The vertex coloring problem has received a lot of attention in the context of synchronous round-based systems where, at each round, a process can send a message to all its neighbors, and receive a message from each of them. Hence, this communication model is particularly suited to point-to-point communication channels. Several vertex coloring algorithms suited to these systems have been proposed. They differ mainly in the number of rounds they require and the number of colors they use. This paper considers a broadcast/receive communication model in which message collisions and message conflicts can occur (a collision occurs when, during the same round, messages are sent to the same process by too many neighbors; a conflict occurs when a process and one of its neighbors broadcast during the same round). This communication model is suited to systems where processes share communication bandwidths. More precisely,the paper considers the case where, during a round, a process may either broadcast a message to its neighbors or receive a message from at most $m$ of them. This captures communication-related constraints or a local memory constraint stating that, whatever the number of neighbors of a process, its local memory allows it to receive and store at most $m$ messages during each round. The paper defines first the corresponding generic vertex multi-coloring problem (a vertex can have several colors). It focuses then on tree networks, for which it presents a lower bound on the number of colors $K$ that are necessary (namely, $K=\lceil\fracΔ{m}\rceil+1$, where $Δ$ is the maximal degree of the communication graph), and an ssociated coloring algorithm, which is optimal with respect to $K$.
In this work we extend the recently proposed synchronous broadcast algorithm amnesiac flooding to the case of intermittent communication channels. In amnesiac flooding a node forwards a received message in the subsequent round. There are several reasons that render an immediate forward of a message impossible: Higher priority traffic, overloaded channels, etc. We show that postponing the forwarding for one or more rounds prevents termination. Our extension overcomes this shortcoming while retaining the advantages of the algorithm: Nodes don't need to memorize the reception of a message to guar…
This paper considers the good-case latency of Byzantine Reliable Broadcast (BRB), i.e., the time taken by correct processes to deliver a message when the initial sender is correct. This time plays a crucial role in the performance of practical distributed systems. Although significant strides have been made in recent years on this question, progress has mainly focused on either asynchronous or randomized algorithms. By contrast, the good-case latency of deterministic synchronous BRB under a majority of Byzantine faults has been little studied. In particular, it was not known whether a goodcase…
This paper considers synchronous discrete-time dynamical systems on graphs based on the threshold model. It is well known that after a finite number of rounds these systems either reach a fixed point or enter a 2-cycle. The problem of finding the fixed points for this type of dynamical system is in general both NP-hard and #P-complete. In this paper we give a surprisingly simple graph-theoretic characterization of fixed points and 2-cycles for the class of finite trees. Thus, the class of trees is the first nontrivial graph class for which a complete characterization of fixed points exists. Th…
We consider the broadcast problem in synchronous radio broadcast models with collision detection. One node of the network is given a message that must be learned by all nodes in the network. We provide a deterministic algorithm that works on the beeping model, which is a restricted version of the radio broadcast model with collision detection. This algorithm improves on the round complexity of previous algorithms. We prove an exactly matching lower bound in the radio broadcast model with collision detection. This shows that the extra power provided by the radio broadcast model with collision d…
This paper considers the problem of achieving exact Byzantine consensus in a synchronous system under a local-broadcast communication model. The nodes communicate with each other via message-passing. The communication network is modeled as an undirected graph, with each vertex representing a node in the system. Under the local-broadcast communication model, when any node transmits a message, all its neighbors in the communication graph receive the message reliably. This communication model is motivated by wireless networks. In this work, we present necessary and sufficient conditions on the un…
In this work we look at Byzantine consensus in asynchronous systems under the local broadcast model. In the local broadcast model, a message sent by any node is received identically by all of its neighbors in the communication network, preventing a faulty node from transmitting conflicting information to different neighbors. Our recent work has shown that in the synchronous setting, network connectivity requirements for Byzantine consensus are lower under the local broadcast model as compared to the classical point-to-point communication model. Here we show that the same is not true in the asy…
Institut de Recherche en Informatique Fondamentale (IRIF) CNRS and Université Paris Cité（基础信息学研究所（IRIF）法国国家科学研究中心与巴黎西岱大学）；Tel Aviv University（特拉维夫大学）；Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）
