[Paper Review] Information-Theoretic Lower Bounds on the Storage Cost of Shared Memory Emulation

Viveck R. Cadambe, Zhiying Wang+1 more|arXiv (Cornell University)|May 22, 2016

Distributed systems and fault tolerance21 references4 citations

TL;DR

This paper establishes information-theoretic lower bounds on the storage cost of emulating atomic shared memory in asynchronous, fault-tolerant distributed systems. It proves that even without server gossip, total storage cost is at least $ 2\frac{N}{N-f+1}\log_2|\mathcal{V}|-o(\log_2|\mathcal{V}|) $, and with gossip, the bound is $ 2\frac{N}{N-f+2}\log_2|\mathcal{V}|-o(\log_2|\mathcal{V}|) $, significantly improving upon prior bounds and applying universally to replication and erasure coding schemes.

ABSTRACT

The focus of this paper is to understand storage costs of emulating an atomic shared memory over an asynchronous, distributed message passing system. Previous literature has developed several shared memory emulation algorithms based on replication and erasure coding techniques. In this paper, we present information-theoretic lower bounds on the storage costs incurred by shared memory emulation algorithms. Our storage cost lower bounds are universally applicable, that is, we make no assumption on the structure of the algorithm or the method of encoding the data. We consider an arbitrary algorithm $A$ that implements an atomic multi-writer single-reader (MWSR) shared memory variable whose values come from a finite set $\mathcal{V}$ over a system of $N$ servers connected by point-to-point asynchronous links. We require that in every fair execution of algorithm $A$ where the number of server failures is smaller than a parameter $f$, every operation invoked at a non-failing client terminates. We define the storage cost of a server in algorithm $A$ as the logarithm (to base 2) of number of states it can take on; the total-storage cost of algorithm $A$ is the sum of the storage cost of all servers. Our results are as follows. (i) We show that if algorithm $A$ does not use server gossip, then the total storage cost is lower bounded by $2 \frac{N}{N-f+1}\log_2|\mathcal{V}|-o(\log_2|\mathcal{V}|)$. (ii) The total storage cost is at least $2 \frac{N}{N-f+2} \log_{2}|\mathcal{V}|-o(\log_{2}|\mathcal{V}|)$ even if the algorithm uses server gossip. (iii) We consider algorithms where the write protocol sends information about the value in at most one phase. We show that the total storage cost is at least $ν^* \frac{N}{N-f+ν^*-1} \log_2( |\mathcal{V}|)- o(\log_2(|\mathcal{V}|),$ where $ν^*$ is the minimum of $f+1$ and the number of active write operations of an execution.

Motivation & Objective

• To establish universal, information-theoretic lower bounds on the storage cost of emulating atomic shared memory in asynchronous, fault-tolerant distributed systems.
• To analyze storage costs without assuming specific algorithmic structures, such as replication or erasure coding, or the use of server gossip.
• To determine the minimal storage cost required for correct emulation under various constraints, including write protocol phase limitations.
• To resolve fundamental open questions about whether storage costs can be reduced below known upper bounds.
• To extend prior bounds by accounting for the number of active write operations and system parameters like N and f.

Proposed method

• Uses information-theoretic analysis to derive lower bounds on server storage cost, defined as the logarithm of the number of possible states per server.
• Applies a proof technique inspired by coding theory, constructing execution traces to derive valency arguments and contradiction-based bounds.
• Considers fair executions with fewer than f server failures, ensuring all non-failing client operations terminate.
• Introduces a novel execution extension argument to show that certain configurations must be valent, leading to contradictions under assumptions of lower storage cost.
• Analyzes three distinct algorithmic classes: no-gossip, with-gossip, and write protocols with at most one phase of value-dependent communication.
• Leverages weak regularity and value-dependent action invariance to derive bounds on the number of distinct values that can be consistently stored.

Experimental results

Research questions

• RQ1 Can the storage cost of atomic shared memory emulation be lower than $ \frac{N}{N-f}\log_2|\mathcal{V}| $, and if so, by how much?
• RQ2 What is the minimal total storage cost required for emulating a multi-writer single-reader shared memory under fault tolerance and asynchrony?
• RQ3 How does the presence or absence of server gossip affect the theoretical lower bound on storage cost?
• RQ4 What is the storage cost lower bound when write protocols are restricted to a single phase of value-dependent communication?
• RQ5 Can the lower bounds be generalized to cover algorithms that send value-dependent messages in multiple phases, such as those handling Byzantine failures?

Key findings

• The first lower bound shows that without server gossip, the total storage cost is at least $ 2\frac{N}{N-f+1}\log_2|\mathcal{V}|-o(\log_2|\mathcal{V}|) $, which is approximately twice the previously known bound of $ \frac{N}{N-f}\log_2|\mathcal{V}| $.
• Even with server gossip, the total storage cost is lower bounded by $ 2\frac{N}{N-f+2}\log_2|\mathcal{V}|-o(\log_2|\mathcal{V}|) $, demonstrating that gossip does not significantly reduce the theoretical minimum storage cost.
• For write protocols that transmit value information in at most one phase, the storage cost is bounded below by $ \nu^*\frac{N}{N-f+\nu^*-1}\log_2|\mathcal{V}|-o(\log_2|\mathcal{V}|) $, where $ \nu^* $ is the minimum of $ f+1 $ and the number of active write operations.
• This third bound is significantly larger than the first two and matches the storage cost achieved by most existing erasure coding-based shared memory algorithms.
• The results apply universally to both replication-based and erasure coding-based algorithms, including regular single-writer single-reader systems.
• The paper conjectures that the bounds still hold even for algorithms with multiple phases of value-dependent messages, provided only one phase carries significant value information, such as in Byzantine-tolerant schemes.