A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials (1995)
| Venue: | B. Caviness and J. Johnson Eds., Springer--Verlag |
| Citations: | 35 - 8 self |
BibTeX
@INPROCEEDINGS{Basu95anew,
author = {Saugata Basu and Richard Pollack and Marie-Françoise Roy},
title = {A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials},
booktitle = {B. Caviness and J. Johnson Eds., Springer--Verlag},
year = {1995},
publisher = {Springer-Verlag}
}
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Abstract
We consider s polynomials P 1 ; : : : ; P s in k ! s variables with coefficients in an ordered domain A contained in a real closed field R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each non-empty sign condition over P 1 ; : : : ; P s . The output is the set of points together with the sign condition at each point. The algorithm uses s(s=k) k d O(k) arithmetic operations in A. The algorithm is nearly optimal in the sense that the size of the output can be as large as s(O(sd=k)) k . Previous algorithms of Canny and Renegar used (sd) O(k) operations [5, 7, 8, 15]. We use either these algorithms in the case s = 1 as a subroutine in our algorithm. As a bonus, our algorithm yields an independent proof of the bound on the number of connected components in all non-empty sign conditions ([14]) and also yields an independent proof of a theorem of Warren 1 Courant Institute of Mathematical Sciences, New York University, N...
