Peter Robinson

Fast byzantine agreement in dynamic networks

We study Byzantine agreement in dynamic networks where topology can change from round to round and nodes can also experience heavy churn (i.e., nodes can join and leave the network continuously over time). Our main contributions are randomized distributed algorithms that achieve almost-everywhere Byzantine agreement with high probability even under a large number of adaptively chosen Byzantine nodes and continuous adversarial churn in a number of rounds that is polylogarithmic in n (where n is the stable network size). We show that our algorithms are essentially optimal (up to polylogarithmic factors) with respect to the amount of Byzantine nodes and churn rate that they can tolerate by showing a lower bound. In particular, we present the following results: 1. An O(log3 n) round randomized algorithmto achieve almost everywhere Byzantine agreement with high probability under a presence of up to O(√n/polylog(n)) Byzantine nodes and up to a churn of O(√n/polylog(n)) nodes per round. We assume that the Byzantine nodes have knowledge about the entire state of network at every round (including random choices made by all the nodes) and can behave arbitrarily. We also assume that an adversary controls the churn - it has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power (but is oblivious to the topology changes from round to round). Our algorithm requires only polylogarithmic in n bits to be processed and sent (per round) by each node. 2. We also present an O(log3 n) round randomized algorithm that has same guarantees as the above algorithm, but works even when the connectivity of the network is controlled by an adaptive adversary (that can choose the topology based on the current states of the nodes). However, this algorithm requires up to polynomial in n bits to be processed and sent (per round) by each node. 3. We show that the above bounds are essentially the best possible, if one wants fast (i.e., polylogarithmic run time) algorithms, by showing that any (randomized) algorithm to achieve agreement in a dynamic network controlled by an adversary that can churn up to Θ(√n log n) nodes per round should take at least a polynomial number of rounds. Our algorithms are the first-known, fully distributed, Byzantine agreement algorithms in highly dynamic networks. We view our results as a step towards understanding the possibilities and limitations of highly dynamic networks that are subject to malicious behavior by a large number of nodes.

Agreement in directed dynamic networks

We study the fundamental problem of achieving consensus in a synchronous dynamic network, where an omniscient adversary controls the unidirectional communication links. Its behavior is modeled as a sequence of directed graphs representing the active (i.e. timely) communication links per round. We prove that consensus is impossible under some natural weak connectivity assumptions, and introduce vertex-stable root components as a--practical and not overly strong--means for circumventing this impossibility. Essentially, we assume that there is a short period of time during which an arbitrary part of the network remains strongly connected, while its interconnect topology keeps changing continuously. We present a consensus algorithm that works under this assumption, and prove its correctness. Our algorithm maintains a local estimate of the communication graphs, and applies techniques for detecting stable network properties and univalent system configurations. Our possibility results are complemented by several impossibility results and lower bounds, which reveal that our algorithm is asymptotically optimal.

Optimal regional consecutive leader election in mobile ad-hoc networks

The regional consecutive leader election (RCLE) problem requires mobile nodes to elect a leader within bounded time upon entering a specific region. We prove that any algorithm requires Ω(Dn) rounds for leader election, where D is the diameter of the network and n is the total number of nodes. We then present a fault-tolerant distributed algorithm that solves the RCLE problem and works even in settings where nodes do not have access to synchronized clocks. Since nodes set their leader variable within O(Dn) rounds, our algorithm is asymptotically optimal with respect to time complexity. Due to its low message bit complexity, we believe that our algorithm is of practical interest for mobile wireless ad-hoc networks. Finally, we present a novel and intuitive constraint on mobility that guarantees a bounded communication diameter among nodes within the region of interest.