We study the broadcast problem on dynamic networks with $n$ processes. The processes communicate in synchronous rounds along an arbitrary rooted tree. The sequence of trees is given by an adversary whose goal is to maximize the number of rounds until at least one process reaches all other processes. Previous research has shown a $\lceil{\frac{3n-1}{2}}\rceil-2$ lower bound and an $O(n\log\log n)$ upper bound. We show the first linear upper bound for this problem, namely $\lceil{(1 + \sqrt 2) n-1}\rceil \approx 2.4n$. Our result follows from a detailed analysis of the evolution of the adjacency…
We present a wait-free algorithm for proper coloring the n nodes of the asynchronous cycle $C_n$, where each crash-prone node starts with its (unique) identifier as input. The algorithm is independent of $n \geq 3$, and runs in $\mathrm{O}(\log^* n)$ rounds in $C_n$. This round-complexity is optimal thanks to a known matching lower bound, which applies even to synchronous (failure-free) executions. The range of colors used by our algorithm, namely $\{0, ..., 4\}$, is optimal too, thanks to a known lower bound on the minimum number of names for which renaming is solvable wait-free in shared-mem…
We provide the first asynchronous distributed algorithms to compute broadcast and minimum spanning tree with $o(m)$ bits of communication, in a graph with $n$ nodes and $m$ edges. For decades, it was believed that $Ω(m)$ bits of communication are required for any algorithm that constructs a broadcast tree. In 2015, King, Kutten and Thorup showed that in the KT1 model where nodes have initial knowledge of their neighbors' identities it is possible to construct MST in $\tilde{O}(n)$ messages in the synchronous CONGEST model. In the CONGEST model messages are of size $O(\log n)$. However, no algo…
We consider Byzantine consensus in a synchronous system where nodes are connected by a network modeled as a directed graph, i.e., communication links between neighboring nodes are not necessarily bi-directional. The directed graph model is motivated by wireless networks wherein asymmetric communication links can occur. In the classical point-to-point communication model, a message sent on a communication link is private between the two nodes on the link. This allows a Byzantine faulty node to equivocate, i.e., send inconsistent information to its neighbors. This paper considers the local broad…
Memory Lower Bounds and Impossibility Results for Anonymous Dynamic Broadcast
School of Computing and Augmented Intelligence and Biodesign Center for Biocomputing, Security and Society Arizona State University（亚利桑那州立大学计算与增强智能学院及生物设计生物计算、安全与社会中心）
This paper considers the Byzantine consensus problem for nodes with binary inputs. The nodes are interconnected by a network represented as an undirected graph, and the system is assumed to be synchronous. Under the classical point-to-point communication model, it is well-known [7] that the following two conditions are both necessary and sufficient to achieve Byzantine consensus among $n$ nodes in the presence of up to $f$ Byzantine faulty nodes: $n \ge 3f+1$ and vertex connectivity at least $2f+1$. In the classical point-to-point communication model, it is possible for a faulty node to equivo…
In this paper, we study the Byzantine lattice agreement problem in synchronous systems. The lattice agreement problem in crash failure model has been studied both in synchronous and asynchronous systems, which leads to the current best upper bound of $O(\log f)$ rounds in both systems. However, very few algorithmic results are known for the lattice agreement problem in Byzantine failure model. The paper [Nowak et al., DISC, 2019] first gives an algorithm for a variant of the lattice agreement problem on cycle-free lattices that tolerates up to $f < n/(h(X) + 1)$ Byzantine faults, where $n$ is …
Leader-based protocols for consensus, i.e., atomic broadcast, allow some processes to unilaterally affect the final order of transactions. This has become a problem for blockchain networks and decentralized finance because it facilitates front-running and other attacks. To address this, order fairness for payload messages has been introduced recently as a new safety property for atomic broadcast complementing traditional agreement and liveness. We relate order fairness to the standard validity notions for consensus protocols and highlight some limitations with the existing formalization. Based…
Consider a single-hop, multi-channel, synchronous radio network in which a source node needs to disseminate a message to all other $n-1$ nodes. An adversary called Eve, which captures environmental noise and potentially malicious interference, aims to disrupt this process via jamming. Assume sending, listening, or jamming on one channel for one time slot costs unit energy. The question is, if Eve spends $T$ units of energy on jamming, can we devise broadcast algorithms in which each node's cost is $o(T)$? Previous results show such resource competitive algorithms do exist in the single-channel…
Distributed vertex coloring is one of the classic problems and probably also the most widely studied problems in the area of distributed graph algorithms. We present a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an $n$-node graph $G$, and where the nodes of $G$ operate in synchronous communication rounds in which they can exchange $O(\log n)$-bit messages over all the edges of $G$. For graphs with maximum degree $Δ$, we show that the $(Δ+1)$-list coloring problem (and therefore also the standard $(Δ+1)$-coloring problem) …
During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of …
It is commonly held that asynchronous consensus is much more complex, difficult, and costly than partially-synchronous algorithms, especially without using common coins. This paper challenges that conventional wisdom with que sera consensus QSC, an approach to consensus that cleanly decomposes the agreement problem from that of network asynchrony. QSC uses only private coins and reaches consensus in $O(1)$ expected communication rounds. It relies on "lock-step" synchronous broadcast, but can run atop a threshold logical clock (TLC) algorithm to time and pace partially-reliable communication at…
The LOCAL model is among the main models for studying locality in the framework of distributed network computing. This model is however subject to pertinent criticisms, including the facts that all nodes wake up simultaneously, perform in lock steps, and are failure-free. We show that relaxing these hypotheses to some extent does not hurt local computing. In particular, we show that, for any construction task $T$ associated to a locally checkable labeling (LCL), if $T$ is solvable in $t$ rounds in the LOCAL model, then $T$ remains solvable in $O(t)$ rounds in the asynchronous LOCAL model. This…
Data dissemination is a fundamental task in distributed computing. This paper studies broadcast problems in various innovative models where the communication network connecting $n$ processes is dynamic (e.g., due to mobility or failures) and controlled by an adversary. In the first model, the processes transitively communicate their ids in synchronous rounds along a rooted tree given in each round by the adversary whose goal is to maximize the number of rounds until at least one id is known by all processes. Previous research has shown a $\lceil{\frac{3n-1}{2}}\rceil-2$ lower bound and an $O(n…
Consensus is a most fundamental task in distributed computing. This paper studies the consensus problem for a set of processes connected by a dynamic directed network, in which computation and communication is lock-step synchronous but controlled by an oblivious message adversary. In this basic model, determining consensus solvability and designing consensus algorithms in the case where it is possible, has been shown to be surprisingly difficult. We present an explicit decision procedure to determine if consensus is possible under a given adversary. This in turn enables us, for the first time,…
Flooding is among the simplest and most fundamental of all distributed network algorithms. A node begins the process by sending a message to all its neighbours and the neighbours, in the next round forward the message to all the neighbours they did not receive the message from and so on. We assume that the nodes do not keep a record of the flooding event. We call this amnesiac flooding (AF). Since the node forgets, if the message is received again in subsequent rounds, it will be forwarded again raising the possibility that the message may be circulated infinitely even on a finite graph. As fa…
The present paper studies local distributed graph problems in highly dynamic networks. Communication and changes of the graph happen in synchronous rounds and our algorithms always, i.e., in every round, satisfy non-trivial guarantees, no matter how dynamic the network is. We define a (in our view) natural generalization of static graph problems to the dynamic graph setting. Throughout the execution of an algorithm we consider a sliding window over the last $T$, e.g., polylogarithmic, rounds. Then, in some round, the feasibility of an output only depends on the topology of the graphs in the cu…
In this paper, we study distributed graph algorithms in networks in which the nodes have a limited communication capacity. Many distributed systems are built on top of an underlying networking infrastructure, for example by using a virtual communication topology known as an overlay network. Although this underlying network might allow each node to directly communicate with a large number of other nodes, the amount of communication that a node can perform in a fixed amount of time is typically much more limited. We introduce the Node-Capacitated Clique model as an abstract communication model, …
Construction and impromptu repair of an MST in a distributed network with o(m) communication
在CONGEST模型中，通信网络是无向图，其$n$节点是处理器，其$m$边是处理器之间的通信链路。在任何给定的时间步，大小为$O(\log n)$的消息可以由每个节点发送到其每个邻居。我们展示了同步模型：如果所有节点在同一轮中开始，并且每个节点知道其ID及其邻居的ID，或者在MST的情况下，知道其入射边的不同权重并且知道$n$，然后有蒙特卡罗算法成功w. h. p.确定最小生成森林（MST）和生成森林（ST）分别使用MST的$O(n \log^2 n/\log\log n)$消息和ST的$O(n \log n )$消息。这些结果与Awerbuch等人的“民间定理”相矛盾，JACM 1990指出，广播树的分布式构造需要$Ω(m)$消息。该下限已在那里和其他论文中针对一些CONGEST模型进行了展示;我们的协议证明了这些模型的局限性。动态分布式网络是一个经历在线边插入或删除。我们还展示了如何修复一个MST或ST在一个动态的网络与异步通信。边删除可以在MST中的$O(n\log n /\log \log n)$预期消息和ST问题的$O(n)$预期消息中处理，而边插入在最坏情况下使用$O(n)$消息。我们称之为“即兴”更新，因为我们假设在边缘更新的处理之间没有额外信息的预处理或存储。以前针对这个问题的算法在每次更新时使用分摊的$o(m)$消息，需要大量的预处理和更新之间的附加本地存储。
Basic synchronous flooding proceeds in rounds. Given a finite undirected (network) graph $G$, a set of sources $I \subseteq G$ initiate flooding in the first round by every node in $I$ sending the same message to all of its neighbours. In each subsequent round, nodes send the message to all of their neighbours from which they did not receive the message in the previous round. Flooding terminates when no node in $G$ sends a message in a round. The question of termination has not been settled - rather, non-termination is implicitly assumed to be possible. We show that flooding terminates on ever…
