Abstract
In this paper, we present efficient geometric algorithms for the discrete constrained 1-D K-means clustering problem and extend our solutions to the continuous version of the problem. One key clustering constraint we consider is that the maximum difference in each cluster cannot be larger than a given threshold. These constrained 1-D K-means clustering problems appear in various applications, especially in intensity-modulated radiation therapy (IMRT). Our algorithms improve the efficiency and accuracy of the heuristic approaches used in clinical IMRT treatment planning.
This research was supported in part by NSF Grant CCF-0515203.
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References
Aggarwal, A., Klawe, M., Moran, S., Shor, P., Wilbur, R.: Geometric applications of a matrix-searching algorithm. Algorithmica 2, 195–208 (1987)
Aggarwal, A., Park, J.: Notes on searching in multidimensional monotone arrays. In: Proc. 29th IEEE Annual Symp. Foundations of Computer Science, pp. 497–512. IEEE Computer Society Press, Los Alamitos (1988)
Aggarwal, A., Schieber, B., Tokuyama, T.: Finding a minimum weight k-link path in graphs with Monge property and applications. In: Proc. 9th Annual ACM Symp. on Computational Geometry, pp. 189–197. ACM Press, New York (1993)
Bär, W., Alber, M., Nüsslin, F.: A variable fluence step clustering and segmentation algorithm for step and shoot IMRT. Physics in Medicine and Biology 46, 1997–2007 (2001)
Bruce, J.D.: Optimum quantization, Sc.D. Thesis, MIT (1964)
Craven, B.D.: Lebesque Measure & Integral, Pitman (1982)
Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review 41, 637–676 (1999)
Gray, R., Neuhoff, D.: Quantization. IEEE Transactions on Information Theory 44, 2325–2383 (1988)
Har-Peled, S., Sadri, B.: How fast is the k-means method? Algorithmica 41, 185–202 (2005)
Hassin, R., Tamir, A.: Improved complexity bounds for location problems on the real line. Operations Research Letters 10, 395–402 (1991)
Lloyd, S.P.: Least squares quantization in PCM, unpublished Bell Laboratories technical note, 1957. IEEE Transactions on Information Theory 28, 129–137 (1982)
Lu, F.S., Wise, G.L.: A further investigation of Max’s algorithm for optimum quantization. IEEE Transactions on Communications 33, 746–750 (1985)
Max, J.: Quantizing for minimum distortion. IEEE Transactions on Information Theory 6, 7–12 (1960)
Mohan, R., Liu, H., Turesson, I., Mackie, T., Parliament, M.: Intensity-modulated radiotherapy: Current status and issues of interest. International Journal of Radiation Oncology, Biology, Physics 53, 1088–1089 (2002)
Monge, G.: Déblai et remblai. Mémories de I’Académie des Sciences, Paris (1781)
Schieber, B.: Computing a minimum-weight k-link path in graphs with the concave Monge property. In: Proc. 6th Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 405–411. ACM Press, New York (1995)
Wu, X.: Optimal quantization by matrix searching. Journal of Algorithms 12, 663–673 (1991)
Wu, Y., Yan, D., Sharpe, M.B., Miller, B., Wong, J.W.: Implementing multiple static field delivery for intensity modulated beams. Medical Physics 28, 2188–2197 (2001)
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Chen, D.Z., Healy, M.A., Wang, C., Xu, B. (2007). Geometric Algorithms for the Constrained 1-D K-Means Clustering Problems and IMRT Applications. In: Preparata, F.P., Fang, Q. (eds) Frontiers in Algorithmics. FAW 2007. Lecture Notes in Computer Science, vol 4613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73814-5_1
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DOI: https://doi.org/10.1007/978-3-540-73814-5_1
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