tree using constant memory per node in a synchronous semi-uniform networks, i.e., networks in which one node is distinguished. We design a synchronous self-stabilizing algorithm that constructs a breadth-first search (BFS) tree in any anonymous semi-uniform network using only a constant number of bits of memory per node. Crucially, our approach operates without any prior knowledge of global network parameters such as maximum degree, diameter, or number of nodes. In contrast to traditional self-stabilizing methods - such as pointer-to-neighbors, distance-toroot, or identifiers - that are unsuitable under strict memory constraints, our solution employs an innovative constant-space token dissemination mechanism. This mechanism effectively eliminates cycles and rectifies errors in the BFS structure, ensuring both correctness and memory efficiency. The proposed algorithm not only meets the stringent requirements of memory-constrained distributed systems, but also opens new avenues for research in the design of self-stabilizing protocols under severe resource limitations: constant space-complexity may not systematically prevent the existence of self-stabilizing algorithms for important non-trivial tasks. 2012 ACM Subject Classification Theory of computation -> Design and analysis of algorithms; Mathematics of computing -> Discrete mathematics
In protocols with asymmetric trust, each participant is free to make its own trust assumptions about others, captured by an asymmetric quorum system. This contrasts with ordinary, symmetric quorum systems and threshold models, where trust assumptions are uniformly shared among participants. Fundamental problems like reliable broadcast and consensus are unsolvable in the asymmetric model if quorum systems satisfy only the classical properties of consistency and availability. As a result, existing solutions introduce stronger assumptions to circumvent this limitation. We show that some requirements used by state-of-the-art approaches are overly restrictive, so much so that they effectively eliminate the benefits of asymmetric trust. To address this, we propose a new approach to characterize asymmetric problems and, building upon it, present an asymmetric asynchronous unauthenticated reliable broadcast algorithm that significantly weakens the assumptions needed to solve the problem. Our techniques are general and can be readily adapted to other core problems in the asymmetric trust setting. 2012 ACM Subject Classification Theory of computation -> Distributed algorithms
We study the exploration problem by mobile agents in Connectivity Time dynamic graphs. The Connectivity Time model was introduced by Michail et al. [JPDC 2014] and is arguably one of the weakest dynamic graph connectivity models. We prove that exploration is impossible in such graphs using ( n-1)( n-2)/2 mobile agents starting from an arbitrary initial configuration, even when agents have full knowledge of system parameters, global communication, full visibility, and infinite memory. We then present an exploration algorithm that uses ( n-1)( n-2)/2 + 1 agents equipped with global communication, 1-hop visibility and O(log n) memory. 2012 ACM Subject Classification Theory of computation -> Distributed algorithms
Total Store Order (TSO) is one of the most popular relaxed memory model in multiprocessor architectures, widely implemented, for example, in Intel's x86 and x64 platforms. It delays write visibility via store buffers, thereby allowing a significant improvement in efficiency. This, however, complicates reasoning about correctness, as executions may violate sequential consistency. We present a semantic framework that provides effective tools that can pinpoint when such synchronization is necessary under TSO. We define a TSO-specific occurs -before relation, adapting Lamport's happens-before to TSO, and prove that events at different sites can be temporally ordered only via an occurs-before chain. Analyzing how fences and RMW5 create these chains lets us identify when they are unavoidable. We present in this BA how these results impact linearizable implementations of registers, capturing information flow and causality in TSO. The full version of this work provides details as well as results regarding the need for synchronization in linearizable implementations of additional objects.
