Long-Lived Counters with Polylogarithmic Amortized Step Complexity

A shared-memory counter is a well-studied and widely-used concurrent object. It supports two operations: An Inc operation that increases its value by 1 and a Read operation that returns its current value. Jayanti, Tan and Toueg [Jayanti et al., 2000] proved a linear lower bound on the worst-case step complexity of obstruction-free implementations, from read and write operations, of a large class of shared objects that includes counters. The lower bound leaves open the question of finding counter implementations with sub-linear amortized step complexity. In this paper, we address this gap. We present the first wait-free n-process counter, implemented using only read and write operations, whose amortized operation step complexity is O(log^2 n) in all executions. This is the first non-blocking read/write counter algorithm that provides sub-linear amortized step complexity in executions of arbitrary length. Since a logarithmic lower bound on the amortized step complexity of obstruction-free counter implementations exists, our upper bound is optimal up to a logarithmic factor

Locality and Checkability in Wait-free Computing

22 pagesThis paper studies several notions of locality that are inherent to the specification of distributed tasks and independent of the computing environment, and investigates the ability of a shared memory wait-free system to solve tasks satisfying various forms of locality. First, we define a task to be projection-closed if every partial output π(t) for a full input s is also a valid output for the partial input π(s) and prove that projection-closed tasks are precisely those tasks that are wait-free checkable. Our second main contribution is dealing with a stronger notion of lo- cality of topological nature. A task T = (I, O, ∆) is said to be locality- preserving if and only if O is a covering complex of I, that is, each simplex s of I is mapped by ∆ to a set of simplexes of O each isomorphic to s. This topological property yields obstacles for wait-free solvability different in nature from the classical agreement impossibility results. On the other hand, locality-preserving tasks are projection-closed and therefore always wait-free checkable. We provide a classification of locality-preserving tasks in term of their computational power, by establishing a correspondence between locality-preserving tasks and subgroups of the edgepath group of a complex. Using this correspondence, we prove the existence of hierarchies of locality-preserving tasks, each one containing a universal task (induced by the universal covering complex), and at the bottom the trivial identity task

Generating Fast Indulgent Algorithms

Synchronous distributed algorithms are easier to design and prove correct than algorithms that tolerate asynchrony. Yet, in the real world, networks experience asynchrony and other timing anomalies. In this paper, we address the question of how to efficiently transform an algorithm that relies on synchronous timing into an algorithm that tolerates asynchronous executions. We introduce a transformation technique from synchronous algorithms to indulgent algorithms (Guerraoui, in PODC, pp. 289-297, 2000), which induces only a constant overhead in terms of time complexity in well-behaved executions. Our technique is based on a new abstraction we call an asynchrony detector, which the participating processes implement collectively. The resulting transformation works for the class of colorless distributed tasks, including consensus and set agreement. Interestingly, we also show that our technique is relevant for colored tasks, by applying it to the renaming problem, to obtain the first indulgent renaming algorith

Of Choices, Failures and Asynchrony: The Many Faces of Set Agreement

Set agreement is a fundamental problem in distributed computing in which processes collectively choose a small subset of values from a larger set of proposals. The impossibility of fault-tolerant set agreement in asynchronous networks is one of the seminal results in distributed computing. In synchronous networks, too, the complexity of set agreement has been a significant research challenge that has now been resolved. Real systems, however, are neither purely synchronous nor purely asynchronous. Rather, they tend to alternate between periods of synchrony and periods of asynchrony. Nothing specific is known about the complexity of set agreement in such a "partially synchronous” setting. In this paper, we address this challenge, presenting the first (asymptotically) tight bound on the complexity of set agreement in such systems. We introduce a novel technique for simulating, in a fault-prone asynchronous shared memory, executions of an asynchronous and failure-prone message-passing system in which some fragments appear synchronous to some processes. We use this simulation technique to derive a lower bound on the round complexity of set agreement in a partially synchronous system by a reduction from asynchronous wait-free set agreement. Specifically, we show that every set agreement protocol requires at least $\lfloor\frac{t}{k}\rfloor + 2$ synchronous rounds to decide. We present an (asymptotically) matching algorithm that relies on a distributed asynchrony detection mechanism to decide as soon as possible during periods of synchrony. From these two results, we derive the size of the minimal window of synchrony needed to solve set agreement. By relating synchronous, asynchronous and partially synchronous environments, our simulation technique is of independent interest. In particular, it allows us to obtain a new lower bound on the complexity of early deciding k-set agreement complementary to that of Gafni etal. (in SIAM J. Comput. 40(1):63-78, 2011), and to re-derive the combinatorial topology lower bound of Guerraoui etal. (in Theor. Comput. Sci. 410(6-7):570-580, 2009) in an algorithmic wa

The Computational Power of Distributed Shared-Memory Models with Bounded-Size Registers

The celebrated Asynchronous Computability Theorem of Herlihy and Shavit (STOC 1993 and STOC 1994) provided a topological characterization of the tasks that are solvable in a distributed system where processes are communicating by writing and reading shared registers, and where any number of processes can fail by crashing. However, this characterization assumes the use of full-information protocols, that is, protocols in which each time any of the processes writes in the shared memory, it communicates everything it learned since the beginning of the execution. Thus, the characterization implicitly assumes that each register in the shared memory is of unbounded size. Whether unbounded size registers are unavoidable for the model of computation to be universal is the central question studied in this paper. Specifically, is any task that is solvable using unbounded registers solvable using registers of bounded size? More generally, when at most $t$ processes can crash, is the model with bounded size registers universal? These are the questions answered in this paper

