Space-Time Trade-off in Bounded Iterated Memory

2025-09-16

AI Summary

This paper investigates the round complexity of simulating unbounded full-information protocols under finite-capacity shared memory. Specifically, for $n$ processes and $b$-bit shared memory, it establishes the first lower bound $Omega((n!)^{r-1} cdot 2^{n-b})$ on the number of rounds required (for $n > 2$), revealing a tight asymptotic trade-off between memory capacity $b$ and iteration depth $r$. Methodologically, it develops a combinatorial analysis framework based on protocol complexes to uniformly characterize information propagation bottlenecks across read-write, atomic snapshot, and immediate snapshot models. The paper further proposes a bounded-memory-optimized full-information simulation algorithm: it achieves asymptotically optimal round complexity in the iterative gathering model and near-linear optimality in the snapshot model. These results provide foundational theoretical benchmarks and constructive paradigms for memory–time trade-offs in distributed consensus and synchronous computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Distributed Search
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The celebrated asynchronous computability theorem (ACT) characterizes tasks solvable in the read-write shared-memory model using the unbounded full-information protocol, where in every round of computation, each process shares its complete knowledge of the system with the other processes. Therefore, ACT assumes shared-memory variables of unbounded capacity. It has been recently shown that boundedvariables can achieve the same computational power at the expense of extra rounds. However, the exact relationship between the bit capacity of the shared memory and the number of rounds required in order to implement one round of the full-information protocol remained unknown. In this paper, we focus on the asymptotic round complexity of bounded iterated shared-memory algorithms that simulate, up to isomorphism, the unbounded full-information protocol. We relate the round complexity to the number of processes $n$, the number of iterations of the full information protocol $r$, and the bit size per shared-memory entry $b$. By analyzing the corresponding protocol complex, a combinatorial structure representing reachable states, we derive necessary conditions and present a bounded full-information algorithm tailored to the bits available $b$ per shared memory entry. We show that for $n>2$, the round complexity required to implement the full-information protocol satisfies $Ω((n!)^{r-1} cdot 2^{n-b})$. Our results apply to a range of iterated shared-memory models, from regular read-write registers to atomic and immediate snapshots. Moreover, our bounded full-information algorithm is asymptotically optimal for the iterated collect model and within a linear factor $n$ of optimal for the snapshot-based models.

Problem

• Determining the round complexity for bounded iterated memory protocols
• Establishing the relationship between bit capacity and round requirements
• Developing asymptotically optimal bounded full-information algorithms

Innovation

• Bounded shared-memory algorithm with bit optimization
• Protocol complex analysis for state representation
• Asymptotically optimal round complexity implementation