Asynchronous byzantine agreement extension studies the message complexity of L-bit multivalued asynchronous byzantine agreement given access to a binary asynchronous Byzantine agreement protocol. We prove that asynchronous byzantine agreement extension can be solved with perfect security and optimal resilience in O( nL + n(2) log n) total communication (in bits) in addition to a single call to a binary asynchronous Byzantine agreement protocol. For L = O( n log n), this gives an asymptotically optimal protocol, resolving a question that remained open for nearly two decades. List decoding is a fundamental concept in theoretical computer science and cryptography, enabling error correction beyond the unique decoding radius and playing a critical role in constructing robust codes, hardness amplification, and secure cryptographic protocols. A key novelty of our perfectly secure and optimally resilient asynchronous byzantine agreement extension protocol is that it uses list decoding - making a striking new connection between list decoding and asynchronous Byzantine agreement. 2012 ACM Subject Classification Security and privacy; Security and privacy -> Cryptography; Security and privacy -> Information-theoretic techniques; Security and privacy -> Distributed systems security
We consider two fundamental communication tasks in arbitrary radio networks: broadcasting (information from one source has to reach all nodes) and gossiping (every node has a message and all messages have to reach all nodes). Nodes are assigned labels that are (not necessarily different) binary strings. Each node knows its own label and can use it as a parameter in the same deterministic algorithm. The length of a labeling scheme is the largest length of a label. The goal is to find labeling schemes of asymptotically optimal length for the above tasks, and to design fast deterministic distributed algorithms for each of them, using labels of optimal length. Our main result concerns broadcasting. We show the existence of a labeling scheme of constant length that supports broadcasting in time O(D + log(2) n), where D is the diameter of the network and n is the number of nodes. This broadcasting time is an improvement over the best currently known O(D log n log(2) n) time of broadcasting with constant-length labels, due to Ellen and Gilbert (SPAA 2020). It also matches the optimal broadcasting time in radio networks of known topology. Hence, we show that appropriately chosen node labels of constant length permit to achieve, in a distributed way, the optimal centralized broadcasting time. This is, perhaps, the most surprising finding of this paper. We are able to obtain our result thanks to a novel methodological tool of propagating information in radio networks, that we call a 2-height respecting tree. Next, we apply our broadcasting algorithm to solve the gossiping problem. We get a gossiping algorithm working in time O(D + Delta log n log(2) n), using a labeling scheme of optimal length O(log Delta), where Delta is the maximum degree. Our time is the same as the best known gossiping time in radio networks of known topology.
We present the delta-Synchronizer, which works in non-synchronous dynamic networks under minimal assumptions. Our model allows for arbitrary topological changes without any guarantee of eventual global or partial stabilization and assumes that nodes are anonymous. This deterministic synchronizer is the first that enables nodes to simulate a dynamic network synchronous algorithm for executions in a semi-synchronous dynamic environment under a weakly-fair node activation scheduler, despite the absence of a global clock, node ids, persistent connectivity or any assumptions about the edge dynamics (in both the synchronous and semi-synchronous environments). We make the following contributions: (1) we extend the definition of synchronizers to networks with arbitrary edge dynamics; (2) we present the first synchronizer from the semi-synchronous to the synchronous model in such networks; and (3) we present non-trivial applications of the proposed synchronizer to existing algorithms. We assume an extension of the Pull communication model by adding a single 1-bit multi-writer atomic register at each edge-port of a node. We show that this extension is needed and that synchronization in our setting is not possible without it. The delta-Synchronizer operates with memory overhead at the nodes that is asymptotically logarithmic on the runtime of the underlying synchronous algorithm being simulated - in particular, it is logarithmic for polynomial-time synchronous algorithms. 2012 ACM Subject Classification Theory of computation. Distributed algorithms
