On Approximating the Weighted Region Problem in Square Tessellations

TL;DR

This work addresses the weighted region problem on a square tessellation and analyzes a grid-graph approximation that places vertices at square centers with 8-neighborhood edges. By introducing a parameterized subgraph on $V_a$ and enforcing diagonal-edge constraints tied to the original shortest path $\pi^{*}$, it derives a per-square bound and optimizes $a$ to balance the competing terms, yielding $C(P^*) \le (\sqrt{2}+1) C(\pi^{*})$. This improves the prior grid-based bound of $2\sqrt{2}$ and demonstrates how a carefully designed discretization can produce near-optimal approximate routing in weighted regions. The results inform fast, practical routing in planar weighted environments and deepen understanding of grid-graph discretizations for geometric shortest-path problems.

Abstract

The weighted region problem is the problem of finding the weighted shortest path on a plane consisting of polygonal regions with different weights. For the case when the plane is tessellated by squares, we can solve the problem approximately by finding the shortest path on a grid graph defined by placing a vertex at the center of each grid. In this note, we show that the obtained path admits $(\sqrt{2}+1)$-approximation. This improves the previous result of $2\sqrt{2}$.

Figures (8)

• Figure 1: The white, gray, and black squares have weights $1$, $2$, and $100$, respectively. The red path $\pi^\ast$ depicts a shortest path from $s$ to $g$, and the black path $P^\ast$ is a shortest path on the grid graph.
• Figure 2: Neighbor vertices of a square.
• Figure 3: Proof of Lemma \ref{['lem:gogocurry2']} when $y'=y$. Only the last case (d) is possible if $v_{xy}, v_{x'y'}\not\in V_a$.
• Figure 8: When $\pi^\ast_{xy}$ goes from $S_{x'y'}$ to $S_{x', y'-1}$.
• Figure 9: When $y'=y-1$.

Theorems & Definitions (9)

• Theorem 3.1
• Lemma 3.2
• proof
• Claim 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof