An algorithm for generalized point location and its applications
Journal of Symbolic Computation
Handbook of theoretical computer science (vol. A)
Point location in arrangements of hyperplanes
Information and Computation
A Polynomial Linear Search Algorithm for the n-Dimensional Knapsack Problem
Journal of the ACM (JACM)
Topological complexity of the range searching
Journal of Complexity
Constraint Databases
A note on point location in arrangements of hyperplanes
Information Processing Letters
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
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Let F"1,...,F"s@?R[X"1,...,X"n] be polynomials of degree at most d, and suppose that F"1,...,F"s are represented by a division free arithmetic circuit of non-scalar complexity size L. Let A be the arrangement of R^n defined by F"1,...,F"s. For any point x@?R^n, we consider the task of determining the signs of the values F"1(x),...,F"s(x) (sign condition query) and the task of determining the connected component of A to which x belongs (point location query). By an extremely simple reduction to the well-known case where the polynomials F"1,...,F"s are affine linear (i.e., polynomials of degree one), we show first that there exists a database of (possibly enormous) size s^O^(^L^+^n^) which allows the evaluation of the sign condition query using only (Ln)^O^(^1^)log(s) arithmetic operations. The key point of this paper is the proof that this upper bound is almost optimal. By the way, we show that the point location query can be evaluated using d^O^(^n^)log(s) arithmetic operations. Based on a different argument, analogous complexity upper-bounds are exhibited with respect to the bit-model in case that F"1,...,F"s belong to Z[X"1,...,X"n] and satisfy a certain natural genericity condition. Mutatis mutandis our upper-bound results may be applied to the sparse and dense representations of F"1,...,F"s.