Abstract
In this paper, we discuss the problem of computing all the integral sequences obtained by rounding an input real valued sequence such that the discrepancy between the input sequence and each output integral sequence is less than one. We show that the number of such roundings is n + 1 if we consider the discrepancy with respect to the set of all subintervals, and give an efficient algorithm to report all of them. Then, we give an optimal method to construct a compact graph to represent the set of global roundings satisfying a weaker discrepancy condition.
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References
T. Asano, T. Matsui, and T. Tokuyama: “On the complexity of the optimal rounding problems of sequences and matrices,” Proceedings of SWAT00, LNCS1851 (2000), pp. 476–489.
T. Asano et al, “Digital Halftoning: Formulation as a combinatorial optimization problem and approximation algorithms based on network flow”, working paper, 2000 November.
J. Beck and V. T. Sös, Discrepancy Theory, in Handbook of Combinatorics Volume II, (ed. R. Graham, M. Grötschel, and L Lovász) 1995, Elsevier.
R. W. Floyd and L. Steinberg: “An adaptive algorithm for spatial gray scale,” SID 75 Digest, Society for Information Display (1975), pp. 36–37.
H. N. Gabow and R. E. Tarjan: “Faster scaling algorithms for network problems,” SIAM J. Comp., 18 (1989), pp. 1013–1036.
D. Gusfield, Algorithms on Strings, Trees and Sequences: Computer science and computational biology, Cambridge U.P. 1997.
W. J. Hsu, “Fibonacci cubes-a new interconnection topology,” IEEE Trans. Parallel and Distributed Systems, 4 (1993) pp.2–12.
D. E. Knuth: “Digital halftones by dot diffusion,” ACM Trans. Graphics, 6-4 (1987), pp. 245–273.
J. O. Limb: “Design of dither waveforms for quantized visual signals,” Bell Syst. Tech. J., 48-7 (1969), pp. 2555–2582.
B. Lippel and M. Kurland: “The effect of dither on luminance quantization of pictures,” IEEE Trans. Commun. Tech., COM-19 (1971), pp.879–888.
V. Rödl and P. Winkler: “Concerning a matrix approximation problem” Crux Mathmaticorum, 1990, pp. 76–79.
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© 2001 Springer-Verlag Berlin Heidelberg
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Sadakane, K., Takki-Chebihi, N., Tokuyama, T. (2001). Combinatorics and Algorithms on Low-Discrepancy Roundings of a Real Sequence. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_14
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DOI: https://doi.org/10.1007/3-540-48224-5_14
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