Asynchrony from Synchrony

While the very first consensus protocols for the synchronous model were designed to match the worst-case lower bound, deciding in exactly t + 1 rounds in all runs, it was soon realized that they could be strictly improved upon by early stopping protocols. These dominate the first ones, by always deciding in at most t + 1 rounds, but often much faster. A protocol is unbeatable if it can't be strictly dominated. Namely, if no protocol Q can decide strictly earlier than P against at least one adversary strategy, while deciding at least as fast as P in all cases. Unbeatability is often a much more suitable notion of optimality for distributed protocols than worst-case performance. Halpern, Moses and Waarts in [17], who introduced this notion, presented a general logic-based transformation of any consensus protocol to an unbeatable protocol that dominates it, and suggested a particular unbeatable consensus protocol. Their analysis is based on a notion of continual common knowledge, which is not easy to work with in practice. Using a more direct knowledge-based analysis, this paper studies unbeatability for both consensus and k-set consensus. We present unbeatable solutions to non-uniform consensus and k-set consensus, and uniform consensus in synchronous messagepassing contexts with crash failures. Our consensus protocol strictly dominates the one suggested in [17], showing that their protocol is in fact beatable. The k-set consensus problem is much more technically challenging than consensus, and its analysis has triggered the development of the topological approach to distributed computing. Worst-case lower bounds for this problem have required either techniques based on algebraic topology [13], or reduction-based proofs [1, 12]. Our proof of unbeatability is purely combinatorial, and is a direct, albeit nontrivial, generalization of the one for consensus. We also present an alternative topological unbeatability proof that allows to understand the connection between the connectivity of protocol complexes and the decision time of processes. All of our protocols make use of a notion of a hidden path of nodes relative to a process i at time m, in which a value unknown to i at m may be seen by others. This is a structure that can implicitly be found in lower bound proofs for consensus going back to the '80s [7]. Its use in our protocols sheds light on the mathematical structure underlying the consensus problem and its variants. For the synchronous model, only solutions to the uniform variant of k-set consensus have been offered. Based on our unbeatable protocols for uniform consensus and for non-uniform k-set consensus, we present a uniform k-set consensus protocol that strictly dominates all known solutions to this problem in the synchronous model.

Journal of The ACM, 1988

The concept of partial synchrony in a distributed system is introduced. Partial synchrony lies between the cases of a synchronous system and an asynchronous system. In a synchronous system, there is a known fixed upper bound A on the time required for a message to be sent from one processor to another and a known fixed upper bound % on the relative speeds of different processors. In an asynchronous system no fixed upper bounds A and @ exist. In one version of partial synchrony, fixed bounds A and Cp exist, but they are not known a priori. The problem is to design protocols that work correctly in the partially synchronous system regardless of the actual values of the bounds A and Cp. In another version of partial synchrony, the bounds are known, but are only guaranteed to hold starting at some unknown time T, and protocols must be designed to work correctly regardless of when time T occurs. Fault-tolerant consensus protocols are given for various cases of partial synchrony and various fault models. Lower bounds that show in most cases that our protocols are optimal with respect to the number of faults tolerated are also given. Our consensus protocols for partially synchronous processors use new protocols for fault-tolerant "distributed clocks" that allow partially synchronous processors to reach some approximately common notion of time.

Abstract The consensus problem is at the heart of solutions related to the development of modern reliable distributed systems. This paper studies necessary and sufficient conditions under which fault-tolerant consensus become solvable in dynamic systems and self-organizing networks.

