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Optimal Net Surface Problems with Applications

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Automata, Languages and Programming (ICALP 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2380))

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Abstract

In this paper, we study an interesting geometric graph called multi-column graph in the d-D space (d ≥ 3), and formulate two combinatorial optimization problems called the optimal net surface problems on such graphs. Our formulations capture a number of important problems such as surface reconstruction with a given topology, medical image segmentation, and metric labeling. We prove that the optimal net surface problems on general d-D multi-column graphs (d ≥ 3) are NP-hard. For two useful special cases of these d-D (d ≥ 3) optimal net surface problems (on the so-called proper ordered multi- column graphs) that often arise in applications, we present polynomial time algorithms. We further apply our algorithms to some surface reconstruction problems in 3-D and 4-D, and some medical image segmentation problems in 3-D and 4-D, obtaining polynomial time solutions for these problems. The previously best known algorithms for some of these applied problems, even for relatively simple cases, take at least exponential time. Our approaches for these applied problems can be extended to higher dimensions.

This research was supported in part by the National Science Foundation under Grant CCR-9988468 and the 21st Century Research and Technology Fund from the State of Indiana.

The work of this author was supported in part by a fellowship from the Center for Applied Mathematics, University of Notre Dame, Notre Dame, Indiana, USA.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Xiaodong Wu & Danny Z. Chen

Authors
  1. Xiaodong Wu
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  2. Danny Z. Chen
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Editor information

Editors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zentrum, ETH Zürich, 8092, Zürich, Switzerland

    Peter Widmayer  & Stephan Eidenbenz  & 

  2. Department of Languages and Sciences of the Computation E.T.S. de Ingeniería Informática, University of Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Francisco Triguero , Rafael Morales  & Ricardo Conejo ,  & 

  3. School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton, BN1 9QN, UK

    Matthew Hennessy

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© 2002 Springer-Verlag Berlin Heidelberg

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Wu, X., Chen, D.Z. (2002). Optimal Net Surface Problems with Applications. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds) Automata, Languages and Programming. ICALP 2002. Lecture Notes in Computer Science, vol 2380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45465-9_88

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  • DOI: https://doi.org/10.1007/3-540-45465-9_88

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43864-9

  • Online ISBN: 978-3-540-45465-6

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