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(MATH) In this paper, we consider the problem of computing the Betti numbers of an arrangement of $n$ compact semi-algebraic sets, $S_1,\ldots,S_n \subset \R^k$, where each $S_i$ 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 $\ell$-th Betti number, $\beta_\ell(\cup_i S_i), 0 \leq \ell \leq k-1$, using $O(n^{\ell+2})$ algebraic operations. Additionally, one has to perform linear algebra on matrices of size bounded by $O(n^{\ell+1})$. All previous algorithms for computing the Betti numbers of arrangements, triangulated the arrangement giving rise to a complex of size $O(n^{2^k})$ in the worst case. To our knowledge this is the first algorithm for computing $\beta_\ell(\cup_i S_i)$ that does not rely on such a global triangulation, and has a graded complexity which depends on $\ell$.