Lecture Notes in Computer Science, 2008

Consider the following problem: a sender S and a receiver R are part of a directed synchronous network and connected through intermediate nodes. Specifically, there exists n node disjoint paths, also called as wires, which are directed from S to R and u wires, which are directed from R to S. Moreover, the wires from S to R are disjoint from the wires directed from R to S. There exists a centralized, static adversary A static t , who has unbounded computing power and who can control at most t wires between S and R in Byzantine fashion. S has a message m S , which we wants to send to R. The challenge is to design a protocol, such that after interacting in phases 5 as per the protocol, R should correctly output m R = m S , except with error probability 2 −Ω(κ) , where κ is the error parameter. This problem is called as statistically reliable message transmission (SRMT). The problem of statistically secure message transmission (SSMT) has an additional requirement that at the end of the protocol, m S should be information theoretically secure from A static t . Desmedt et.al have given the necessary and sufficient condition for the existence of SRMT and SSMT protocols in the above settings. They also presented an SSMT protocol, satisfying their characterization. The authors in claimed that their protocol is efficient and has polynomial computational and communication complexity. However, we show that it is not so. That is, we specify an adversary strategy, which may cause the protocol to have exponential computational and communication complexity 6 . We then present 6 This is an extended, modified and elaborate version of new and efficient SRMT and SSMT protocols, satisfying the characterization of . Finally we show that the our proposed protocols are communication optimal by deriving lower bound on the communication complexity of SRMT and SSMT protocols. As far our knowledge is concerned, our protocols are the first communication optimal SRMT and SSMT protocols in directed networks.

IACR Cryptol. ePrint Arch., 2018

Fault-tolerant distributed consensus is a fundamental problem in secure distributed computing. In this work, we consider the problem of distributed consensus in directed graphs tolerating crash failures. Tseng and Vaidya (PODC’15) presented necessary and sufficient condition for the existence of consensus protocols in directed graphs. We improve the round and communication complexity of their protocol. Moreover, we prove that our protocol requires the optimal number of communication rounds, required by any protocol belonging to a restricted class of crash-tolerant consensus protocols in directed graphs.

University of Michigan - Deep Blue, 2022

Graphs are important and useful mathematical representations that can be used to model different objects in various research disciplines. For example, graphs are used for representing structures of chemical compounds in mathematical chemistry, representing phylogenetic trees in evolutionary biology, the small world concepts in social networks, and topology of a computer network. There are several properties on a graph for us to study, such as connectivity, reachability and distances. In this thesis, we study three fun algorithmic challenges and apply graph algorithms and data structures to improve them. The first challenge is to design a data structure that solves the fully dynamic graph connectivity problem. For any sequence of edge insertions and deletions to a graph, our data structure performs updates in amortized randomized O(log n(log log n) 2) time, improving Thorup's result by a O(log log n) factor for the first time since 2000. Our result leaves a small O((log log n) 2) gap to the Ω(log n) lower bound given by Pǎtraşcu and Demaine in 2006. The second topic we study includes various structures on graphs concerning distances and reachability, such as spanners, emulators, hopsets and shortcutting sets. A (β,)-hopset is a set of weighted edges added to an undirected weighted graph so that distances between any pair of vertices can then be (1 +)-approximated using a path with at most β edges (hops). We applied Thorup and Zwick's sublinear additive emulator construction to build a hopset. The construction, parameterized by k > 0, has an optimal tradeoff between the hopset size O(n 1+ 1 2 k+1 −1), the hop bound β = O(k/) k , and the approximation ratio (1 +). This optimal tradeoff is established by the work of Abboud, Bodwin, and Pettie. A shortcutting set is a reachability-preserving set of directed edges whose addition to a directed unweighted graph reduces the diameter. We show that there is a Ω(n 1/6) diameter lower bound on any shortcutting set of O(n) size. The lower bound techniques can also be applied to undirected weighted graphs. In particular, we provide a +Ω(n 1/11) additive distortion lower bound for O(n)-size spanners and a +Ω(n 1/18) additive distortion lower bound for O(n)-size emulators. The third problem we solve is the asynchronous Byzantine agreement problem. In distributed computing, reaching consensus among all n processes is a challengx ing task, especially when a subset of processes are secretly corrupted and malicious behavior may occur. We provide a protocol that solves Byzantine agreement in an asynchronous distributed network model under perhaps the most challenging assumptions. Although we assume a fully-connected authenticated network ensuring that once a message is sent it will be delivered, the protocol has to work against a full-information (no cryptography for hiding information) and adaptive (corrupting processes over time) adversary. Most Byzantine agreement protocols are developed under weaker adversary models. We propose the first polynomial time protocol that allows up to (n − 1)/4 faulty processes, improving King and Saia's 2013 result from (1.14 × 10 −9)n faulty processes. Unlike the first two topics in this thesis, the asynchronous Byzantine agreement problem is not obviously related to graphs. But interestingly, one building block in our protocol is a fractional matching algorithm with a continuous Lipschitz guarantee. xi

