Atomic Snapshots from Small Registers

Existing n-process implementations of atomic snapshots from registers use large registers. We consider the problem of implementing an m-component snapshot from small, Theta(log(n))-bit registers. A natural solution is to consider simulating the large registers. Doing so straightforwardly can significantly increase the step complexity. We introduce the notion of an interruptible read and show how it can reduce the step complexity of simulating the large registers in the snapshot of Afek et al. In particular, we show how to modify a recent large register simulation to support interruptible reads. Using this modified simulation, the step complexity of UPDATE and SCAN changes from Theta(n*m) to Theta(n*m+m*w), instead of Theta(n*m*w), if each component of the snapshot consists of Theta(w*log(n)) bits. We also show how to modify a limited-use snapshot to use small registers when the number of UPDATE operations is in n^{O(1)}. In this case, we change the step complexity of UPDATE from Theta((log(n))^3) to O(w + (log(n))^2*log(m)) and the step complexity of SCAN from Theta(log(n)) to O(m*w + log(n))

A Complexity-Based Hierarchy for Multiprocessor Synchronization

For many years, Herlihy's elegant computability based Consensus Hierarchy has been our best explanation of the relative power of various types of multiprocessor synchronization objects when used in deterministic algorithms. However, key to this hierarchy is treating synchronization instructions as distinct objects, an approach that is far from the real-world, where multiprocessor programs apply synchronization instructions to collections of arbitrary memory locations. We were surprised to realize that, when considering instructions applied to memory locations, the computability based hierarchy collapses. This leaves open the question of how to better capture the power of various synchronization instructions. In this paper, we provide an approach to answering this question. We present a hierarchy of synchronization instructions, classified by their space complexity in solving obstruction-free consensus. Our hierarchy provides a classification of combinations of known instructions that seems to fit with our intuition of how useful some are in practice, while questioning the effectiveness of others. We prove an essentially tight characterization of the power of buffered read and write instructions.Interestingly, we show a similar result for multi-location atomic assignments

Why Extension-Based Proofs Fail

We introduce extension-based proofs, a class of impossibility proofs that includes valency arguments. They are modelled as an interaction between a prover and a protocol. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve k-set agreement among n > k > 1 processes in a wait-free manner in certain asynchronous models. However, it was unknown whether proofs based on simpler techniques were possible. We show that this impossibility result cannot be obtained for one of these models by an extension-based proof and, hence, extension-based proofs are limited in power.Comment: This version of the paper is for the NIS model. Previous versions of the paper are for the NIIS mode

On the Space Complexity of Colourless Tasks

In this thesis, we prove lower bounds on the number of registers needed to solve colourless tasks in asynchronous shared memory systems. Many fundamental synchronization tasks, such as consensus, k-set agreement, and epsilon-approximate agreement, are colourless. We show that it is possible to transform any nondeterministic solo-terminating algorithm (including any randomized wait-free algorithm) into an obstruction-free algorithm that uses the same number of registers. This result extends to algorithms using any finite number of deterministic objects that support read operations. Hence, we can focus on proving lower bounds for obstruction-free algorithms. We prove a tight lower bound on the number of registers needed to solve obstruction-free consensus. We also prove the first non-constant lower bounds on the number of registers needed to solve obstruction-free k-set agreement and obstruction-free epsilon-approximate agreement. The bound for k-set agreement is asymptotically tight when k is a constant and the bound for epsilon-approximate agreement is asymptotically tight when epsilon is sufficiently small. To prove these bounds, we introduce a new technique, revisionist simulations. This technique allows us to prove a general theorem that yields lower bounds on the number of registers needed to solve any colourless task in an obstruction-free manner. Finally, we define the class of extension-based proofs and show that no extension-based proof can establish the impossibility of deterministically solving k-set agreement among n > k > 1 processes in a wait-free manner using registers.Ph.D

