Abstract
We describe an algorithm for computing the zero-th and the first Betti numbers of the union of n simply connected compact semi-algebraic sets in ℝk, where each such set is defined by a constant number of polynomials of constant degrees. The complexity of the algorithm is O(n 3). We also describe an implementation of this algorithm in the particular case of arrangements of ellipsoids in ℝ3 and describe some of our results.
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Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition. I. The basic algorithm. SIAM J. Comput. 13(4), 865–877 (1984)
Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition. II. An adjacency algorithm for the plane. SIAM J. Comput. 13(4), 878–889 (1984)
Arnon, D.S., Collins, G.E., McCallum, S.: An adjacency algorithm for cylindrical algebraic decompositions of three-dimensional space. J. Symbolic Comput. 5(1-2), 163–187 (1988)
Basu, S.: Computing the Betti numbers of arrangements via spectral sequences. J. Comput. System Sci. 67(2), 244–262 (2003); Special issue on STOC 2002 (Montreal, QC)
Basu, S.: Computing the first few Betti numbers of semi-algebraic sets in single exponential time (2005) (preprint)
Basu, S.: Computing the top Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time. In: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing (2005)
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2003)
Basu, S., Pollack, R., Roy, M.-F.: Computing the first Betti number and the connected components of semi-algebraic sets. In: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing (2005)
Brown, C.W.: An overview of QEPCAD B: a tool for real quantifier elimination and formula simplification. Journal of Japan Society for Symbolic and Algebraic Computation 10(1), 13–22 (2003)
Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.: Persistence barcodes for shapes. In: SGP 2004: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometric Processing, pp. 124–135. ACM Press, New York (2004)
Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. Theoretical Computer Science 84, 77–105 (1991)
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Edelsbrunner, H.: The union of balls and its dual shape. Discrete Comput. Geom. 13(3-4), 415–440 (1995)
Fortuna, E., Gianni, P., Parenti, P., Traverso, C.: Algorithms to compute the topology of orientable real algebraic surfaces. J. Symbolic Comput. 36(3-4), 343–364 (2003)
Halperin, D.: Arrangements. In: Handbook of discrete and computational geometry, CRC Press Ser. Discrete Math. Appl., pp. 389–412. CRC, Boca Raton (1997)
Koltun, V.: Sharp bounds for vertical decompositions of linear arrangements in four dimensions. Discrete Comput. Geom. 31(3), 435–460 (2004)
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Basu, S., Kettner, M. (2005). Computing the Betti Numbers of Arrangements in Practice. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_2
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DOI: https://doi.org/10.1007/11555964_2
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