On the Complexity of Implementing Certain Classes of Shared Objects

We consider shared memory systems in which asynchronous processes cooperate with each other by communicating via shared data objects, such as counters, queues, stacks, and priority queues. The common approach to implementing such shared objects is based on locking: To perform an operation on a shared object, a process obtains a lock, accesses the object, and then releases the lock. Locking, however, has several drawbacks, including convoying, priority inversion, and deadlocks. Furthermore, lock-based implementations are not fault-tolerant: if a process crashes while holding a lock, other processes can end up waiting forever for the lock. Wait-free linearizable implementations were conceived to overcome most of the above drawbacks of locking. A wait-free implementation guarantees that if a process repeatedly takes steps, then its operation on the implemented data object will eventually complete, regardless of whether other processes are slow, or fast, or have crashed. In this thesis, we first present an efficient wait-free linearizable implementation of a class of object types, called closed and closable types, and then prove time and space lower bounds on wait-free linearizable implementations of another class of object types, called perturbable types. (1) We present a wait-free linearizable implementation of n-process closed and closable types (such as swap, fetch&add, fetch&multiply, and fetch&L, where L is any of the boolean operations and, or, or complement) using registers that support load-link (LL) and store-conditional (SC) as base objects. The time complexity of the implementation grows linearly with contention, but is never more than O(log ^2 n). We believe that this is the first implementation of a class of types (as opposed to a specific type) to achieve a sub-linear time complexity. (2) We prove linear time and space lower bounds on the wait-free linearizable implementations of n-process perturbable types (such as increment, fetch&add, modulo k counter, LL/SC bit, k-valued compare&swap (for any k \u3e= n), single-writer snapshot) that use resettable consensus and historyless objects (such as registers that support read and write) as base objects. This improves on some previously known Omega(sqrt{n}) space complexity lower bounds. It also shows the near space optimality of some known wait-free linearizable implementations

An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.Comment: 46 pages, with large margins. Includes quant-ph/0004072 plus 30 pages of new material, mostly on fault-toleranc

Uncertainty relations for multiple measurements with applications

Uncertainty relations express the fundamental incompatibility of certain observables in quantum mechanics. Far from just being puzzling constraints on our ability to know the state of a quantum system, uncertainty relations are at the heart of why some classically impossible cryptographic primitives become possible when quantum communication is allowed. This thesis is concerned with strong notions of uncertainty relations and their applications in quantum information theory. One operational manifestation of such uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be "locked". Furthermore, knowing the key, it is possible to recover, that is "unlock", the message. We give new efficient constructions of bases satisfying strong uncertainty relations leading to the first explicit construction of an information locking scheme. We also give several other applications of our uncertainty relations both to cryptographic and communication tasks. In addition, we define objects called QC-extractors, that can be seen as strong uncertainty relations that hold against quantum adversaries. We provide several constructions of QC-extractors, and use them to prove the security of cryptographic protocols for two-party computations based on the sole assumption that the parties' storage device is limited in transmitting quantum information. In doing so, we resolve a central question in the so-called noisy-storage model by relating security to the quantum capacity of storage devices.Comment: PhD Thesis, McGill University, School of Computer Science, 158 pages. Contains arXiv:1010.3007 and arXiv:1111.2026 with some small addition

Efficient and Practical Constructions of LL/SC Variables

Over the past decade, LL/SC have emerged as the most suitable synchronization instructions for the design of lock-free algorithms. However, current architectures do not support these instructions; instead, they support either CAS or RLL/RSC (e.g. POWER4, MIPS, SPARC, IA-64). To bridge this gap, this paper presents two efficient wait-free algorithms for implementing 64-bit LL/SC objects from 64-bit CAS or RLL/RSC objects. Our first algorithm is practical: it has a small, constant time complexity (of 4 for LL and 5 for SC) and a space overhead of only 4 words per process. This algorithm uses unbounded sequence numbers. For theoretical interest, we also present a more complex bounded algorithm that still guarantees constant time complexity and O(1) space overhead per process. The LL/SC primitive is free of the well-known ABA problem that afflicts CAS. By efficiently implementing LL/SC words from CAS words, this work presents an efficient general solution to the ABA problem

Randomized versus Deterministic Implementations of Concurrent Data Structures

One of the key trends in computing over the past two decades has been increased distribution, both at the processor level, where multi-core architectures are now the norm, and at the system level, where many key services are currently distributed overmultiple machines. Thus, understanding the power and limitations of computing in a concurrent, distributed setting is one of the major challenges in Computer Science. In this thesis, we analyze the complexity of implementing concurrent data structures in asynchronous shared memory systems. We focus on the complexity of a classic distributed coordination task called renaming, in which a set of processes need to pick distinct names from a small set of identifiers. We present the first tight bounds for the time complexity of this problem, both for deterministic and randomized implementations, solving a long-standing open problem in the field. For deterministic algorithms, we prove a tight linear lower bound; for randomized solutions, we provide logarithmic upper and lower bounds on time complexity. Together, these results show an exponential separation between deterministic and randomized renaming solutions. Importantly, the lower bounds extend to implementations of practical shared-memory data structures, such as queues, stacks, and counters. From a technical perspective, this thesis highlights new connections between the distributed renaming problem and other fundamental objects, such as sorting networks, mutual exclusion, and counters. In particular, we show that sorting networks can be used to obtain optimal randomized solutions to renaming, and that, in turn, the existence of these solutions implies a linear lower bound on the complexity of the problem. In sum, the results in this thesis suggest that deterministic implementations of shared-memory data structures do not scale well in terms of worst-case time complexity. On the positive side, we emphasize randomization as a natural alternative, which can circumvent the deterministic lower bounds with high probability. Thus, a promising direction for future work is to extend our randomized renaming techniques to obtain efficient implementations of concurrent data structures

Quantum computing 40 years later

Forty years ago, Richard Feynman proposed harnessing quantum physics to build a more powerful kind of computer. Realizing Feynman's vision is one of the grand challenges facing 21st century science and technology. In this article, we'll recall Feynman's contribution that launched the quest for a quantum computer, and assess where the field stands 40 years later.Comment: 49 pages. To appear in Feynman Lectures on Computation, 2nd edition, published by Taylor & Francis Group, edited by Anthony J. G. Hey. (v2) typos correcte

A Wait-free Queue with Poly-logarithmic Worst-case Step Complexity

In this work, we introduce a novel linearizable wait-free queue implementation. Linearizability and lock-freedom are standard requirements for designing shared data structures. To the best of our knowledge, all of the existing linearizable lock-free queues in the literature have a common problem in their worst case, called the CAS Retry Problem. We show that our algorithm avoids this problem with the helping mechanism which we use and has a worst-case running time better than prior lock-free queues. The amortized number of steps for an Enqueue or Dequeue in our algorithm is O(log^2 p + log q), where p is the number of processes and q is the size of the queue when the operation is linearized

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum