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.

) Elizabeth Borowsky (borowsky@hpl.hp.com) Hewlett-Packard Laboratories Palo-Alto, CA 94303 U.S.A. Eli Gafni (eli@cs.ucla.edu) Computer Science Department University of California, Los Angeles Los Angeles, CA 90024 U.S.A. July 1, 1996 Abstract In a sequence of two pioneering papers Herlihy and Shavit characterized waitfree shared-memory computations. The derivation of the characterization involves homology for the necessary conditions, and complex geometry arguments for the sufficiency. This paper gives an alternative proof of the conditions using familiar algorithmic arguments. Our only reliance on geometry is the use of a corollary to the simplicial approximation. Furthermore, this paper is the first to present another consequence of the relation between distributed algorithms and topology: that certain theorems in topology are naturally proven by distributed algorithms interpretations. Our techniques can be extended to characterize models that are more complex than the wait-free...

In a shared-memory distributed system, n independent asynchronous processes communicate by reading and writing to shared memory. An algorithm is adaptive (to total contention) if its step complexity depends only on the actual number, k, of active processes in the execution; this number is unknown in advance and may change in different executions of the algorithm. Adaptive algorithms are inherently wait-free, providing fault-tolerance in the presence of an arbitrary number of crash failures and different processes' speed. A wait-free adaptive collect algorithm with O(k) step complexity is presented, together with its applications in wait-free adaptive algorithms for atomic snapshots, immediate snapshots and renaming. Keywords: contention-sensitive complexity, wait-free algorithms, asynchronous sharedmemory systems, read/write registers, atomic snapshots, immediate atomic snapshots, renaming. Work supported by the fund for the promotion of research in the Technion. y Department of Computer Science, The Technion, Haifa 32000, Israel. hagit@cs.technion.ac.il. z Department of Computer Science, The Technion, Haifa 32000, Israel. leonf@cs.technion.ac.il. x Computer Science Department, UCLA. eli@cs.ucla.edu. 1

Abstract. This paper introduces a simple notion of layering as a tool for analyzingwell-behaved runs of a given model of distributed computation. Using layering, a model-independent analysis of the consensus problem is performed and then applied to provinglower bounds and impossibility results for consensus in a number of familiar and less familiar models. The proofs are simpler and more direct than existingones, and they expose a unified structure to the difficulty of reachingconsensus. In particular, the proofs for the classical synchronous and asynchronous models now follow the same outline. A new notion of connectivity amongstates in runs of a consensus protocol, called potence connectivity, is introduced. This notion is more general than previous notions of connectivity used for this purpose and plays a key role in the uniform analysis of consensus.

The Cambridge Research Laboratory was founded in 1987 to advance the state of the art in both core computing and human-computer interaction, and to use the knowledge so gained to support the Company’s corporate objectives. We believe this is best accomplished through interconnected pursuits in technology creation, advanced systems engineering, and business development. We are actively investigating scalable computing; mobile computing; vision-based human and scene sensing; speech interaction; computer-animated synthetic persona; intelligent information appliances; and the capture, coding, storage, indexing, retrieval, decoding, and rendering of multimedia data. We recognize and embrace a technology creation model which is characterized by three major phases: Freedom: The life blood of the Laboratory comes from the observations and imaginations of our research staff. It is here that challenging research problems are uncovered (through discussions with customers, through interactions with others in the Corporation, through other professional interactions, through reading, and the like) or that new ideas are born. For any such problem or idea, this phase culminates in the nucleation of a project team around a well articulated central research question and the outlining of a research plan. Focus: Once a team is formed, we aggressively pursue the creation of new technology based on

A simulation of a wait-free, atomic, single-writer multi-reader register in an asynchronous message passing system is presented. The simulation can withstand the failure of up to half of the processors, and requires O(n) messages (for each read or write operation), assuming there are n+ 1 processors in the system. It improves on the previous simulation, which requires O(n 2 ) messages (for each read or write operation). The message complexity of the new simulation is within a constant factor of the optimum. The new simulation improves the complexity of algorithms for the following problems in the message-passing model in the presence of processor failures: multi-writer multireader registers, concurrent time-stamp systems, `-exclusion, atomic snapshots, randomized consensus, implementation of data structures, as well as improved fault-tolerant algorithms for any solvable decision task. Keywords: fault-tolerance, shared memory, message passing, wait-free algorithms, processor failures,...

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 " ? 0 we show matching upper and lower bounds of \Theta(log(s=")) 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\Gammaire n) steps. These results are used to show a separation between the wait-free single-writer mult...

We consider the the relative power of two important synchronization problems: set agreement and renaming. We show that renaming is strictly weaker than set agreement, in a round-by-round model of computation. We introduce new techniques, including previously unknown connections between properties of manifolds and computation, as well as novel “symmetry-breaking” constructions.

This paper discusses results that say what cannot be computed in certain environments or when insucient resources are available. A comprehensive survey would require an entire book. As in Nancy Lynch's excellent 1989 paper, \A Hundred Impossibility Proofs for Distributed Computing" [86], we shall restrict ourselves to some of the results we like best or think are most important. Our aim is to give you the avour of the results and some of the techniques that have been used. We shall also mention some interesting open problems and provide an extensive list of references. The focus will be on results from the past decade.

We introduce a simple notion of layering that provides a tool for defining submodels of a given model of distributed computation. We describe two layerings, the synchronic and the permutation layering, and show that they induce appropriate submodels of several asynchronous models of computation. The synchronic layering applies to the synchronous model too. We perform a model-independent analysis of the consensus problem in terms of abstract connectivity properties of layering functions. By defining particular layerings in specific models, we derive several popular (and some new) lower bounds and impossibility results for consensus in various classical models. These results are often stronger in the sense that they apply to the submodel induced by the layering. The proofs obtained in this way are also simpler and more direct than existing ones. Moreover, the analysis is done in a uniform fashion and demonstrates the fundamental common structure of the consensus problem in the presence ...