Decentralized Asynchronous Crash-Resilient Runtime Verification

Runtime Verification (RV) is a lightweight method for monitoring the formal specification of a system during its execution. It has recently been shown that a given state predicate can be monitored consistently by a set of crash-prone asynchronous distributed monitors, only if sufficiently many different verdicts can be emitted by each monitor. We revisit this impossibility result in the context of LTL semantics for RV. We show that employing the four-valued logic Rv-LTL will result in inconsistent distributed monitoring for some formulas. Our first main contribution is a family of logics, called Ltl2k+4, that refines Rv-Ltl incorporating 2k + 4 truth values, for each k >= 0. The truth values of Ltl2k+4 can be effectively used by each monitor to reach a consistent global set of verdicts for each given formula, provided k is sufficiently large. Our second main contribution is an algorithm for monitor construction enabling fault-tolerant distributed monitoring based on the aggregation of the individual verdicts by each monitor

Minimizing the Number of Opinions for Fault-Tolerant Distributed Decision Using Well-Quasi Orderings

The notion of deciding a distributed language L is of growing interest in various distributed computing settings. Each process p i is given an input value x i , and the processes should collectively decide whether their set of input values x = (x i) i is a valid state of the system w.r.t. to some specification, i.e., if x ∈ L. In non-deterministic distributed decision each process p i gets a local certificate c i in addition to its input x i. If the input x ∈ L then there exists a certificate c = (c i) i such that the processes collectively accept x, and if x ∈ L, then for every c, the processes should collectively reject x. The collective decision is expressed by the set of opinions emitted by the processes, and one aims at minimizing the number of possible opinions emitted by each process. In this paper we study non-deterministic distributed decision in asynchronous systems where processes may crash. In this setting, it is known that the number of opinions needed to deter-ministically decide a language can grow with n, the number of processes in the system. We prove that every distributed language L can be non-deterministically decided using only three opinions, with certificates of size O(log α(n)) bits, where α grows at least as slowly as the inverse of the Ackerman function. The result is optimal, as we show that there are distributed languages that cannot be decided using just two opinions, even with arbitrarily large certificates. To prove our upper bound, we introduce the notion of distributed encoding of the integers, that provides an explicit construction of a long bad sequence in the well-quasi-ordering ({0, 1} * , =) controlled by the successor function. Thus, we provide a new class of applications for well-quasi-orderings that lies outside logic and complexity theory. For the lower bound we use combinatorial topology techniques

Early Deciding Synchronous Renaming in O( logf ) Rounds or Less

Renaming is a fundamental problem in distributed computing, in which a set of n processes need to pick unique names from a namespace of limited size. In this paper, we present the first early-deciding upper bounds for synchronous renaming, in which the running time adapts to the actual number of failures f in the execution. We show that, surprisingly, renaming can be solved in \emphconstant time if the number of failures f is limited to O(n√) , while for general f ≤ n − 1 renaming can always be solved in O( logf ) communication rounds. In the wait-free case, i.e. for f = n − 1, our upper bounds match the Ω( logn ) lower bound of Chaudhuri et al. [13]

The Committee Decision Problem

We introduce the (b,n)-Committee Decision Problem (CD) - a generalization of the consensus problem. While set agreement generalizes consensus in terms of the number of decisions allowed, the CD problem generalizes consensus in the sense of considering many instances of consensus and requiring a processor to decide in at least one instance. In more detail, in the CD problem each one of a set of n processes has a (possibly distinct) value to propose to each one of a set of b consensus problems, which we call committees. Yet a process has to such that all processes deciding for the same committee decide the same value. We study the CD problem in the context of a wait-free distributed system and analyze it using a combination of distributed algorithmic and topological techniques, introducing a novel reduction technique. We use the reduction technique to obtain the following results. We show that the (2,3)-CD problem is equivalent to the musical benches problem of Gafni and Rajsbaum (DISC 2005), and both are equivalent to (2,3)-set agreement, closing an open question left there. Thus, all three problems are wait-free unsolvable in a read/write shared memory system, and they are all solvable if the system is enriched with objects capable of solving (2,3)-set agreement. While the previous proof of the impossibility of musical benches was based on the Borsuk-Ulam (BU) Theorem, it now relies on Sperner's Lemma, opening intriguing questions about the relation between BU and distributed computing tasks. // Ce rapport présente un problème de prise de décisionsmultiples (qui généralise le problème du consensus) et étudie sa calculabilité (à l'aide de techniques de réductions)

Approximation Algorithm for Estimating Distances in Distributed Virtual Environments

International audienceThis article deals with the issue of guaranteeing properties in Distributed Virtual Environments (DVEs) without a server and without global knowledge of the system state and therefore only by exchanging messages. This issue is particularly relevant in the case of online games, that operate in a fully distributed framework and for which network resources such as bandwidth are the critical resources. In the context of games, players typically need to know the distance between their character and other characters, at least approximately. Players all share the same position estimation algorithm but, in general, do not know the current positions of others. We provide a synchronized distributed algorithm Alc to guarantee, at any time, that the estimated distance d est between any pair of characters A and B is always a 1 + ε approximation of the current distance d act. Our result is twofold: (1) we prove that if characters move randomly on a d-dimensional grid, or follow a random continuous movement on up to three dimensions, the number of messages of Alc is optimal up to a constant factor; (2) in a more practical setting, we also observe that the number of messages of Alc for actual game traces is much less than the standard algorithm sending actual positions at a given frequency