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Efficient Algorithm for Box Folding

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WALCOM: Algorithms and Computation (WALCOM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11355))

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Abstract

For a given polygon P and a polyhedron Q, the folding problem asks if Q can be obtained from P by folding it. This simple problem is quite complicated, and there is no known efficient algorithm that solves this problem in general. In this paper, we focus on the case that Q is a box, and the size of Q is not given. That is, input of the box folding problem is a polygon P, and it asks if P can fold to boxes of certain sizes. We note that there exist an infinite number of polygons P that can fold into three boxes of different sizes. In this paper, we give a pseudo polynomial time algorithm that computes all possible ways of folding of P to boxes.

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Notes

  1. 1.

    We do not give the formal definition of curvature. Intuitively, it indicates the quantity of paper around the point measured by its angle. See [6] for further details.

  2. 2.

    For sake of simplicity, we do not define the labels of points in P on an edge shared by two rectangles of Q. We also do not define the label of a point p corresponding to the vertex \(v_i\) of Q.

  3. 3.

    Some readers may consider the first phase is enough. However, we have not yet checked if some particles of polygons cause overlap on a face of Q. In other words, we have to check each face is made by particles of polygons by gluing without overlap or hole.

  4. 4.

    The number of polyomino of area 30 is 2368347037571252.

References

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Acknowledgements

A part of this research is supported by JSPS KAKENHI Grant Number JP17H06287 and 18H04091.

Author information

Authors and Affiliations

  1. Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Sakura, Saitama, 338-8570, Japan

    Koichi Mizunashi & Takashi Horiyama

  2. School of Information Science, Japan Advanced Institute of Science and Technology (JAIST), 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

    Ryuhei Uehara

Authors
  1. Koichi Mizunashi
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  2. Takashi Horiyama
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  3. Ryuhei Uehara
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Corresponding author

Correspondence to Ryuhei Uehara .

Editor information

Editors and Affiliations

  1. Indian Institute of Technology Guwahati, Guwahati, India

    Gautam K. Das

  2. Indian Institute of Technology Guwahati, Guwahati, India

    Partha S. Mandal

  3. Indian Statistical Institute, Kolkata, India

    Krishnendu Mukhopadhyaya

  4. Gunma University, Kiryu, Japan

    Shin-ichi Nakano

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Mizunashi, K., Horiyama, T., Uehara, R. (2019). Efficient Algorithm for Box Folding. In: Das, G., Mandal, P., Mukhopadhyaya, K., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2019. Lecture Notes in Computer Science(), vol 11355. Springer, Cham. https://doi.org/10.1007/978-3-030-10564-8_22

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  • DOI: https://doi.org/10.1007/978-3-030-10564-8_22

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10563-1

  • Online ISBN: 978-3-030-10564-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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