Discovering elements of a hidden set, also known as Group Testing (GT), is a well-established area in which one party tries to discover elements hidden by the other party by asking queries and analyzing feedback. The feedback is a function of the intersection of the query with the hidden set - in our case, it is a classical double-threshold function, which returns i if the intersection is a singleton i and "null" otherwise (i.e., when the intersection is empty or of size at least 2). In this work, we enhance GT by two features. First, we introduce a local feedback framework to this problem: each hidden element is an "autonomous" element and can analyze feedback itself, but only for the queries to which it belongs. The goal is to design a deterministic non-adaptive sequence of queries that enables each non-hidden element to learn about all other hidden elements. We show that, surprisingly, this task requires substantially more queries than the classic group testing -- by proving a super-cubic (in terms of the number of hidden elements) lower bound and by constructing a specific query sequence of slightly longer length. Such a query system is also an extension of a well-known superimposed code, in a way that the decoding can be done only by the owners of the codewords. Second, we extend the results to the model where elements may belong to certain clusters and retrieving them could be done only via queries avoiding elements from "interfering" clusters. The main challenge is in not knowing which interfering clusters are non-empty (and thus, need to be avoided) and how to speed up the retrieval process by asking queries across many clusters. Our algorithms can be generalized to other feedback functions, to adversarial/stochastic fault-prone scenarios, implemented in a distributed setting and applied to the information theory and codes.

We study the problem of distributed construction of efficient de-centralized communication schedules for ad-hoc wireless networks. We consider a model which is close to real scenarios: (1) the SINR interference model, which covers most important distinctive features of contemporary wireless communication, such as signal fading, collisions and accumulation of signal, and (2) any underlying metric of bounded growth, which includes Euclidean space with certain type of obstacles. Most of efficient solutions in the SINR model rely on probabilistic algorithms, assuming access of nodes to the sources of independent truly random bits, which in practice is hard to get by wireless devices.In this paper we show that key efficient de-centralized communication abstraction primitives, such as aggregation and broadcast schedules (which are bases of many other communication tasks), can be efficiently built by distributed algorithms in SINR networks without using any randomization. In particular, the length of the built communication schedules are asymptotically as efficient as those significantly relying on randomization, and close to the theoretic lower bounds. Importantly, the time (round) complexity of our solutions grows only logarithmically with respect to the growth of the density of a network, which makes them scalable in real-life scenarios.

Radio Networks (RN) is one of the fundamental models for network communication where nodes can broadcast messages locally but their simultaneous transmissions can interfere with each other at their shared neighbors. This work focuses on performing the very fundamental primitive of Local Broadcast, in spite of the interferences. We investigate to what extent local knowledge, called advice, relating to the 2-local domination number γ₂ may speed up Local Broadcast. Specifically for each node and some dominating set, knowledge about some neighboring dominating node and the local number among the neighbors of that dominating node. We show that such advice is sufficient to build an efficient oblivious transmission schedule. Along those lines, we present three algorithms trading the level of adaptiveness (from oblivious to adaptive) for bits of advice per node (from O(log (Δγ₂)) to 1). All our algorithms complete Local Broadcast in Õ(Δγ₂²) rounds, where Δ is the maximum degree of the network. On the side of lower bounds, we show that, for each quasi-adaptive deterministic Local Broadcast algorithm, there is some RN that requires Ω(min{(min{Δ,γ₂}/log n)²,n}) communication rounds, where n is the number of network nodes. In quasi-adaptive protocols nodes may stop executing once its computational task is completed. To the best of our knowledge, this is the first (nearly) quadratic Local Broadcast (same message for all neighbors) lower bound in the RN model. Our lower bound is stronger than previous works in multiple ways: i) it is nearly quadratically better than the best known general lower bound for this class of algorithms, ii) it applies to a wider class of algorithms than previous work for fully oblivious, iii) it achieves similar time lower bound than previous work proved for a much more demanding Local Broadcast where each node sends a possibly different message to each neighbor, and iv) it takes into account the local domination parameter γ₂.

