A worst-case bound for topology computation of algebraic curves

摘要

Computing the topology of an algebraic plane curve e means computing a combinatorial graph that is isotopic to G and thus represents its topology in H2. We prove that, for a polynomial of degree n with integer coefficients bounded by 2~P, the topology of the induced curve can be computed with O(n~8p(n 4- p)) bit operations (0 indicates that we omit logarithmic factors). Our analysis improves the previous best known complexity bounds by a factor of n2. The improvement is based on new techniques to compute and refine isolating intervals for the real roots of polynomials, and on the consequent amortized analysis of the critical fibers of the algebraic curve.