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

V. A. Vaishampayan, M. F. Bollauf

TL;DR

The nearest lattice point problem in <inline-formula> <tex-math notation="LaTeX">$\mathbb {R}^{n}$ </tex-math></inline-formula> is formulated in a distributed network with nodes to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node communication.

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.

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

TL;DR

The nearest lattice point problem in <inline-formula> <tex-math notation="LaTeX"> </tex-math></inline-formula> is formulated in a distributed network with nodes to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node communication.

Abstract

The nearest lattice point problem in is formulated in a distributed network with 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 . 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.

Paper Structure

This paper contains 23 sections, 5 theorems, 42 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Lattice Basis and Source Model
  4. Lattice Basis
  5. Source Model
  6. Description of the Interactive Single Round Protocol
  7. Geometric Structure of $\mathcal{B}\bigcap \mathcal{V}$
  8. Stage-II Protocol Description: Further Details
  9. Decoder
  10. Error Probability and Communication Rate Calculations
  11. Error Probability
  12. Optimizing the Action of Node-2
  13. A Partially Optimized Expression for $P_e$
  14. Communication Rates
  15. Optimization of the Partition
  16. ...and 8 more sections

Key Result

Theorem 1

For any partition of $\mathcal{B}^{(2)}$ into intervals $\mathcal{I}_{W_1,Z_1,Z_2}(w_1,z_1,z_2),~z_2=-k_{w_1,z_1},\ldots,k_{w_1,z_1}$, for given $W_1=w_1\neq 0$ and $Z_1=z_1$ where $\dot{u}$ and $\dot{l}$ are the (constant) slopes of $u(x_1)$ and $l(x_1)$ for $x_1\in \mathcal{I}_{W_1,Z_1}(w_1,z_1)$. Further, equality holds when $k_{w_1,z_1}=1$ and the two horizontal cuts are located at $x_2=u(x_1^

Figures (13)

  • Figure 1: Setup for $n=2$. Two sensors are connected by noiseless links and have partial information about a vector $\tilde{\bm{X}}$. The nodes collaborate in order to find the nearest point in the lattice $\Lambda$. (a) Sensors and communication network (b) signal space visualization in $\mathbb{R}^2$.
  • Figure 2: The Babai partition (rectangular partition with solid lines) and the Voronoi partition (hexagonal partition with dotted lines) for a lattice in $\mathbb{R}^2$. The first stage of the algorithm determines the cell of the Babai partition which contains $\bm{x}$. The second stage reclassifies $\bm{x}$ according to the Voronoi partition. The shaded triangular regions illustrate the points that must be reclassified in the second stage.
  • Figure 3: $\mathcal{B}$ and intersecting Voronoi cells for a given lattice. Coordinates of lattice basis vectors $\bm{v}_1$, $\bm{v}_2$ and face intersection points $\bm{a},\bm{b},\ldots,\bm{h}$, of Babai and Voronoi cells are shown. Faces of the Voronoi cell $\mathcal{V}$ are determined by the relevant vectors, $\pm \bm{v}_1$, $\pm \bm{v}_2$ and $\pm (\bm{v}_2-\bm{v}_1)$. Coordinates of cell boundary intersection points: $\bm{b}=(-1+\rho \cos \theta,\rho \sin \theta)/2$, $\bm{c}=(\rho \cos \theta,\rho \sin \theta)/2$, $\bm{d}=(1,(\rho-\cos \theta)/\sin \theta)/2$, $\bm{e}=(1,-(\rho-\cos \theta)/\sin \theta)/2$, $\bm{f}=-\bm{b}$, $\bm{g}=-\bm{c}$, $\bm{h}=-\bm{d}$ and $\bm{a}=-\bm{e}$.
  • Figure 4: Operational description of the two-stage protocol for finding an approximation to ${\bm{y}}_{v}(\tilde{\bm x})$. At the end of Stage-I, both nodes have determined ${\bm{y}}_{np}(\tilde{\bm x})$. Stage-II then refines the result of Stage-I, resulting in the lattice point $\bm{y}_\Pi(\tilde{\bm{x}})$.
  • Figure 5: The collective action of encoding and decoding operations in Stage-I (top) and Stage-II (bottom). In Stage-II the vector $\bm{x}$ always lies in the Babai cell $\mathcal{B}$.
  • ...and 8 more figures

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Theorem 5
  • proof
  • Remark 2