The distributed Data Retrieval (DR) model consists of k peers connected by a complete peer-to-peer communication network, and a trusted external data source that stores an array X of n bits ( n >> k). Up to ssk of the peers might fail in any execution (for beta is an element of [0, 1)). Peers can obtain the information either by inexpensive messages passed among themselves or through expensive queries to the source array X. In the DR model, we focus on designing protocols that minimize the number of queries performed by any nonfaulty peer (a measure referred to as query complexity) while maximizing the resilience parameter beta. The Download problem requires each nonfaulty peer to correctly learn the entire array X. Earlier work on this problem focused on synchronous communication networks and established several deterministic and randomized upper and lower bounds. Our work is the first to extend the study of distributed data retrieval to asynchronous communication networks. We address the Download problem under both the Byzantine and crash failure models. We present query-optimal deterministic solutions in an asynchronous model that can tolerate any fixed fraction beta < 1 of crash faults. In the Byzantine failure model, it is known that deterministic protocols incur a query complexity of Omega(n) per peer, even under synchrony. We extend this lower bound to randomized protocols in the asynchronous model for beta >= 1/2, and further show that for beta >= 1/2, a randomized protocol exists with near-optimal query complexity. To the best of our knowledge, this is the first work to address the Download problem in asynchronous communication networks. 2012 ACM Subject Classification Theory of computation. Distributed algorithms
We consider an asynchronous network of n message-sending parties, up to t of which are byzantine. We study approximate agreement, where the parties obtain approximately equal outputs in the convex hull of their inputs. In their seminal work, Abraham, Amit and Dolev [OPODIS '04] solve this problem in R with the optimal resilience t < n/3with a protocol where each party reliably broadcasts a value in every iteration. This takes Theta(n(2)) messages per reliable broadcast, or Theta(n(3)) messages per iteration. In this work, we forgo reliable broadcast to achieve asynchronous approximate agreement against t < n/3faults with quadratic communication. In a tree with the maximum degree Delta and the centroid decomposition height h, we achieve edge agreement in at most 6h+ 1 rounds with O(n(2)) messages of size O(log Delta + log h) per round. We do this by designing a 6-round multivalued 2-graded consensus protocol and using it to recursively reduce the task to edge agreement in a subtree with a smaller centroid decomposition height. Then, we achieve edge agreement in the infinite path Z, again with the help of 2-graded consensus. Finally, we show that our edge agreement protocol enables s-agreement in R in 6 log(2) M/epsilon+O(log log M/epsilon) rounds with O(n(2) log M/epsilon) messages and O(n(2) log M/epsilon log log M/epsilon) bits of communication, where M is the maximum non-byzantine input magnitude.
We consider a stochastic network model consisting of a set of n synchronous processes communicating by message passing. In each round, processes send messages directly to each other over a complete communication graph. The processes do not fail, but messages can be lost. Each message is delivered with probability p, for a given parameter p is an element of [0, 1]. We study the following optimization version of approximate agreement in this model. We assume that processes start with binary input values, execute an algorithm for a fixed number of rounds, and decide values in [0, 1] satisfying the usual validity requirement stating that if all processes start with the same input value, then they should all decide that value. We propose deterministic algorithms that minimize the expected discrepancy, namely, the expected maximum distance between the decided values. We also present lower bounds on the expected discrepancy, which demonstrate the optimality of our algorithms for two processes. Finally, we present applications of our algorithms to solve randomized consensus and randomized approximate agreement. 2012 ACM Subject Classification Theory of computation -> Probabilistic computation; Theory of computation -> Distributed computing models; Theory of computation -> Random network models; Theory of computation -> Distributed algorithms
We give new, improved bounds for approximating the sparsest cut value or in other words the conductance phi of a graph in the CONGEST model. As our main result, we present an algorithm running in O(log(2) n/phi) rounds in which every vertex outputs a value phi satisfying phi < phi < root 2.01 phi. In most regimes, our algorithm improves significantly over the previously fastest algorithm for the problem [Chen, Meierhans, Probst Gutenberg, Saranurak; SODA 25]. Additionally, our result generalizes to k-way conductance. We obtain these results, by approximating the eigenvalues of the normalized Laplacian matrix L := I-Deg(-1/2) A Deg(-1/2), where, A is the adjacency matrix and Deg is the diagonal matrix with the weighted degrees on the diagonal. We show our algorithms are near-optimal by proving a lower bound for computing the smallest non-trivial eigenvalue of L, even in the stronger LOCAL model The previous state of the art sparsest cut algorithm is in the technical realm of expander decompositions. Our algorithms, on the other hand, are relatively simple and easy to implement. At the core, they rely on the well-known power method, which comes down to repeatedly multiplying the Laplacian with a vector. This operation can be performed in a single round in the CONGEST model. All our algorithms apply to weighted, undirected graphs. Our lower bounds apply even in unweighted graphs.