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+log² 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(Dlog n + log² 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 + Δlog n + log² n), using a labeling scheme of optimal length O(log Δ), where Δ is the maximum degree. Our time is the same as the best known gossiping time in radio networks of known topology.

Stability is an important issue in order to characterize the performance of a network, and it has become a major topic of study in the last decade. Roughly speaking, a communication network system is said to be stable if the number of packets waiting to be delivered (backlog) is finitely bounded at any one time. In this paper we introduce a number of routing scheduling algorithms which, making use of certain knowledge about the network's structure, guarantee stability for certain injection rates. First, we introduce two new families of combinatorial structures, which we call universally strong selectors and generalized universally strong selectors, that are used to provide a set of transmission schedules. Making use of these structures, we propose two local-knowledge packet-oblivious routing scheduling algorithms. The first proposed routing scheduling algorithm only needs to know some upper bounds on the number of links and on the network's degree, and is asymptotically optimal regarding the injection rate for which stability is guaranteed. The second proposed routing scheduling algorithm is close to be asymptotically optimal, but it only needs to know an upper bound on the number of links. For such algorithms, we also provide some results regarding both the maximum latencies and queue lengths. Furthermore, we also evaluate how the lack of global knowledge about the system topology affects the performance of the routing scheduling algorithms.

We consider the fundamental problems of size discovery and topology recognition in radio networks modeled by simple undirected connected graphs. Size discovery calls for all nodes to output the number of nodes in the graph, called its size, and in the task of topology recognition each node has to learn the topology of the graph and its position in it. We do not assume collision detection: in case of a collision, node v does not hear anything (except the background noise that it also hears when no neighbor transmits). The time of a deterministic algorithm for each of the above problems is the worst-case number of rounds it takes to solve it. Nodes have labels which are (not necessarily different) binary strings. Each node knows its own label and can use it when executing the algorithm. The length of a labeling scheme is the largest length of a label. For size discovery, we construct a labeling scheme of length O(log logΔ) (which is known to be optimal, even if collision detection is available) and we design an algorithm for this problem using this scheme and working in time O(log² n), where n is the size of the graph. We also show that time complexity O(log² n) is optimal for the problem of size discovery, whenever the labeling scheme is of optimal length O(log logΔ). For topology recognition, we construct a labeling scheme of length O(logΔ), and we design an algorithm for this problem using this scheme and working in time O (DΔ+min(Δ²,n)), where D is the diameter of the graph. We also show that the length of our labeling scheme is asymptotically optimal.

Shah, Shin and Tetali [25] initiated the study of queuing systems on the medium access channel with arbitrary interference graph. The graph models dependencies between the channel members -- any two connected by an edge are dependent. This problem could be also re-stated as a query system with dependent elements or local scheduling of dependent packets/jobs. In short, if two dependent units, also called stations, want to transmit (or be in the same query, or do jobs in parallel), there is a conflict and none of them succeeds. The problem is particularly challenging, if the units need to make their decisions locally, without any coordination. Prior the paper by Shah, Shin and Tetali [FOCS 2011], the main focus was on the clique graph -- even in this simple topology, many problems related to stability remain open. While the solution by Shah, Shin and Tetali [FOCS 2011] is semi-local, as nodes make use of information about interfering neighbors in the graph, we provide the first purely local stable algorithms. In particular, in our algorithms stations make their decision whether to transmit or not based only on looking at their local queues. Based only on this feature, and without using a priori knowledge of topology, we design an algorithm that allows all stations for implicit transferring information to and from all other stations. We use it as a tool for developing universally stable algorithms for the problem of queuing messages -- i.e., guaranteeing bounded queues when, on average, less than one independent set is injected per round. The first one is stable in adversarial (worst-case) sense, the second one -- in stochastic sense (average-case). We also prove optimality of our algorithm in adversarial sense: no algorithm can be stable against injections with rate ρ=1 (when, on average, one independent set is injected per round). Moreover, for the class of non-adaptive protocols, we show that it is not possible to achieve stability for any injection rate Ω(1/log m), where m is the size of the largest clique in the interference graph. We extend the result by showing that such protocols are not stable on some interference graphs with no cliques of size at least 3 even for injection rates Ω(√[3](log n)/n). This result separates the model with general interference graphs from the classical model of a multiple access channel (in which the interference graph is a clique), in which there exist non-adaptive algorithms stable for adversarial injection rates O(1/log2 n).

Millimeter wave communication (mmWave) allows high-speed access to the radio channel. Given the highly-directional nature of mmWave, dense deployments can be implemented with a macro base station serving many micro base stations, rather than connecting micro base stations directly to the core network as in legacy cellular systems. Moreover, micro base stations may cooperate in relaying packets to other micro base stations. Relays and spatial reuse speed up communication, but increase the complexity of scheduling. In this work, we study the mmWave wireless backhaul scheduling problem in the described architecture, assuming stochastic arrival of packets at the macro base station to be delivered to micro base stations. We present various results concerning system stability, defined as a bounded expected queue sizes of macro base station and micro base stations, under different patterns of random traffic. In particular, that almost all admissible arrival patterns could be handled by some universally stable algorithms, while non-admissible arrival patterns do not allow stability for any algorithm.

We investigate routing on networks modeled as multiple access channels, when packets are injected continually. An energy cap is a component of the system, understood as a bound on the number of stations that can be switched on simultaneously. Each packet is injected into some station and needs to be delivered to its destination station via the channel. A station has to be switched on in order to receive a packet when it is heard on the channel. Each station manages when it is switched on and off by way of a programmable wake-up mechanism, which is scheduled by a routing algorithm. Packet injection is governed by adversarial models that determine upper bounds on injection rates and burstiness. We develop deterministic distributed routing algorithms and assess their performance in the worst-case sense. An algorithm knows the number of stations but does not know the adversary. One of the algorithms maintains bounded queues for the maximum injection rate 1 subject only to the energy cap 3. This energy cap is provably optimal, in that obtaining the same throughput with the energy cap 2 is impossible. We give algorithms subject to the minimum energy cap 2 that have latency polynomial in the total number of stations~n for each fixed adversary of injection rate less than 1. An algorithm is k-energy-oblivious if at most k stations are switched on in a round and for each station the rounds when it will be switched on are determined in advance. We give a k-energy-oblivious algorithm that has packet delay O(n) for adversaries of injection rates less than (k-1)/(n-1), and show that there is no k-energy-oblivious stable algorithm against adversaries with injection rates greater than k/n. An algorithm routes directly when each packet makes only one hop from the station into which it is injected straight to its destination. We give a k-energy-oblivious algorithm routing directly, which has latency O(n^2/k) for adversaries of sufficiently small injection rates that are O(k^2/n^2). We develop a k-energy-oblivious algorithm routing directly, which is stable for injection rate k(k-1)/n(n-1), and show that no k-energy-oblivious algorithm routing directly can be stable against adversaries with injection rates greater than k(k-1)/n(n-1).

In this work we study stability of local memoryless packet scheduling policies in a distributed system of n nodes/queues under contention. The local policies at nodes may only access their current local queues, and have no other feedback from the underlying distributed system. Moreover, their memory is limited to some basic parameters. The packets arrive at queues according to arrival patterns controlled by an adversary restricted only by injection rate rho and burstiness b, or driven by a stochastic process; the former model analyzes worst-case stability while the latter - average case. We assume that the underlying distributed system is a classic shared channel, in which no two packets could be successfully scheduled (and removed from queues) at the same time. We show that there is a local memoryless scheduling policy which is both adversarially and stochastically stable for injection rates Omega(1/log n). Another algorithm achieves even higher - constant - stable injection rate, but only for a bounded range of burstiness. The first algorithm is utilizing properties of interleaved ultra-selectors, for which we prove stronger properties than known so far, while the second one is based on entirely new concept of selector with thresholds, unlike previously considered binary selectors/codes in the literature. Note that popular Backoff algorithms, some of which achieve stability for constant (stochastic) injection rates [Johan Håstad et al., 1996], use memory to record current state (e.g., the number of unsuccessful transmissions or the result of random sampling in each window) as well as randomization and feedback from the channel; unlike solutions in this work, which are memoryless and do not rely on randomization or channel feedback (thus, could be used independently from the link layer protocols). {}

We consider deterministic distributed communication in wireless ad hoc networks of identical devices in the SINR model without predefined infrastructure. Most algorithmic results in this model rely on additional features or capabilities, e.g., randomization, access to geographic coordinates, power control, carrier sensing with various precision of measurements, and interference cancellation. We study a pure scenario, when no such features are available.

We present a distributed randomized algorithm finding Minimum Spanning Tree (MST) of a given graph in O(1) rounds, with high probability, in the congested clique model. The input graph in the congested clique model is a graph of n nodes, where each node initially knows only its incident edges. The communication graph is a clique with limited edge bandwidth: each two nodes (not necessarily neighbours in the input graph) can exchange O(log n) bits. As in previous works, the key part of the MST algorithm is an efficient Connected Components (CC) algorithm. However, unlike the former approaches, we do not aim at simulating the standard Boruvka's algorithm, at least at initial stages of the CC algorithm. Instead, we develop a new technique which combines connected components of sample sparse subgraphs of the input graph in order to accelerate the process of uncovering connected components of the original input graph. More specifically, we develop a sparsification technique which reduces an initial CC problem in O(1) rounds to its two restricted instances. The former instance has a graph with maximal degree O(log log n) as the input – here our sample-combining technique helps. In the latter instance, a partition of the input graph into O(n/ log log n) connected components is known. This gives an opportunity to apply previous algorithms to determine connected components in O(1) rounds. Our result addresses a problem proposed by Lotker et al. [SPAA 2003; SICOMP 2005] and improves over previous O(log* n) algorithm of Ghaffari et al. [PODC 2016], and O(log log log n) algorithm of Hegeman et al. [PODC 2015]. It also determines Θ(1) round complexity in the congested clique for MST, as well as other graph problems, including bipartiteness, cut verification, s-t connectivity, and cycle containment.

We study stability of local packet scheduling policies in a distributed system of n nodes. The local policies at nodes may only access their local queues, and have no other feedback from the underlying distributed system. The packets arrive at queues according to arrival patterns controlled by an adversary restricted only by injection rate rho and burstiness b. In this work, we assume that the underlying distributed system is a shared channel, in which in order to get rid of a packet from the queue, a node needs to schedule it for transmission on the channel and no other packet is scheduled for transmission at the same time. We show that there is a local adaptive scheduling policy with relatively small memory, which is universally stable on a shared channel, that is, it has bounded queues for any rho<1 and b >= 0. On the other hand, without memory the maximal stable injection rate is O(1/log n). We show a local memoryless (non-adaptive) scheduling policy based on novel idea of ultra strong selectors which is stable for slightly smaller injection c/log^2 n, for some constant c>0.

We consider the problems of online and stochastic packet queueing in a distributed system of n nodes with queues, where the communication between the nodes is done via a multiple access channel. In the online setting, in each round, an arbitrary number of packets can be injected to nodes’ queues. Two measures of performance are considered: the total number of packets in all queues, called the total load , and the maximum queue size, called the maximum load . We develop a deterministic distributed algorithm that is asymptotically optimal with respect to both complexity measures, in a competitive way. More precisely, the total load of our algorithm is bigger than the total load of any other algorithm, including centralized online solutions, by only an additive term of O ( n 2 ), whereas the maximum queue size of our algorithm is at most n times bigger than the maximum queue size of any other algorithm, with an extra additive O ( n ). The optimality for both measures is justified by proving the corresponding lower bounds, which also separates nearly exponentially distributed solutions from the centralized ones. Next, we show that our algorithm is also stochastically stable for any expected injection rate smaller or equal to 1. This is the first solution to the stochastic queueing problem on a multiple access channel that achieves such stability for the (highest possible) rate equal to 1.

