Discretized Approaches to Schematization

TL;DR

This paper explores discretized methods for creating schematic shapes in cartography by overlaying plane graphs on polygons, analyzing two approaches—map matching and face selection—and proving their computational hardness.

Contribution

It introduces two discretized approaches for schematic shape computation and proves NP-hardness of their optimization problems under various conditions.

Findings

NP-hardness of approximating minimal Fréchet distance cycles

NP-hardness of optimal face set selection in grid and tiling graphs

Complexity persists even with area preservation and specified turn sequences

Abstract

To produce cartographic maps, simplification is typically used to reduce complexity of the map to a legible level. With schematic maps, however, this simplification is pushed far beyond the legibility threshold and is instead constrained by functional need and resemblance. Moreover, stylistic geometry is often used to convey the schematic nature of the map. In this paper we explore discretized approaches to computing a schematic shape $S$ for a simple polygon $P$. We do so by overlaying a plane graph $G$ on $P$ as the solution space for the schematic shape. Topological constraints imply that $S$ should describe a simple polygon. We investigate two approaches, simple map matching and connected face selection, based on commonly used similarity metrics. With the former, $S$ is a simple cycle $C$ in $G$ and we quantify resemblance via the Fr\'echet distance. We prove that it is NP-hard to compute a cycle that approximates the minimal Fr\'echet distance over all simple cycles in a plane graph $G$. This result holds even if $G$ is a partial grid graph, if area preservation is required and if we assume a given sequence of turns is specified. With the latter, $S$ is a connected face set in $G$, quantifying resemblance via the symmetric difference. Though the symmetric difference seems a less strict measure, we prove that it is NP-hard to compute the optimal face set. This result holds even if $G$ is full grid graph or a triangular or hexagonal tiling, and if area preservation is required. Moreover, it is independent of whether we allow the set of faces to have holes or not.