A singly exponential stratification scheme for real semi-algebraic varieties and its applications
Theoretical Computer Science
Handbook of discrete and computational geometry
Different bounds on the different Betti numbers of semi-algebraic sets
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
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FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
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FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Computing the betti numbers of arrangements in practice
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
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In this paper, we consider the problem of computing the Betti numbers of an arrangement of n compact semi-algebraic sets, S1,...,Sn ⊂ Rk, where each Si is described using a constant number of polynomials with degrees bounded by a constant. Such arrangements are ubiquitous in computational geometry. We give an algorithm for computing lth Betti number, β(∪i=1n Si), 0 ≤ ℓ ≤ k - 1 using O(nℓ+2) algebraic operations. Additionally, one has to perform linear algebra on integer matrices of size bounded by O(nℓ+2). All previous algorithms for computing the Betti numbers of arrangements, triangulated the whole arrangement giving rise to a complex of size O(n2k) in the worst case. Thus, the complexity of computing the Betti numbers (other than the zeroth one) for these algorithms was O(n2k). To our knowledge this is the first algorithm for computing βℓ(∪i=1n Si) that does not rely on such a global triangulation, and has a graded complexity which depends on ℓ.