Abstract
This paper generalizes the multidimensional searching scheme of Dobkin and Lipton [SIAM J. Comput. 5(2), pp. 181–186, 1976] for the case of arbitrary (as opposed to linear) real algebraic varieties. Let d,r be two positive constants and let P 1,...,P n be n rational r-variate polynomials of degree ≤d. Our main result is an \(O(n^{2^{r + 6} } )\) data structure for computing the predicate [∃i (1≤i≤n)|P i (x)=0] in O(log n) time, for any x∈E r. The method is intimately based on a decomposition technique due to Collins [Proc. 2nd GI Conf. on Automata Theory and Formal Languages, pp. 134–183, 1975]. The algorithm can be used to solve problems in computational geometry via a locus approach. We illustrate this point by deriving an o(n 2) algorithm for computing the time at which the convex hull of n (algebraically) moving points in E 2 reaches a steady state.
This research was supported in part by NSF grants MCS 83-03925 and the Office of Naval Research and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146 and ARPA Order No. 4786.
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Chazelle, B. (1985). Fast searching in a real algebraic manifold with applications to geometric complexity. In: Ehrig, H., Floyd, C., Nivat, M., Thatcher, J. (eds) Mathematical Foundations of Software Development. CAAP 1985. Lecture Notes in Computer Science, vol 185. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-15198-2_9
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DOI: https://doi.org/10.1007/3-540-15198-2_9
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