NDLI: Computational Aspects of the Gromov–Hausdorff Distance and its Application in Non-rigid Shape Matching

Content Provider	Springer Nature Link
Author	Schmiedl, Felix
Copyright Year	2017
Abstract	The Gromov–Hausdorff distance of two compact metric spaces is a measure for how far the spaces are from being isometric and has been extensively studied in the field of metric geometry. In recent years, a lot of attention has been devoted to computational aspects of the Gromov–Hausdorff distance. One of the most prominent applications is the problem of shape matching, where the goal is to decide whether two shapes given as polygonal meshes are equivalent under certain invariances. Therefore, many methods have been proposed which heuristically estimate the Gromov–Hausdorff distance of metric spaces induced by the shapes. However, the computational complexity of computing the Gromov–Hausdorff distance has not yet been thoroughly investigated. We show that—under standard complexity theoretic assumptions—determining the Gromov–Hausdorff distance of two finite metric spaces cannot be approximated within any reasonable bound in polynomial time. Furthermore, we discover attributes of metric spaces which have a major impact on the complexity of an instance. This enables us to develop an approximation algorithm which is fixed parameter tractable with respect to corresponding parameters.
Ending Page	880
Page Count	27
Starting Page	854
File Format	PDF
ISSN	01795376
e-ISSN	14320444
Journal	Discrete & Computational Geometry
Issue Number	4
Volume Number	57
Language	English
Publisher	Springer US
Publisher Date	2017-04-07
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Computational Mathematics and Numerical Analysis Approximation Non-rigid shape matching Computer graphics; computational geometry Combinatorial complexity of geometric structures Gromov–Hausdorff distance Approximation algorithms Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Combinatorics Computational complexity
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
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3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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