. This paper introduces a general formulation of atomic snapshot memory, a shared memory partitioned into words written (updated) by individual processes, or instantaneously read (scanned) in its entirety. This paper presents three wait-free implementations of atomic snapshot memory. The first implementation in this paper uses unbounded (integer) fields in these registers, and is particularly easy to understand. The second implementation uses bounded registers. Its correctness proof follows the ideas of the unbounded implementation. Both constructions implement a single-writer snapshot memory, in which each word may be updated by only one process, from single-writer, n-reader registers. The third algorithm implements a multi-writer snapshot memory from atomic n-writer, n-reader registers, again echoing key ideas from the earlier constructions. All operations require \Theta(n 2 ) reads and writes to the component shared registers in the worst case. Categories and Subject Discriptors:...

We give necessary and sufficient combinatorial conditions characterizing the computational tasks that can be solved by N asynchronous processes, up to t of which can fail by halting. The range of possible input and output values for an asynchronous task can be associated with a high-dimensional geometric structure called a simplicial complex. Our main theorem characterizes computability y in terms of the topological properties of this complex. Most notably, a given task is computable only if it can be associated with a complex that is simply connected with trivial homology groups. In other words, the complex has “no holes!” Applications of this characterization include the first impossibility results for several long-standing open problems in distributed computing, such as the “renaming ” problem of Attiya et. al., the “k-set agreement ” problem of Chaudhuri, and a generalization of the approximate agreement problem. 1

We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebraic and combinatorial topology, in which a task's possible input and output values are each associated with high-dimensional geometric structures called simplicial complexes. We characterize computability in terms of the topological properties of these complexes. This characterization has a surprising geometric interpretation: a task is solvable if and only if the complex representing the task's allowable inputs can be mapped to the complex representing the task's allowable outputs by a function satisfying certain simple regularity properties. Our formalism thus replaces the "operational" notion of a wait-free decision task, expressed in terms of interleaved computations unfolding ...

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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.

1 1 Introduction 2 2 Related Work 5 3 Preliminaries 7 3.1 The Asynchronous Shared-Memory Model : : : : : : : : : : : : : : : : : : : 7 3.2 Sensitivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 4 The Left/Right Algorithm 11 4.1 The General Scheme : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 4.2 The Left/Right Algorithm : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 4.2.1 Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 4.2.2 The code : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 4.2.3 Correctness of the Algorithm : : : : : : : : : : : : : : : : : : : : : : 16 4.2.4 Analysis of the Algorithm : : : : : : : : : : : : : : : : : : : : : : : : 18 4.3 Inherently Asymmetric Data Structures : : : : : : : : : : : : : : : : : : : : 21 5 The Decision Algorithm 23 5.1 Monotone Paths : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 5.1.1 One Phase :...

It is often important for the correct processes in a distributed system to reach agreement, despite the presence of some faulty processes. Byzantine agreement (BA) is a paradigm problem that attempts to isolate the key features of reaching agreement. We focus here on the number of messages required to reach BA, with particular emphasis on the number of messages required in the failure-free runs, since these are the ones that occur most often in practice. The number of messages required is sensitive to the types of failures considered. In earlier work, Amdur et al. [1990] established tight upper and lower bounds on the worst- and average-case number of messages required in failurefree runs for crash failures. We provide tight upper and lower bounds for all remaining types of failures that have been considered in the literature on the BA problem: receiving omission, sending omission and general omission failures, as well as arbitrary failures with or without message authentication. We ...

A snapshot scan algorithm takes an "instantaneous" picture of a region of shared memory that may be updated by concurrent processes. Many complex shared memory algorithms can be greatly simplified by structuring them around the snapshot scan abstraction. Unfortunately, the substantial decrease in conceptual complexity is quite often counterbalanced by an increase in computational complexity. In this paper, we introduce the notion of a weak snapshot scan, a slightly weaker primitive that has a more efficient implementation. We propose the following methodology for using this abstraction: first, design and verify an algorithm using the more powerful snapshot scan, and second, replace the more powerful but less efficient snapshot with the weaker but more efficient snapshot, and show that the weaker abstraction nevertheless suffices to ensure the correctness of the enclosing algorithm. We give two examples of algorithms whose performance can be enhanced while retaining a simple m...

The computer industry is currently examining the use of strong synchronization operations such as double compare-and-swap (DCAS) as a means of supporting non-blocking synchronization on tomorrow's multiprocessor machines. However, before such a strong primitive will be incorporated into hardware design, its utility needs to be proven by developing a body of effective non-blocking data structures using DCAS. As part of this effort, we present two new linearizable non-blocking implementations of concurrent deques using the DCAS operation. The first uses an array representation, and improves on former algorithms by allowing uninterrupted concurrent access to both ends of the deque while correctly handling the difficult boundary cases when the deque is empty or full. The second uses a linked-list representation, and is the first non-blocking unbounded-memory deque implementation. It too allows uninterrupted concurrent access to both ends of the deque.

We address the problem of asynchronous consensus. In view of the ~\Omega ) lower bound for consensus with the standard adversary scheduler [2], we examine the problem with other adversary models. We define the value-oblivious scheduler, which at all times has full knowledge of the entire state of system except for the actual random values generated and manipulated by the program, as long as they do not affect the dynamics of the system. We argue that the value-oblivious adversary model faithfully captures the possible sources of asynchrony in real-world systems. We present a randomized algorithm that achieves consensus in O(n log n) total work in the presence of a valueoblivious scheduler. Total work is defined as the total number of steps performed by all processors collectively. The amortized work per processor is thus O(log n). The expected contention on any register is O(1) and with high probability never do more than O(log n) processors try to access the same register concurrently.