Hilbert curves for efficient exploratory landscape analysis neighbourhood sampling

TL;DR

This work investigates Hilbert space-filling curves as a means to generate spatially correlated samples for exploratory landscape analysis, addressing the need for local neighbourhood information while maintaining broad search-space coverage. It demonstrates that Hilbert curve sampling delivers saliency in information-content features comparable to Latin hypercube sampling but at a substantially lower computational cost, and that Hilbert-curve ordering of existing samples is significantly faster than nearest-neighbour ordering while preserving feature quality. The study also shows that stochastic variants of Hilbert curves can introduce randomness without compromising coverage, and that ordering via Hilbert curves provides smoother step sizes. Overall, Hilbert curves offer a scalable, efficient alternative for both sampling and ordering in landscape analysis, with practical implications for automated algorithm design and performance prediction.

Abstract

Landscape analysis aims to characterise optimisation problems based on their objective (or fitness) function landscape properties. The problem search space is typically sampled, and various landscape features are estimated based on the samples. One particularly salient set of features is information content, which requires the samples to be sequences of neighbouring solutions, such that the local relationships between consecutive sample points are preserved. Generating such spatially correlated samples that also provide good search space coverage is challenging. It is therefore common to first obtain an unordered sample with good search space coverage, and then apply an ordering algorithm such as the nearest neighbour to minimise the distance between consecutive points in the sample. However, the nearest neighbour algorithm becomes computationally prohibitive in higher dimensions, thus there is a need for more efficient alternatives. In this study, Hilbert space-filling curves are proposed as a method to efficiently obtain high-quality ordered samples. Hilbert curves are a special case of fractal curves, and guarantee uniform coverage of a bounded search space while providing a spatially correlated sample. We study the effectiveness of Hilbert curves as samplers, and discover that they are capable of extracting salient features at a fraction of the computational cost compared to Latin hypercube sampling with post-factum ordering. Further, we investigate the use of Hilbert curves as an ordering strategy, and find that they order the sample significantly faster than the nearest neighbour ordering, without sacrificing the saliency of the extracted features.

Figures (8)

• Figure 1: Visualisation of a $2D$ Hilbert curve of order $p = \{1,2,3\}$.
• Figure 2: Illustration in 2D of adding stochasticity to a Hilbert curve
• Figure 3: Effect of randomisation strategy on step size. Sample was generated from an $8^{th}$ order Hilbert curve in 2D space resulting in a total of 65 535 points. The $x$-axis represents the distance between two consecutive points.
• Figure 4: Comparison of time (in seconds) to generate samples and calculate information content metrics for Hilbert curve sampling and LHS using nearest neighbour and random ordering. Trendlines are polynomials of order 2.
• Figure 5: Visualisation of sample ordering using Hilbert curve (left) and nearest neighbour (right) ordering strategies.