We consider the problem of scheduling packets of different lengths via k directed parallel communication links. The links are prone to simultaneous errors --- if an error occurs, all links are affected. Dynamic packet arrivals and errors are modelled by a worst-case adversary. The goal is to optimize competitive throughput of online scheduling algorithms. Two types of failures are considered: jamming, when currently scheduled packets are simply not delivered, and crashes, when additionally the channel scheduler crashes losing its current state. For the former, milder type of failures, we prove an upper bound on competitive throughput of 3/4 - 1/(4k) for odd values of k, and 3/4 - 1/(4k+4) for even values of k. On constructive side, we design an online algorithm that, for packets of two different lengths, matches the upper bound on competitive throughput. To compare, scheduling on independent channels, that is, when adversary could cause errors on each channel independently, reaches throughput of 1/2. This shows that scheduling under simultaneous jamming is provably more efficient than scheduling under channel-independent jamming. In the setting with crash failures we prove a general upper bound for competitive throughput of (√5-1)/2 and design an algorithm achieving it for packets of two different lengths. This result has two interesting implications. First, simultaneous crashes are significantly stronger than simultaneous jamming. Second, due to the above mentioned upper bound of 1/2 on throughput under channel-independenterrors, scheduling under simultaneous crashes is significantly stronger than channel-independent crashes, similarly as in the case of jamming errors.

Recently, very fast deterministic and randomized algorithms have been obtained for connectivity and minimum spanning tree in the unicast congested clique. In contrast, no solution faster than a simple parallel implementation of the Boruvka's algorithm has been known for both problems in the broadcast congested clique. In this announcement, we present the first sub-logarithmic deterministic algorithm for connected components in the broadcast congested clique.

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 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 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.

We consider the task of assigning time slots on a user-dependent and time-varying wireless channel. This scheduling problem occurs in cellular networks due to the presence of channel fading and user mobility. We introduce a simple notion of global fairness, where each of n users is guaranteed a 1/(n + ε) fraction of its total possible throughput, for some approximation parameter ε ≥ 0, and study its limitations from theoretical and experimental perspectives. We formally prove that a slight modification of the standard proportional fair algorithm satisfies the global fairness constraint. To the best of our knowledge, this is the first formal analysis providing global fairness property to the channel in any execution and any channel conditions. As confirmed by our simulations, our global fairness constraint is in fact satisfied by a wide class of algorithms. Our framework allows optimization of an arbitrary metric subject to the global fairness constraint. In particular, we have analyzed a variant of the provably fair algorithm that optimizes the total throughput. It turned out that the channel utilization of this algorithm is significantly better than that of the classical Proportional Fair algorithm.

This paper studies the task of setting up ad hoc wireless networks. In such networks, it is often the case that nodes become active at different times, without coordination or knowledge about network topology. We consider the following tasks: wake-up, clock synchronization, leader election, and multimessage broadcast. We show how to achieve these goals in scalable O(D polylog(n)) time. As a tool we define and give a solution to a quasi-backbone problem, which aims to set up transmission probabilities at nodes in a way that they can be efficiently used to solve other tasks. Our results are obtained by minimalistic algorithms, which do not require power control or carrier sensing capabilities, use very small energy, local computation and memory. Moreover, unlike many previous work, they remain scalable even if the network is not highly connected.

Multiple access channel is a communication model in which many users, also called stations, could exchange information. Since it offers limited capacity, some information sent through it might be lost due to signal interference (collision). Therefore, successful message delivery to a station requires breaking symmetry on the channel. In this work we consider the channel-synchronization problem on non-synchronized channels: assuming stations with messages wake-up dynamically on the channel, what is the minimum (expected) time needed for all stations to receive at least one message each.

In this work we address the question how important is the knowledge of geometric location and network density to the efficiency of (distributed) wireless communication in ad hoc networks. We study fundamental communication task of broadcast and develop well-scalable, randomized algorithms that do not rely on GPS information, and which efficiency formulas do not depend on how dense the geometric network is. We consider two settings: with and without spontaneous wake-up of nodes. In the former setting, in which all nodes start the protocol at the same time, our algorithm accomplishes broadcast in O(D log n + log2 n) rounds under the SINR model, with high probability (whp), where D is the diameter of the communication graph and n is the number of stations. In the latter setting, in which only the source node containing the original message is active in the beginning, we develop a slightly slower algorithm working in O(D log2 n) rounds whp. Both algorithms are based on a novel distributed coloring method, which is of independent interest and potential applicability to other communication tasks under the SINR wireless model.

Growing context-sensitive grammars were introduced in 1986 as a restricted variant of context-sensitive grammars, where all productions are length increasing. Several interesting properties of these grammars have been shown since then, including polynomial time complexity of the membership problem and machine model characterizations. Various characterizations of the model, efficient recognition algorithm and the properties of its deterministic variant (possessing a characterization by string-rewriting systems) justify the practical value. Moreover, as pointed out by McNaughton in 1999, growing context-sensitive grammars complement the Chomsky hierarchy in a very natural way. This article reviews results on this topic and proposes some open problems.

Abstract Quite often algorithms designed for no‐collision‐detection radio networks use a hidden form of collision detection: it is assumed that a station can simultaneously send and listen. If it cannot hear its own message, apparently the message has been scrambled by another station sending at the same time. Industrial standard IEEE 802.11 says that a station can either send or listen to a radio channel at a given time, but not both. In order to relate the industrial standard and theoretical algorithms we consider a weak radio network model with no collision detection in which a station cannot simultaneously send and receive signals. Otherwise we talk about a strong model. In this paper we consider a measure called energy cost (or ‘power consumption’) which is equal to the maximum over all stations of the number of steps in which the station is sending or listening. We show that computational power of weak and strong single‐hop radio networks differ substantially in the deterministic case: deterministic leader election requires $\Omega(\log n)$ energy cost in the weak model and can be solved by a practical algorithm with $O(\sqrt{\log n})$ energy cost in the strong model. By contrast, we present a very efficient randomized simulation of strong radio networks by weak ones, with preprocessing that requires $O(n)$ steps and has energy cost $O(\log \log n)$ . Copyright © 2003 John Wiley & Sons, Ltd.

We present energy efficient algorithms for leader election in single channel single-hop radio networks with no collision detection. We present a deterministic solution with sublogarithmic energy cost (the best previous result was O(logn)) and show a double logarithmic lower bound. We prove that this lower bound holds in a randomized case, in a certain sense. For the case, when the number n of active stations can be approximated in advance, we show a randomized algorithm with energy consumption O(log ∗ n) that yields a result with high probability (the best previous result was O(loglogn)). 1.

We consider systems consisting of a constant number of finite automata communicating via messages. We assume that the automata are asynchronous, but the answers given by the system must be always correct. We examine computational power of such systems by inspecting the number of messages exchanged. This is motivated by the fact that communication volume is one of the most important complexity measures. We show that any asynchronous system of finite automata that exchanges o(n) messages is able to recognize regular languages only. This is much different than in the case of synchronous systems considered before (where already a constant number of messages suffices to recognize some non-regular languages). We show that asynchronous and synchronous systems may differ significantly in their computational power also for tasks requiring ( n) messages. We consider a language Ltrans consisting of words of the form A#A T , where A T denotes transposition of matrix A and the matrices are written row by row. While it is easy to see thatLtrans can be recognized withO(n) messages by a synchronous system of finite automata, we show thatLtrans requires ( n 3=2 = log 2 n) messages on any asynchronous system.