We investigate the step complexity of the Leader Election problem (and implementing the corresponding test-and-set object) in asynchronous shared memory, where processes communicate through registers supporting atomic read and write and must coordinate so that a single process becomes the leader. Determining tight step complexity bounds for solving this problem is one of the key open problems in the theory of shared memory distributed computing. The best known algorithm is a randomized tournament-tree, which has worst-case expected step complexity O(log N) for N processes. There are provably no deterministic wait-free algorithms, and only restricted lower bounds are known for obstruction-free and randomized wait-free algorithms. We introduce a new lower bound that establishes an Omega( log N / log log N+ log Q ) step complexity for any obstruction-free Leader Election algorithm, where N is the number of processes, and 2 <= Q <= N is a bound on the value contention, which we define as the maximum number of different values that processes can be simultaneously poised to write to the same register in any execution of the algorithm. Our result is strictly stronger than previous bounds based on write contention. In particular, it implies new lower bounds on step complexity that depend on register size. 2012 ACM Subject Classification Theory of computation -> Shared memory algorithms; Theory of computation -> Lower bounds and information complexity; Theory of computation -> Concurrency
The maximum independent set problem is a classic optimization problem in graph theory that has also been studied quite intensively in the distributed setting. Although the problem is hard to approximate within reasonable factors in general, there are good approximation algorithms known for several sparse graph families. In the present paper, we consider deterministic distributed CONGEST algorithms for the weighted version of the problem in trees and graphs of bounded arboricity (i.e., hereditary sparse graphs). For trees, we prove that the task of deterministically computing a (1 - epsilon) -approximate solution to the maximum weight independent set (MWIS) problem has a tight Theta(log* (n)/ epsilon) complexity. The lower bound already holds on unweighted oriented paths. On the upper bound side, we show that the bound can be achieved even in unrooted trees. For graphs G = (V, E) of arboricity beta > 1, we give two algorithms. If the sum of all node weights is w(V), we show that for any epsilon > 0, an independent set of weight at least (1 - epsilon) center dot omega(v)/4 beta can be computed in O(log(2)(beta/5)/epsilon + log* n) rounds. This result is obtained by a direct application of the local rounding framework of Faour, Ghaffari, Grunau, Kuhn, and Rozhon [SODA '23]. We further show that for any epsilon > 0, an independent set of weight at least (1 - epsilon) center dot omega(v)/2 beta+1 can be computed in O(log(3)(beta) center dot log(1/epsilon)/epsilon(2) center dot log n) rounds. For epsilon = omega(1/root beta), this significantly improves on a recent result of Gil [OPODIS '23], who showed that a 1/[(2 + epsilon)/beta]-approximation to the MWIS problem can be computed in O(beta/epsilon center dot log n) rounds. As an intermediate step to our result, we design an algorithm to compute an independent set of total weight at least (1 - epsilon) center dot Sigma(v is an element of V) omega(v)/deg(v))+1 in time O(log(3)(Delta) center dot log(1/epsilon)/s vertical bar log* n), where Delta is the maximum degree of the graph.
Scalability is a central concern of Byzantine Fault Tolerant (BFT) distributed protocols. The ubiquitous approach to work around the well-known Dolev-Reischuk Omega(n(2)) communication complexity lower bound is to use a random selection process to draw a hopefully small committee from a population of agents to run the communication-heavy protocol. We propose a notion of hierarchical consensus that combines two sub-protocols: an optimistic primary sub-protocol that can tolerate less than 1/2 failures and a fallback secondary protocol that can tolerate less than 1/3 failures; we achieve the higher failure threshold by requiring a weaker notion of liveness for the primary. This distinction between the level of fault tolerance between primary and secondary is reflected in the size of committees implementing these protocols. For a population of agents with close to 2/3 of honest agents, we need to select a committee with hundreds of agents to reach the level of tolerance expected for the primary, whereas we need thousands to reach the level expected for the secondary with a very small probability of error.. Our hierarchical construct is such that if the primary comes to a decision, it can simply propagate it to the secondary protocol, so it does not need to properly engage in an agreement protocol independently. Our architecture is flexible and allows us to use our technique for most protocols that are based on random sampling. By studying hierarchical protocols, we discovered new theoretical results of independent interest. Specifically, the ability to handover from a primary protocol requires a new Justifiability property that allows agents to pre-decide on a value, such that if the protocol decides, it must be on that pre-decided value. 2012 ACM Subject Classification Networks -> Network protocol design; Theory of computation -> Distributed algorithms
