We survey results from distributed computing that show tasks to be impossible, either outright or within given resource bounds, in various models The parameters of the models considered include synchrony, fault-tolerance, different communication media, and randomization The resource bounds refer to time, space and message complexity These results are useful in understanding the inherent difficulty of individual problems and in studying the power of different models of distributed computing There is a strong emphasis in our presentation on explaining the wide variety of techniques that are used to obtain the results described

Hundreds of impossibility results for distributed computing

Loop agreement is a family of wait-free tasks that includes instances of set agreement and approximate agreement tasks. A task G implements task F if one can construct a solution to F from a solution to G, possibly followed by access to a read/write memory. Loop agreement tasks form a lattice under this notion of implementation.This paper presents a classification of loop agreement tasks. Each loop agreement task can be assigned an algebraic signature consisting of a finitely presented group G and a distinguished element g in G. This signature characterizes the task's power to implement other tasks. If F and G are loop agreement tasks with respective signatures 〈F,f〉 and 〈G,g〉, then F implements G if and only if there exists a group homomorphism h : F → G carrying f to g.

A classification of wait-free loop agreement tasks

This paper introduces the use of topological models and methods, formerly used to analyze computability, as tools for the quantification and classification of asynchronous complexity We present the first asynchronous complexity theorem, applied to decision tasks in the iterated immediate snapshot (IIS) model of Borowsky and Gafni We do so by introducing a novel form of topological tool called the nonuniform chromatic subdivision Building on the framework of Herlihy and Shavit's topological computability model, our theorem states that the time complexity of any asynchronous algorithm is directly proportional to the level of nonuniform chromatic subdivisions necessary to allow a simplicial map from a task's input complex to its output complex To show the power of our theorem, we use it to derive a new tight bound on the time to achieve n process approximate agreement in the IIS model: $\bigl\lceil \log_d \frac{\max\_input - \min\_input}{\epsilon} \bigr\rceil$, where d = 3 for two processes and d = 2 for three or more This closes an intriguing gap between the known upper and lower bounds implied by the work of Aspnes and Herlihy More than the new bounds themselves, the importance of our asynchronous complexity theorem is that the algorithms and lower bounds it allows us to derive are intuitive and simple, with topological proofs that require no mention of concurrency at all

Toward a Topological Characterization of Asynchronous Complexity

This paper introduces the use of topological models and methods, formerly used to analyze computability, as tools for the quantification and classification of asynchronous complexity. We present the first asynchronous complexity theorem, applied to decision tasks in the iterated immediate snapshot (IIS) model of Borowsky and Gafni. We do so by introducing a novel form of topological tool called the nonuniform chromatic subdivision. Building on the framework of Herlihy and Shavit's topological computability model, our theorem states that the time complexity of any asynchronous algorithm is directly proportional to the level of nonuniform chromatic subdivisions necessary to allow a simplicial map from a task's input complex to its output complex. To show the power of our theorem, we use it to derive a new tight bound on the time to achieve n process approximate agreement in the IIS model: logd max input−min input � , where d = 3 for two processes and d = 2 for three or more. This closes an intriguing gap between the known upper and lower bounds implied by the work of Aspnes and Herlihy. More than the new bounds themselves, the importance of our asynchronous complexity theorem is that the algorithms and lower bounds it allows us to derive are intuitive and simple, with topological proofs that require no mention of concurrency at all.

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Towards a topological characterization of asynchronous complexity

A wait-free hierarchy ACM Transactions on Programming Languages and Systems, 11 (1991), pp. 124--149; Proceedings of the 12th ACM Symposium on Principles of Distributed Computing, 1993, pp. 145--158] classifies object types on the basis of their strength in supporting wait-free implementations of other types. Such a hierarchy is robust if it is impossible to implement objects of types that it classifies as "strong" by combining objects of types that it classifies as "weak." We prove that if nondeterministic types are allowed, the only wait-free hierarchy that is robust is the trivial one, which lumps all types into a single level. In particular, the consensus hierarchy (the most closely studied wait-free hierarchy) is not robust. Our result implies that, in general, it is not possible to determine the power of a concurrent system that supports a given set of primitive object types by reasoning about the power of each primitive type in isolation.

All of Us Are Smarter than Any of Us: Nondeterministic Wait-Free Hierarchies Are Not Robust

Agreement problems are central to the study of wait-free protocols for shared memory distributed systems. We examine two specific issues arising out of this study.
We consider the complexity of the wait-free approximate agreement problem in an asynchronous shared memory comprised of only single-bit multi-writer multi-reader registers. For real-valued inputs of magnitude at most s and a real-valued accuracy requirement $\varepsilon>0$ we show matching upper and lower bounds of $\Theta(\log(s/\varepsilon))$ steps and shared registers. For inputs drawn from any fixed finite range this is significantly better than the best possible algorithm for single-writer multi-reader registers, which, for n processes, requires $\Omega(\log\ n)$ steps. These results are used to show a separation between the wait-free single-writer multi-reader and wait-free multi-writer multi-reader models of computation.
The consensus hierarchy characterizes the strength of a shared object by its ability to solve the consensus problem in a wait-free manner. One important application of a hierarchy classifying the power of objects is to compare the power of systems offering different collections of objects. Ideally, a hierarchy should reduce the task of determining the strength of an architecture supporting shared memory distributed systems to the problem of determining the strength of each type of shared object supported by the architecture. Informally, a hierarchy that allows this is robust. Several variations of the consensus hierarchy have appeared in the literature, and it has been shown that all but one of them are not robust. The remaining hierarchy, named $h\sbsp{m}{r},$ has been the subject of considerable research. We show that, in a natural setting, the consensus hierarchy $h\sbsp{m}{r}$ is not robust.

Eric Schenk

Papers

Computability and complexity results for agreement problems in shared-memory distributed systems