Combinatorial and Asymptotical Results on the Neighborhood Grid

TL;DR

This paper investigates the combinatorial properties of the neighborhood grid data structure used in spatial simulations, correcting previous assumptions, extending its analysis to various sizes and dimensions, and exploring state uniqueness and maximization.

Contribution

It corrects prior assumptions about uniqueness, generalizes the neighborhood grid to multiple sizes and dimensions, and provides a partial classification of its states using combinatorial formulas.

Findings

Identified that the previous uniqueness assumption does not hold.

Extended the neighborhood grid concept to arbitrary sizes and dimensions.

Provided a partial classification of states using the hook-length formula.

Abstract

In various application fields, such as fluid-, cell-, or crowd-simulations, spatial data structures are very important. They answer nearest neighbor queries which are instrumental in performing necessary computations for, e.g., taking the next time step in the simulation. Correspondingly, various such data structures have been developed, one being the \emph{neighborhood grid}. In this paper, we consider combinatorial aspects of this data structure. Particularly, we show that an assumption on uniqueness, made in previous works, is not actually satisfied. We extend the notions of the neighborhood grid to arbitrary grid sizes and dimensions and provide two alternative, correct versions of the proof that was broken by the dissatisfied assumption. Furthermore, we explore both the uniqueness of certain states of the data structure as well as when the number of these states is maximized. We provide a partial classification by using the hook-length formula for rectangular Young tableaux. Finally, we conjecture how to extend this to all 2-dimensional cases.