Abstract
(I) Given a cuttable polygon P drawn on a piece of planar material Q, we show how to cut P out of Q by a (small) segment saw with a total length no more than 2.5 times the optimal. We revise the algorithm of Demaine et al. (2001) so as to achieve this ratio.
(II) We prove that any collection \(\mathcal{R}\) of n disjoint axis-parallel rectangles is cuttable by at most 4n rays and present an algorithm that runs in O(n logn) time for computing a suitable cutting sequence. In particular the same result holds for cutting with an arbitrary segment saw (of any length).
(III) In contrast, we show that there exist collections of disjoint rectangles (in arbitrary orientations) that are uncuttable by a segment saw. We also present various uncuttable collections of disjoint polygons, including triangles.
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References
Ahmed, S.I., Islam, M.A., Hasan, M.: Cutting a cornered convex polygon out of a circle. Journal of Computers 5(1), 4–11 (2010)
Alon, N., Katchalski, M., Pulleyblank, W.R.: Cutting disjoint disks by straight lines. Discrete & Computational Geometry 4, 239–243 (1989)
Bereg, S., Dăescu, O., Jiang, M.: A PTAS for cutting out polygons with lines. Algorithmica 53, 157–171 (2009)
Bereg, S., Dumitrescu, A., Pach, J.: Sliding disks in the plane. International Journal of Computational Geometry & Applications 15(8), 373–387 (2008)
de Bruijn, N.G.: Aufgaben 17 and 18 (in Dutch). Nieuw Archief voor Wiskunde 2, 67 (1954)
Bhadury, J., Chandrasekaran, R.: Stock cutting to minimize cutting length. European Journal of Operations Research 88, 69–87 (1996)
Chandrasekaran, R., Dăescu, O., Luo, J.: Cutting out polygons, In. In: Proceedings of the 17th Canadian Conference on Computational Geometry, pp. 183–186 (2005)
Czyzowicz, J., Rivera-Campo, E., Urrutia, J.: Separating convex sets in the plane. Discrete & Computational Geometry 7, 189–195 (1992)
Czyzowicz, J., Rivera-Campo, E., Urrutia, J.: Separation of convex sets. Discrete Applied Mathematics 51(3), 325–328 (1994)
Dăescu, O., Luo, J.: Cutting out polygons with lines and rays. International Journal of Computational Geometry and Applications 16, 227–248 (2006)
Demaine, E.D., Demaine, M.L., Kaplan, C.S.: Polygons cuttable by a circular saw. Computational Geometry: Theory and Applications 20, 69–84 (2001)
Dumitrescu, A.: An approximation algorithm for cutting out convex polygons. Computational Geometry: Theory and Applications 29, 223–231 (2004)
Dumitrescu, A.: The cost of cutting out convex n-gons. Discrete Applied Mathematics 143, 353–358 (2004)
Dumitrescu, A., Hasan, M.: Cutting out polygons with a circular saw. International Journal of Computational Geometry and Applications 23(2), 127–139 (2013)
Dumitrescu, A., Jiang, M.: On reconfiguration of disks in the plane and other related problems. Computational Geometry: Theory and Applications 46(3), 191–202 (2013)
Fejes Tóth, L., Heppes, A.: Über stabile Körpersysteme (in German). Compositio Mathematica 15, 119–126 (1963)
Guibas, L., Yao, F.F.: On translating a set of rectangles. In: Computational Geometry. Advances in Computing Research, vol. 1, pp. 61–67. JAI Press, London (1983)
Hope, K., Katchalski, M.: Separating plane convex sets. Math. Scand. 66, 44–46 (1990)
O’Rourke, J.: Computational Geometry in C, 2nd edn. Cambridge University Press (1998)
Overmars, M.H., Welzl, E.: The complexity of cutting paper. In: Proceedings of the 1st Annual ACM Symposium on Computational Geometry, pp. 316–321 (1985)
Pach, J., Tardos, G.: Cutting glass. Discrete & Computational Geometry 24, 481–495 (2000)
Pach, J., Tardos, G.: Separating convex sets by straight lines. Discrete Mathematics 241(1-3), 427–433 (2001)
Tan, X.: Approximation Algorithms for Cutting Out Polygons with Lines and Rays. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 534–543. Springer, Heidelberg (2005)
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Dumitrescu, A., Ghosh, A., Hasan, M. (2015). On Collections of Polygons Cuttable with a Segment Saw. In: Ganguly, S., Krishnamurti, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2015. Lecture Notes in Computer Science, vol 8959. Springer, Cham. https://doi.org/10.1007/978-3-319-14974-5_6
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DOI: https://doi.org/10.1007/978-3-319-14974-5_6
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