On the Bit Complexity of Iterated Memory

TL;DR

The paper investigates the minimal memory required per process to simulate $r$ rounds of the full-information iterated immediate snapshot (IIS) protocol in an asynchronous shared-memory setting. Using a combinatorial-topology framework, it characterizes when a bounded-memory IIS (B-IIS) protocol is equivalent to the full-information IIS via distinguishability under an encoding, and derives tight asymptotic bounds on the per-process bit complexity: $\Theta(r\,n\log n)$ for $n\ge 3$ and $\Theta(1)$ (with 2 bits) for $n=2$. The core technical contributions are a precise analysis of the $\mathrm{f}$-vector of iterated chromatic subdivisions and the introduction of indistinguishability graphs to connect encoding size with the protocol’s topological structure, yielding both lower and upper bounds and explicit constructions. The results illuminate the feasibility of memory-bounded simulations for wait-free tasks, including approximate agreement, and open avenues for applying these methods to other tasks and subdivisions.

Abstract

Computability, in the presence of asynchrony and failures, is one of the central questions in distributed computing. The celebrated asynchronous computability theorem (ACT) characterizes the computing power of the read-write shared-memory model through the geometric properties of its protocol complex: a combinatorial structure describing the states the model can reach via its finite executions. This characterization assumes that the memory is of unbounded capacity, in particular, it is able to store the exponentially growing states of the full-information protocol. In this paper, we tackle an orthogonal question: what is the minimal memory capacity that allows us to simulate a given number of rounds of the full-information protocol? In the iterated immediate snapshot model (IIS), we determine necessary and sufficient conditions on the number of bits an IIS element should be able to store so that the resulting protocol is equivalent, up to isomorphism, to the full-information protocol. Our characterization implies that $n\geq 3$ processes can simulate $r$ rounds of the full-information IIS protocol as long as the bit complexity per process is $Θ(r n \log n)$. Two processes, however, can simulate any number of rounds of the full-information protocol using only $2$ bits per process, which implies, in particular, that just $2$ bits per process are sufficient to solve $\varepsilon$-agreement for arbitrarily small $\varepsilon$.

Figures (4)

• Figure 1: Iterative application of the standard chromatic subdivision to $\Delta^2$. Leftmost diagram illustrate the $\Delta^2$ simplex. The diagram in the middle shows $\mathrm{Ch}\ {\Delta^2}$, which is subdivided again resulting in the rightmost image $\mathrm{Ch}^2\ {\Delta^2}$.
• Figure 2: Commutative diagram illustrating the equivalence between $\Xi_b$ and $\mathrm{Ch}$ over an input complex $\mathcal{I}$ after $r$ iterations. Dashed arrows indicate the existence of an isomorphism.
• Figure 3: Example of a simplicial complex along with its corresponding indistinguishability graph of the green-labeled process. The relative positions of the green vertices are preserved in the drawing.
• Figure 4: From top to bottom, the input complex, protocol complex for one round, and the protocol complex for two rounds of the bounded approximate agreement algorithm. Numbers next to the vertices indicate the internal state of the process. Green arrows illustrate the next_state function executed in each round for the blue process.

Theorems & Definitions (49)

• definition thmcounterdefinition: Distinguishability of a vertex
• definition thmcounterdefinition: Distinguishability of a chromatic simplicial complex
• theorem thmcountertheorem: Characterization of $\Xi_b$
• proof
• corollary thmcountercorollary
• definition thmcounterdefinition: Indistinguishability graph of a chromatic complex
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary
• proof