Sublinear Bounds for Randomized Leader Election

This paper concerns randomized leader election in synchronous distributed networks. A distributed leader election algorithm is presented for complete n-node networks that runs in O(1) rounds and (with high probability) takes only \(O(\sqrt{n}\log^{3/2} n)\) messages to elect a unique leader (with high probability). This algorithm is then extended to solve leader election on any connected non-bipartite n-node graph G in O(τ(G)) time and \(O(\tau(G)\sqrt{n}\log^{3/2} n)\) messages, where τ(G) is the mixing time of a random walk on G. The above result implies highly efficient (sublinear running time and messages) leader election algorithms for networks with small mixing times, such as expanders and hypercubes. In contrast, previous leader election algorithms had at least linear message complexity even in complete graphs. Moreover, super-linear message lower bounds are known for time-efficient deterministic leader election algorithms. Finally, an almost-tight lower bound is presented for randomized leader election, showing that \(\Omega(\sqrt n)\) messages are needed for any O(1) time leader election algorithm which succeeds with high probability. It is also shown that Ω(n 1/3) messages are needed by any leader election algorithm that succeeds with high probability, regardless of the number of the rounds. We view our results as a step towards understanding the randomized complexity of leader election in distributed networks.

Storage and search in dynamic peer-to-peer networks

We study robust and efficient distributed algorithms for searching, storing, and maintaining data in dynamic Peer-to-Peer (P2P) networks. P2P networks are highly dynamic networks that experience heavy node churn (i.e., nodes join and leave the network continuously over time). Our goal is to guarantee, despite high node churn rate, that a large number of nodes in the network can store, retrieve, and maintain a large number of data items. Our main contributions are fast randomized distributed algorithms that guarantee the above with high probability even under high adversarial churn. In particular, we present the following main results: 1. A randomized distributed search algorithm that with high probability guarantees that searches from as many as n - o(n) nodes (n is the stable network size) succeed in O(log n )-rounds despite O(n/log1+δn) churn, for any small constant δ > 0, per round. We assume that the churn is controlled by an oblivious adversary (that has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power, but is oblivious to the random choices made by the algorithm). 2. A storage and maintenance algorithm that guarantees, with high probability, data items can be efficiently stored (with only θ(log n) copies of each data item) and maintained in a dynamic P2P network with churn rate up to O(n/log1+δn) per round. Our search algorithm together with our storage and maintenance algorithm guarantees that as many as n - o(n) nodes can efficiently store, maintain, and search even under O(n/log1+δn) churn per round. Our algorithms require only polylogarithmic in n bits to be processed and sent (per round) by each node. To the best of our knowledge, our algorithms are the first-known, fully-distributed storage and search algorithms that provably work under highly dynamic settings (i.e., high churn rates per step). Furthermore, they are localized (i.e., do not require any global topological knowledge) and scalable. A technical contribution of this paper, which may be of independent interest, is showing how random walks can be provably used to derive scalable distributed algorithms in dynamic networks with adversarial node churn.

On the Complexity of Universal Leader Election

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 article 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 nontrivial 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 article 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 nodes 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 our lower bound technique gives rise to an Ω(m) message lower bound for randomized broadcast algorithms. An interesting fundamental problem is whether both upper bounds (messages and time) can be reached simultaneously in the randomized setting for all graphs. The answer is known to be negative in the deterministic setting. We answer this problem partially by presenting a randomized algorithm that matches both complexities in some cases. This already separates (for some cases) randomized algorithms from deterministic ones. As first steps towards the general case, we present several universal leader election algorithms with bounds that tradeoff messages versus time. We view our results as a step towards understanding the complexity of universal leader election in distributed networks.

DEX: Self-Healing Expanders

We present a fully-distributed self-healing algorithm DEX, that maintains a constant degree expander network in a dynamic setting. To the best of our knowledge, our algorithm provides the first efficient distributed construction of expanders - whose expansion properties hold deterministically - that works even under an all-powerful adaptive adversary that controls the dynamic changes to the network (the adversary has unlimited computational power and knowledge of the entire network state, can decide which nodes join and leave and at what time, and knows the past random choices made by the algorithm). Previous distributed expander constructions typically provide only probabilistic guarantees on the network expansion which rapidly degrade in a dynamic setting, in particular, the expansion properties can degrade even more rapidly under adversarial insertions and deletions. Our algorithm provides efficient maintenance and incurs a low overhead per insertion/deletion by an adaptive adversary: only

Gracefully Degrading Consensus and k-Set Agreement in Directed Dynamic Networks

O(log n)

Regional consecutive leader election in mobile ad-hoc networks

rounds and

A time- and message-optimal distributed algorithm for minimum spanning trees

O(log n)