The Squishy Grid Problem

2025-07-30

AI Summary

This paper investigates asymptotically isometric embeddings of integer grid graphs into Euclidean space with respect to Euclidean distance. We propose three novel construction paradigms: (1) a deterministic embedding based on aperiodic pinwheel tilings; (2) a hierarchical highway topology; and (3) random edge-weight assignment drawn from a two-point distribution. The first two yield explicit convergence rates, achieving multiplicative distortion bounds of $1 + 1/Theta(log^xi log D)$ and $1 + 1/Theta(D^{1/9})$, respectively, where $D$ denotes the graph diameter. The third method employs an extremely simple random distribution and empirically attains approximation error within 1% of Euclidean distance. Integrating geometric tiling theory, recursive embedding design, and probabilistic analysis, our work substantially advances the metric representability of grid graphs and establishes a new paradigm for approximating continuous geometry via discrete structures.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment $w:E o mathbb{R}_{ge 0}$, and hope to have grid distances be asymptotically isometric to Euclidean distances, that is, for all grid points $u,v$, $mathrm{dist}_w(u,v) = (1pm o(1))|u-v|_2$. We give three methods for solving this problem, each attractive in its own way. * Our first construction is based on an embedding of the recursive, non-periodic pinwheel tiling of Radin and Conway into the integer grid. Distances in the pinwheel graph are asymptotically isometric to Euclidean distances, but no explicit bound on the rate of convergence was known. We prove that the multiplicative distortion of the pinwheel graph is $(1+1/Θ(log^ξlog D))$, where $D$ is the Euclidean distance and $ξ=Θ(1)$. The pinwheel tiling approach is conceptually simple, but can be improved quantitatively. * Our second construction is based on a hierarchical arrangement of "highways." It is simple, achieving stretch $(1 + 1/Θ(D^{1/9}))$, which converges doubly exponentially faster than the pinwheel tiling approach. * The first two methods are deterministic. An even simpler approach is to sample the edge weights independently from a common distribution $mathscr{D}$. Whether there exists a distribution $mathscr{D}^*$ that makes grid distances Euclidean, asymptotically and in expectation, is major open problem in the theory of first passage percolation. Previous experiments show that when $mathscr{D}$ is a Fisher distribution, grid distances are within 1% of Euclidean. We demonstrate experimentally that this level of accuracy can be achieved by a simple 2-point distribution that assigns weights 0.41 or 4.75 with probability 44% and 56%, respectively.

Problem

• Approximating Euclidean distances using infinite grid graphs
• Optimizing edge-weight assignment for asymptotic isometry
• Comparing deterministic and probabilistic methods for distance accuracy

Innovation

• Pinwheel tiling for asymptotically isometric grid distances
• Hierarchical highways for faster convergence stretch
• Two-point distribution for accurate Euclidean approximation