The Squishy Grid Problem

TL;DR

This work investigates how to weight the infinite integer grid so that grid distances approximate Euclidean distances asymptotically. It presents two deterministic constructions: (i) embedding Radin–Conway's nonperiodic pinwheel tiling yields a multiplicative distortion of $(1+1/\Theta(\log^ξ\log D))$ with $ξ=\Theta(1)$, and (ii) a hierarchical highway design achieves additive error $O(\|u-v\|_2^{8/9})$ and near-Euclidean distances. In addition, it studies randomized edge weights, showing simple 2-point and 3-point discrete distributions can empirically approximate Euclidean distances to about $0.75\%$ and $0.62\%$ error, respectively, improving upon previous Fisher-distribution results. The results connect to first-passage percolation through time constants and limit shapes, and they provide both theoretical bounds and practical distributions for near-Euclidean grid metrics, while highlighting open questions about subpolynomial additive error and exact limit shapes. Overall, the paper advances deterministic and probabilistic strategies for approximating Euclidean metric geometry on grids with potential applications in scalable routing and geometric embeddings.

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 $w:E\to \mathbb{R}_{\ge 0}$, and hope to have grid distances be asymptotically isometric to Euclidean distances, that is, for all grid points $u,v$, $\mathrm{dist}_w(u,v) = (1\pm o(1))\|u-v\|_2$. 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 $(1+1/Θ(\log^ξ\log D))$, 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.

Figures (11)

• Figure 1: (A)$\mathscr{T}_1$, containing $\mathscr{T}_0$ in red. (B)$\mathscr{T}_2$, containing $\mathscr{T}_1$ in red, and $\mathscr{T}_0$ in blue.
• Figure 2: Two triangles within $T$. The hypotenuses of $T,A,B$ have angles $-\gamma,-2\gamma,0$, respectively.
• Figure 3: A grid-path that tracks $\ell$.
• Figure 4: Illustrations of various cases in Step 2. (A)$\ell$ and $s$ intersect. The segment $\ell\cap \operatorname{Fat}(s,k_i)$ is removed from $\ell$. (B)$\ell$ and $s$ do not intersect. Left: $\overline{AB} \geq k_i$ and $\overline{AB}$ is removed from $\ell$. Right: $\overline{AB} < k_i$ and $B'$ is such that $\overline{AB'}=k_i$ is removed from $\ell$.
• Figure 5: Step 2 detours from the proof of \ref{['lem:highway-lower-upper-bound']}. (A) The case when segment $s$ intersects $\ell^*$. (B) When segment $s$ does not intersect $\ell^*$, but $\operatorname{Fat}(s,k_i)$ does.

Theorems & Definitions (20)

• Theorem 2.1
• proof
• Corollary 2.3
• Theorem 2.4: Baker baker1977-exposition
• Lemma 2.1
• proof
• proof
• Theorem 2.5
• proof
• Theorem 3.1