Achieving processor cooperation in the presence of faults is a major problem in distributed systems. Popular paradigms such as Byzantine agreement have been studied principally in the context of a complete network. Indeed, Dolev [J. Algorithms, 3 (1982), pp. 14-30] and Hadzilacos [Issues of Fault Tolerance in Concurrent Computations, Ph.D. thesis, Harvard University, Cambridge, MA, 1984] have shown that fl(t) connectivity is necessary if the requirement is that all nonfaulty processors decide unanlmously, where is the number of faults to be tolerated. We believe that in forseeable technologies the number of faults will grow with the size of the network while the degree will remain practically fixed. We therefore raise the question whether it is possible to avoid the connectivity requirements by slightly lowering our expectations. In many practical situations we may be willing to "lose" some correct processors and settle for cooperation between the vast majority of the processors. Thus motivated, we present a general simulation technique by which vertices (processors) in almost any network of bounded degree can simulate an algorithm designed for the complete network. The simulation has the property that although some correct processors may be cut off from the majority of the network by faulty processors, the vast majority of the correct processors will be able to communicate among themselves undisturbed by the (arbitrary) behavior of the faulty nodes.

Lecture Notes in Computer Science, 1991

A self stabilizing protocol for constructing a rooted spanning tree in an arbitrary asynchronous network of processors that communicate through sha~ed memory is presented. The processors have unique identifiers but are otherwise identical. The network topology is assumed to be dynamic, that is, edges can join or leave the computation before it eventually stabilizes.

Journal of the ACM, 2003

This article introduces and explores the condition-based approach to solve the consensus problem in asynchronous systems. The approach studies conditions that identify sets of input vectors for which it is possible to solve consensus despite the occurrence of up to f process crashes. The first main result defines acceptable conditions and shows that these are exactly the conditions for which a consensus protocol exists. Two examples of realistic acceptable conditions are presented, and proved to be maximal, in the sense that they cannot be extended and remain acceptable. The second main result is a generic consensus shared-memory protocol for any acceptable condition. The protocol always guarantees agreement and validity, and terminates (at least) when the inputs satisfy the condition with which the protocol has been instantiated, or when there are no crashes. An efficient version of the protocol is then designed for the message passing model that works when f < n/2, and it is shown that no such protocol exists when f ≥ n/2. It is also shown how the protocol's safety can be traded for its liveness. Abstract Devices]: Models of Computation-relations A preliminary version of this article appeared in

Lecture Notes in Computer Science, 2014

Two mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds using a deterministic algorithm. Each agent has a different label, which it can use in the execution of the algorithm, but it does not know the label of the other agent. Agents do not know any bound on the size of the network. In each round an agent decides if it remains idle or if it wants to move to one of the adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault in a given round, it remains in the current node, regardless of its decision. If it planned to move and the fault happened, the agent is aware of it. We consider three scenarios of fault distribution: random (independently in each round and for each agent with constant probability 0 < p < 1), unbounded adversarial (the adversary can delay an agent for an arbitrary finite number of consecutive rounds) and bounded adversarial (the adversary can delay an agent for at most c consecutive rounds, where c is unknown to the agents). The quality measure of a rendezvous algorithm is its cost, which is the total number of edge traversals. For random faults, we show an algorithm with cost polynomial in the size n of the network and polylogarithmic in the larger label L, which achieves rendezvous with very high probability in arbitrary networks. By contrast, for unbounded adversarial faults we show that rendezvous is not feasible, even in the class of rings. Under this scenario we give a rendezvous algorithm with cost O(n ), where is the smaller label, working in arbitrary trees, and we show that Ω( ) is the lower bound on rendezvous cost, even for the two-node tree. For bounded adversarial faults, we give a rendezvous algorithm working for arbitrary networks, with cost polynomial in n, and logarithmic in the bound c and in the larger label L.