In this paper, we study the leader election problem in oriented ring networks under content-oblivious asynchronous message-passing systems, where an adversary may arbitrarily corrupt message contents. Frei et al. (DISC 2024) recently presented a uniform terminating leader election algorithm for oriented rings in this setting, with message complexity O(nID(max)) on a ring of size n, where IDmax is the largest identifier in the system. In this paper, we investigate the message complexity of leader election in this model, showing that no uniform algorithm can solve the problem if each process is limited to sending a constant number of messages in one direction. Interestingly, this limitation hinges on the uniformity assumption. In the non-uniform setting - where processes know an upper bound U >= n on the ring size - we present an algorithm with message complexity O(nUID(min)), in which each process sends O(UIDmin) messages clockwise and only three messages counter-clockwise. Here, IDmin is the smallest identifier in the system. This dependence on the identifiers compares favorably with the dependence on IDmax of Frei et al. (DISC 2024). We also show a non-uniform algorithm where each process sends O( U log IDmin) messages in one direction and O(log IDmin) in the other. The factor log IDmin is optimal, matching the lower bound of Frei et al. (DISC 2024). Finally, in the anonymous setting, we propose a randomized algorithm where each process sends only O(log(2) U) messages, with a success probability of 1 - U-c. 2012 ACM Subject Classification Theory of computation -> Distributed algorithms
We study the problem of Strong Byzantine Agreement and establish tight upper and lower bounds on communication complexity, parameterized by the actual number of Byzantine faults. Specifically, for a system of n parties tolerating up to t Byzantine faults, out of which only f = t are actually faulty, we obtain the following results: In the partially synchronous setting, we present the first Byzantine Agreement protocol that achieves adaptive communication complexity of O( n + t center dot f) words, which is asymptotically optimal. Our protocol has an optimal resilience of t < n/3. In the asynchronous setting, we prove a lower bound of Omega( n + t(2)) on the expected number of messages, and design an almost matching protocol with an optimal resilience that solves agreement with O(( n + t(2)) center dot log n) words. Our main technical contribution in the asynchronous setting is the utilization of a bipartite expander graph that allows for low-cost information dissemination. 2012 ACM Subject Classification Theory of computation. Distributed algorithms
We consider the Download problem in the Data Retrieval Model, introduced in DISC'24, where a distributed set of peers, some of which may be Byzantine, seek to learn n bits of data stored at a trustworthy external data source. Each bit of data can be learned by a peer either through a direct and costly query of the source or through other peers that have already learned it; the goal is to design a collaborative protocol that reduces the query complexity defined as the maximum number of bits queried by any honest peer. We begin with a randomized protocol for the Download problem that achieves optimal query complexity, up to a logarithmic factor. For a stronger "dynamic" adversary that can change the set of Byzantine peers from one round to the next, we achieve optimality (within log factors) for both query complexity (in expectation) and time complexity, but with larger messages. In broadcast communication, where all peers (including Byzantine peers) are required to send the same message to all peers, we achieve (up to log factors) an optimal trade-off between query complexity, time complexity, and message size with the dynamic adversary. All of our protocols can tolerate any constant fraction beta < 1 of Byzantine peers. 2012 ACM Subject Classification Theory of computation -> Distributed algorithms
We show that self-consistency can be a crucial property for autonomous mobile robots. Specifically, we consider the task of gathering three robots, placed adversarially in distinct locations in the Euclidean plane, in a single point. We assume the natural scheduler ROUNDROBIN, which activates the robots in turns. An activated robot perceives all robot locations in an adversarially scaled, rotated, and mirrored Cartesian coordinate system with itself at the origin and then moves wherever it wants. We show that this task cannot be solved in the default robot model (without any consistency guarantees and no multiplicity detection) but becomes feasible if we assume self-consistency (i.e., no changes between the different activations of the same robot) of either the unit length (i.e., no scaling) or the compass (i.e., no rotating) by providing explicit algorithms.
