Interactive Nearest Lattice Point Search in a Distributed Setting: Two Dimensions

TL;DR

This paper investigates distributed algorithms for the nearest lattice point problem in two dimensions, analyzing error probabilities and communication costs for single-round and multi-round protocols, highlighting the efficiency of different lattice structures.

Contribution

It introduces a detailed analysis of error probabilities and communication costs for distributed nearest lattice point search, emphasizing the impact of lattice structure and communication rounds.

Findings

Error probability depends on lattice structure.

Zero error requires infinite communication bits.

Hexagonal lattice is most communication-expensive.

Abstract

The nearest lattice point problem in $\mathbb{R}^n$ is formulated in a distributed network with $n$ nodes. The objective is to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node communication. Algorithms with a single as well as an unbounded number of rounds of communication are considered for the case $n=2$. For the algorithm with a single round, expressions are derived for the error probability as a function of the total number of communicated bits. We observe that the error exponent depends on the lattice structure and that zero error requires an infinite number of communicated bits. In contrast, with an infinite number of allowed communication rounds, the nearest lattice point can be determined without error with a finite average number of communicated bits and a finite average number of rounds of communication. In two dimensions, the hexagonal lattice, which is most efficient for communication and compression, is found to be the most expensive in terms of communication cost.