Space Lower Bounds for the Signal Detection Problem

Many shared memory algorithms have to deal with the problem of determining whether the value of a shared object has changed in between two successive accesses of that object by a process when the responses from both are the same. Motivated by this problem, we define the signal detection problem, which can be studied on a purely combinatorial level. Consider a system with n+1 processes consisting of n readers and one signaller. The processes communicate through a shared blackboard that can store a value from a domain of size m. Processes are scheduled by an adversary. When scheduled, a process reads the blackboard, modifies its contents arbitrarily, and, provided it is a reader, returns a Boolean value. A reader must return true if the signaller has taken a step since the reader\u27s preceding step; otherwise it must return false. Intuitively, in a system with n processes, signal detection should require at least n bits of shared information, i.e., m >= 2^n. But a proof of this conjecture remains elusive. We prove a lower bound of m >= n^2, as well as a tight lower bound of m >= 2^n for two restricted versions of the problem, where the processes are oblivious or where the signaller always resets the blackboard to the same fixed value. We also consider a one-shot version of the problem, where each reader takes at most two steps. In this case, we prove that it is necessary and sufficient that the blackboard can store m=n+1 values

A complexity-based classification for multiprocessor synchronization

Abstract For many years, Herlihy’s elegant computability-based Consensus Hierarchy has been our best explanation of the relative power of various objects. Since real multiprocessors allow the different instructions they support to be applied to any memory location, it makes sense to consider combining the instructions supported by different objects, rather than considering collections of different objects. Surprisingly, this causes Herlihy’s computability-based hierarchy to collapse. In this paper, we suggest an alternative: a complexity-based classification of the relative power of sets of multiprocessor synchronization instructions, captured by the minimum number of memory locations of unbounded size that are needed to solve obstruction-free consensus when using different sets of instructions

Patterns and determinants of wood physical and mechanical properties across major tree species in China

The physical and mechanical properties of wood affect the growth and development of trees, and also act as the main criteria when determining wood usage. Our understanding on patterns and controls of wood physical and mechanical properties could provide benefits for forestry management and bases for wood application and forest tree breeding. However, current studies on wood properties mainly focus on wood density and ignore other wood physical properties. In this study, we established a comprehensive database of wood physical properties across major tree species in China. Based on this database, we explored spatial patterns and driving factors of wood properties across major tree species in China. Our results showed that (i) compared with wood density, air-dried density, tangential shrinkage coefficient and resilience provide more accuracy and higher explanation power when used as the evaluation index of wood physical properties. (ii) Among life form, climatic and edaphic variables, life form is the dominant factor shaping spatial patterns of wood physical properties, climatic factors the next, and edaphic factors have the least effects, suggesting that the effects of climatic factors on spatial variations of wood properties are indirectly induced by their effects on species distribution

Why extension-based proofs fail

It is impossible to deterministically solve wait-free consensus in an asynchronous system. The classic proof uses a valency argument, which constructs an infinite execution by repeatedly extending a finite execution. We introduce extension-based proofs, a class of impossibility proofs that are modelled as an interaction between a prover and a protocol and that include valency arguments. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve k-set agreement among n > k ≥ 2 processes in a wait-free manner. However, it was unknown whether proofs based on simpler techniques were possible. We show that this impossibility result cannot be obtained by an extension-based proof and, hence, extension-based proofs are limited in power

Brief Announcement: Why Extension-Based Proofs Fail

We introduce extension-based proofs, a class of impossibility proofs that includes valency arguments. They are modelled as an interaction between a prover and a protocol. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve k-set agreement among n > k ≥ 2 processes in a wait-free manner. However, it was unknown whether proofs based on simpler techniques were possible. We explain why this impossibility result cannot be obtained by an extension-based proof and, hence, extension-based proofs are limited in power

Why extension-based proofs fail

We introduce extension-based proofs, a class of impossibility proofs that includes valency arguments. They are modelled as an interaction between a prover and a protocol. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve -set agreement among processes or approximate agreement on a cycle of length 4 among processes in a wait-free manner in asynchronous models where processes communicate using objects that can be constructed from shared registers. However, it was unknown whether proofs based on simpler techniques were possible. We show that these impossibility results cannot be obtained by extension-based proofs in the iterated snapshot model and, hence, extension-based proofs are limited in power