Abstract
We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake’s problem is ”length-tractable”: if the snake is ”fat”, i.e., its length/width ratio is small, the shortest path can be computed in polynomial time.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIJCOMP 29(3), 912–953 (2000)
Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Computing Surveys 30, 412–458 (1998)
Alterovitz, R., Branicky, M.S., Goldberg, K.Y.: Motion planning under uncertainty for image-guided medical needle steering. Int. J. Rob. Res. 27(11-12), 1361–1374 (2008)
Arkin, E.M., Mitchell, J.S.B., Polishchuk, V.: Maximum thick paths in static and dynamic environments. Comp. Geom. Theory and Appl. 43(3), 279–294 (2010)
Asano, T., Kirkpatrick, D., Yap, C.K.: d 1-optimal motion for a rod. In: SoCG 1996 (1996)
Asano, T., Kirkpatrick, D., Yap, C.K.: Minimizing the trace length of a rod endpoint in the presence of polygonal obstacles is NP-hard. In: CCCG 2003 (2003)
Barcia, J., Diaz-Banez, J., Gomez, F., Ventura, I.: The anchored Voronoi diagram: static and dynamic versions and applications. In: EuroCG 2003 (2003)
Bereg, S., Kirkpatrick, D.: Curvature-bounded traversals of narrow corridors. In: SoCG 2005 (2005)
Chew, L.P.: Planning the shortest path for a disc in O(n 2 logn) time. In: SoCG 1985 (1985)
Chowdhury, D.: Molecular motors: Design, mechanism, and control. Computing in Science and Engineering 10, 70–77 (2008)
Cook IV, A.F., Wenk, C., Daescu, O., Bitner, S., Cheung, Y.K., Kurdia, A.: Visiting a sequence of points with a bevel-tip needle. In: LATIN 2010 (2010)
Efrat, A., Sharir, M.: A near-linear algorithm for the planar segment center problem. Discrete & Computational Geometry 16, 239–257 (1996)
Hu, T., Kahng, A., Robins, G.: Optimal robust path planning in general environments. IEEE Transactions on Robotics and Automation 9, 775–784 (1993)
Lee, J.Y., Choset, H.: Sensor-based planning for a rod-shaped robot in three dimensions: Piecewise retracts of R3 x S2. Int. J. Rob. Res. 24(5), 343–383 (2005)
Lichtenstein, D.: Planar formulae and their uses. SIJCOMP 11(2), 329–343 (1982)
Pach, J., Tardos, G.: Forbidden patterns and unit distances. In: SoCG 2005 (2005)
Sifrony, S., Sharir, M.: A new efficient motion-planning algorithm for a rod in two-dimensional polygonal space. Algorithmica 2, 367–402 (1987)
Veigel, C., Coluccio, L.M., Jontes, J.D., Sparrow, J.C., Milligan, R.A., Molloy, J.: The motor protein myosin-I produces its working stroke in two steps. Nature 398, 530–533 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kostitsyna, I., Polishchuk, V. (2010). Simple Wriggling Is Hard Unless You Are a Fat Hippo. In: Boldi, P., Gargano, L. (eds) Fun with Algorithms. FUN 2010. Lecture Notes in Computer Science, vol 6099. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13122-6_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-13122-6_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13121-9
Online ISBN: 978-3-642-13122-6
eBook Packages: Computer ScienceComputer Science (R0)