Abstract
We study the problem of determining an optimal bipartition {A,B} of a set X of n points in ℝ2 that minimizes the sum of the sample variances of A and B, under the size constraints |A| = k and |B| = n − k. We present two algorithms for such a problem. The first one computes the solution in \(O(n\sqrt[3]{k}\log^2 n)\) time by using known results on convex-hulls and k-sets. The second algorithm, for an input X ⊂ ℝ2 of size n, solves the problem for all \(k=1,2,\ldots,\lfloor n/2\rfloor\) and works in O(n 2 logn) time.
This research has been supported by project PRIN #H41J12000190001 “Automata and formal languages: mathematical and applicative aspects”.
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References
Aloise, D., Deshpande, A., Hansen, P., Popat, P.: NP-hardness of Euclidean sum-of-squares clustering. Machine Learning 75, 245–249 (2009)
Basu, S., Davidson, I., Wagstaff, K.: Constrained Clustering: Advances in Algorithms, Theory, and Applications. Chapman and Hall/CRC (2008)
Bertoni, A., Goldwurm, M., Lin, J., Pini, L.: Size-constrained 2-Clustering in the Plane with Manhattan Distance. In: Proc. 15th Italian Conference on Theoretical Computer Science. CEUR Workshop Proceedings, vol. 1231, pp. 33–44. CEUR-WS.org (2014) ISSN 1613-0073
Bertoni, A., Goldwurm, M., Lin, J., Saccà, F.: Size Constrained Distance Clustering: Separation Properties and Some Complexity Results. Fundamenta Informaticae 115(1), 125–139 (2012)
Bishop, C.: Pattern Recognition and Machine Learning. Springer (2006)
Dasgupta, S.: The hardness of k-means clustering. Technical Report CS2007-0890, Department of Computer Science and Engineering, University of California, San Diego (2007)
Dey, T.: Improved Bounds for Planar k-Sets and Related Problems. Discrete & Computational Geometry 19(3), 373–382 (1998)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. EATCS monographs on theoretical computer science. Springer (1987)
Erdős, P., Lovász, L., Simmons, A., Straus, E.G.: Dissection graphs of planar point sets. In: A Survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), pp. 139–149. North-Holland, Amsterdam (1973)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edn. Springer (2009)
Inaba, M., Katoh, N., Imai, H.: Applications of weighted voronoi diagrams and randomization to variance-based k-clustering (extended abstract). In: Proceedings of the Tenth Annual Symposium on Computational Geometry, SCG 1994, USA, pp. 332–339 (1994)
Lin. J.: Exact algorithms for size constrained clustering. PhD Thesis, Dottorato di ricerca in Matematica, Statistica e Scienze computationali, Università degli Studi di Milano. Ledizioni Publishing (2013)
MacQueen, J.B.: Some method for the classification and analysis of multivariate observations. In: Proceedings of the 5th Berkeley Symposium on Mathematical Structures, pp. 281–297 (1967)
Mahajan, M., Nimbhorkar, P., Varadarajan, K.: The planar k-means problem is NP-hard. Theoretical Computer Science 442, 13–21 (2012)
Novick, B.: Norm statistics and the complexity of clustering problems. Discrete Applied Mathematics 157, 1831–1839 (2009)
Overmars, M.H., van Leeuwen, J.: Maintenance of configurations in the plane. J. Comput. Syst. Sci. 23(2), 166–204 (1981)
Preparata, F., Shamos, M.: Computational geometry: an introduction. Texts and monographs in computer science. Springer (1985)
Theodoridis, S., Koutroumbas, K.: Pattern Recognition. Academic Press, Elsevier (2009)
Vattani, A.: K-means requires exponentially many iterations even in the plane. In: Proceedings of the 25th Symposium on Computational Geometry (SoCG) (2009)
Wagstaff, K., Cardie, C.: Clustering with instance-level constraints. In: Proc. of the 17th Intl. Conf. on Machine Learning, pp. 1103–1110 (2000)
Zhu, S., Wang, D., Li, T.: Data clustering with size constraints. Knowledge-Based Systems 23(8), 883–889 (2010)
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Bertoni, A., Goldwurm, M., Lin, J. (2015). Exact Algorithms for 2-Clustering with Size Constraints in the Euclidean Plane. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, JJ., Wattenhofer, R. (eds) SOFSEM 2015: Theory and Practice of Computer Science. SOFSEM 2015. Lecture Notes in Computer Science, vol 8939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46078-8_11
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