IWCIA 2006: Combinatorial Image Analysis pp 415-429

# Shortest Paths in a Cuboidal World

• Fajie Li
• Reinhard Klette
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4040)

## Abstract

Since 1987 it is known that the Euclidean shortest path problem is NP-hard. However, if the 3D world is subdivided into cubes, all of the same size, defining obstacles or possible spaces to move in, then the Euclidean shortest path problem has a linear-time solution, if all spaces to move in form a simple cube-curve. The shortest path through a simple cube-curve in the orthogonal 3D grid is a minimum-length polygonal curve (MLP for short). So far only one general and linear (only with respect to measured run times) algorithm, called the rubberband algorithm, was known for an approximative calculation of an MLP. The algorithm is basically defined by moves of vertices along critical edges (i.e., edges in three cubes of the given cube-curve). A proof, that this algorithm always converges to the correct MLP, and if so, then always (provable) in linear time, was still an open problem so far (the authors had successfully treated only a very special case of simple cube-curves before). In a previous paper, the authors also showed that the original rubberband algorithm required a (minor) correction.

This paper finally answers the open problem: by a further modification of the corrected rubberband algorithm, it turns into a provable linear-time algorithm for calculating the MLP of any simple cube-curve.

The paper also presents an alternative provable linear-time algorithm for the same task, which is based on moving vertices within faces of cubes.

For a disticntion, we call the modified original algorithm now the edge-based rubberband algorithm, and the second algorithm is the face-based rubberband algorithm; the time complexity of both is in $${\cal O}(m)$$, where m is the number of critical edges of the given simple cube-curve.

## Keywords

Short Path Apply Procedure Consecutive Vertex Simple Cube Polygonal Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Bülow, T., Klette, R.: Digital curves in 3D space and a linear-time length estimation algorithm. IEEE Trans. Pattern Analysis Machine Intell. 24, 962–970 (2002)
2. 2.
Canny, J., Reif, J.H.: New lower bound techniques for robot motion planning problems. In: Proc. IEEE Conf. Foundations Computer Science, pp. 49–60 (1987)Google Scholar
3. 3.
Choi, J., Sellen, J., Yap, C.-K.: Approximate Euclidean shortest path in 3-space. In: Proc. ACM Conf. Computational Geometry, pp. 41–48. ACM Press, New York (1994)Google Scholar
4. 4.
Coeurjolly, D., Debled-Rennesson, I., Teytaud, O.: Segmentation and length estimation of 3D discrete curves. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Dagstuhl Seminar 2000. LNCS, vol. 2243, pp. 299–317. Springer, Heidelberg (2002)
5. 5.
Dror, M., Efrat, A., Lubiw, A., Mitchell, J.: Touring a sequence of polygons. In: Proc. STOC, pp. 473–482 (2003)Google Scholar
6. 6.
Ficarra, E., Benini, L., Macii, E., Zuccheri, G.: Automated DNA fragments recognition and sizing through AFM image processing. IEEE Trans. Inf. Technol. Biomed. 9, 508–517 (2005)
7. 7.
Jonas, A., Kiryati, N.: Length estimation in 3-D using cube quantization. J. Math. Imaging and Vision 8, 215–238 (1998)
8. 8.
Karavelas, M.I., Guibas, L.J.: Static and kinetic geometric spanners with applications. In: Proc. ACM-SIAM Symp. Discrete Algorithms, pp. 168–176 (2001)Google Scholar
9. 9.
Klette, R., Bülow, T.: Critical edges in simple cube-curves. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 467–478. Springer, Heidelberg (2000)
10. 10.
Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)
11. 11.
Li, F., Klette, R.: Minimum-length polygon of a simple cube-curve in 3D space. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 502–511. Springer, Heidelberg (2004)
12. 12.
Li, F., Klette, R.: The class of simple cube-curves whose MLPs cannot have vertices at grid points. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 183–194. Springer, Heidelberg (2005)
13. 13.
Li, F., Klette, R.: Minimum-Length Polygons of First-Class Simple Cube-Curve. In: Gagalowicz, A., Philips, W. (eds.) CAIP 2005. LNCS, vol. 3691, pp. 321–329. Springer, Heidelberg (2005)
14. 14.
Li, F., Klette, R.: Analysis of the rubberband algorithm. Technical Report CITR-TR-175, Computer Science Department, The University of Auckland, Auckland, New Zealand (2006), www.citr.auckland.ac.nz
15. 15.
Li, T.-Y., Chen, P.-F., Huang, P.-Z.: Motion for humanoid walking in a layered environment. In: Proc. Conf. Robotics Automation, vol. 3, pp. 3421–3427 (2003)Google Scholar
16. 16.
Luo, H., Eleftheriadis, A.: Rubberband: an improved graph search algorithm for interactive object segmentation. In: Proc. Int. Conf. Image Processing, vol. 1, pp. 101–104 (2002)Google Scholar
17. 17.
Sklansky, J., Kibler, D.F.: A theory of nonuniformly digitized binary pictures. IEEE Trans. Systems Man Cybernetics 6, 637–647 (1976)
18. 18.
Sloboda, F., Zaťko, B., Klette, R.: On the topology of grid continua. In: Proc. Vision Geometry, SPIE, vol. 3454 (1998)Google Scholar
19. 19.
Sloboda, F., Zaťko, B., Stoer, J.: On approximation of planar one-dimensional grid continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp. 113–160. Springer, Singapore (1998)Google Scholar
20. 20.
Sun, C., Pallottino, S.: Circular shortest path on regular grids. CSIRO Math. Information Sciences, CMIS Report No. 01/76, Australia (2001)Google Scholar
21. 21.
Talbot, M.: A dynamical programming solution for shortest path itineraries in robotics. Electr. J. Undergrad. Math. 9, 21–35 (2004)Google Scholar
22. 22.
Wolber, R., Stäb, F., Max, H., Wehmeyer, A., Hadshiew, I., Wenck, H., Rippke, F., Wittern, K.: Alpha-Glucosylrutin: Ein hochwirksams Flavonoid zum Schutz vor oxidativem Stress. J. German Society Dermatology 2, 580–587 (2004)Google Scholar