Abstract
We consider the problem of covering polygons, without any acute interior angles, with rectangles. The rectangles must lie entirely within the polygon and it is preferable to cover the polygon with as few rectangles as possible. Let P be an arbitrary hole-free input polygon, with n vertices, coverable by rectangles. Let μ(P) denote the minimum number of rectangles required to cover P. In this paper we show, by using new techniques, that it is possible to construct a covering within an O(α(n)) approximation factor in O(n+μ(P)) time, where α(n) is the extremely slowly growing inverse of Ackermann's function. This improves the Ω(n 0.49...) worst-case approximation factor in time O(n log n+μ(P)) known before.
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References
B.M. Chazelle, Computational Geometry and Convexity, Ph.D. Thesis, Department of Computer Science, Yale University, New Haven, CT, 1979.
J.C. Culberson and R.A. Reckhow, Covering Polygon is Hard, Journal of Algorithms, 17:2–44, 1994.
J. Snoeyink, C.A. Wang and F. Chin, Finding the Medial Axis of a Simple Polygon in Linear-Time, ISAAC '95, Cairns, Australia, 1995 (LNCS 1006 Springer-Verlag).
D. Franzblau, Performance Guarantees on a Sweep-Line Heuristic for Covering Rectilinear Polygons with Rectangles, SIAM J. Disc. Math., 2:307–321, 1989.
A. Hegediis, Algorithms for covering polygons by rectangles, Computer Aided Design, vol. 14, no. 5, 1982.
S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel Sequences and of Generalized Path Compression Schemes, Combinatorica, 6:151–177, 1986.
J.M. Keil, Minimally Covering a Horizontally Convex Orthogonal Polygon, Proceedings 2nd Annual Symposium Computational Geometry, pp.43–51, 1986.
D.G. Kirkpatrick, Efficient computation of continuous skeletons, 20th Annual IEEE Symposium on Foundation of Computer Science, 1979.
C. Levcopoulos, A Fast Heuristic for Covering Polygons by Rectangles, Proceedings FCT'85, Cottbus, GDR, 1985 (LNCS 199, Springer-Verlag).
C. Levcopoulos; Improved Bounds for Covering Ceneral Polygons with Rectangles, FST & TCS'87 Pune, India, 1987 (LNCS 287, Springer-Verlag).
C. Levcopoulos and J. Gudmundsson, Close Approximations of Minimum Rectangular Coverings, FST & TCS'96 Hyderabad, India, 1996 (LNCS 1180, Springer).
C. Levcopoulos and J. Gudmundsson, A Linear-Time Heuristic for Minimum Rectangular Coverings, LU-CS-TR:96-170, Dept. of Comp. Sci., Lund University, 1996.
W.J. Masek, Some NP-complete set covering problems, manuscript, MIT, 1979.
J. O'Rourke, The decidability of Covering by Convex Polygons, Report JHU-EE 82-1, Dept. Elect. Engrg. Comp. Sci., Johns Hopkins University, Baltimore, 1982.
J. O'Rourke and K.J. Supowit, Some NP-hard Polygon Decomposition Problems, IEEE Transactions on Information Theory, vol. IT-29, pp.181–190, 1983.
F.P. Preparata and M.I. Shamos, Computational Geometry, New York, Springer-Verlag, 1985.
A. Wiernik, Planar Realization of Nonlinear Davenport-Schinzel Sequences by Segments, 27th IEEE Symposium on Computer Science, 1986.
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© 1997 Springer-Verlag Berlin Heidelberg
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Levcopoulos, C., Gudmundsson, J. (1997). A linear-time heuristic for minimum rectangular coverings (Extended abstract). In: Chlebus, B.S., Czaja, L. (eds) Fundamentals of Computation Theory. FCT 1997. Lecture Notes in Computer Science, vol 1279. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0036193
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DOI: https://doi.org/10.1007/BFb0036193
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