The Complexity of the Hausdorff Distance

Jungeblut, Paul; Kleist, Linda; Miltzow, Tillmann

doi:10.48550/arxiv.2112.04343

Cited by 2 publications

(2 citation statements)

References 37 publications

(52 reference statements)

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“…In spatial sciences, distance is an important parameter to estimate the relative position of spatial objects [49]. The distance can also be used as a metric to examine how similar or dissimilar two point sets (or geometric objects in spatial sciences) are [50]. Generally, three types of distance metrics, including maximum, minimum, and centroid, are defined to measure the degrees of resemblance between two spatial objects:…”

Section: Similaritymentioning

confidence: 99%

A Novel Method for Extracting and Analyzing the Geometry Properties of the Shortest Pedestrian Paths Focusing on Open Geospatial Data

Hosseini¹,

Tong

Lim

et al. 2023

IJGI

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Unlike car navigation, where almost all vehicles can traverse every route, one route might not be optimal or even suitable for all pedestrians. Route geometry information, including tortuosity, twists and turns along roads, junctions, and road slopes, among others, matters a great deal for specific types of pedestrians, particularly those with limited mobility, such as wheelchair users and older adults. Offering practical routing services to these users requires that pedestrian navigation systems provide further information on route geometry. Therefore, this article proposes a novel method for extracting and analyzing the geometry properties of the shortest pedestrian paths, with a focus on open geospatial data across four aspects: (a) similarity, (b) route curviness, (c) road turns and intersections, and (d) road gradients. Deriving from the Hausdorff distance, a metric called the “dissimilarity ratio” was developed, allowing us to determine whether pairs of routes show any tendencies to be similar to each other. Using the “sinuosity index”, a segment-based technique quantified the route curviness based on the number and degree of the road turns along the route. Moreover, relying upon open elevation data, the road gradients were extracted to identify routes offering smoother motion and better accessibility. Lastly, the road turns and intersections were investigated as pedestrian convenience and safety indicators. A local government area of Greater Sydney in Australia was chosen as the case study. The analysis was implemented on OpenStreetMap (OSM) shortest pedestrian paths against Google Maps as a benchmark for real-world commercial applications. The similarity analysis indicated that over 90% of OSM routes were identical or roughly similar to Google Maps. In addition, while Spearman’s rank correlation showed a direct relationship between route curviness and route length, rS(758) = 0.92, p < 0.001, OSM, on average, witnessed more tortuous routes and, consequently, shorter straight roads between turns. However, OSM routes could be more suitable for pedestrians when the frequency of intersections and road slopes are at the center of attention. Finally, the devised metrics in this study, including the dissimilarity ratio and sinuosity index, showed their practicability in translating raw values into meaningful indicators.

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Section: Similaritymentioning

confidence: 99%

A Novel Method for Extracting and Analyzing the Geometry Properties of the Shortest Pedestrian Paths Focusing on Open Geospatial Data

Hosseini¹,

Tong

Lim

et al. 2023

IJGI

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“…There have been several works on designing approximation algorithms for computing the Hausdorff distance between convex polygons [7], curves [10], images [35], meshes [6] or point cloud data [54]; see also [3,4,37,39]. There are relatively few [44] known exact formula for the Hausdorff distance between sets.…”

Section: Related Workmentioning

confidence: 99%

Hausdorff Distance between Norm Balls and their Linear Maps

Haddad¹,

Halder²

2022

Preprint

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We consider the problem of computing the (two-sided) Hausdorff distance between the unit p 1 and p 2 norm balls in finite dimensional Euclidean space for 1 < p 1 < p 2 ≤ ∞, and derive a closed-form formula for the same. When the two different norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to the unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.

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The Complexity of the Hausdorff Distance

Abstract: We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class ∀∃

Cited by 2 publications

References 37 publications

A Novel Method for Extracting and Analyzing the Geometry Properties of the Shortest Pedestrian Paths Focusing on Open Geospatial Data

A Novel Method for Extracting and Analyzing the Geometry Properties of the Shortest Pedestrian Paths Focusing on Open Geospatial Data

Hausdorff Distance between Norm Balls and their Linear Maps

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