A worst-case bound for topology computation of algebraic curves
Computing the topology of an algebraic plane curve C means computing a combinatorial graph that is isotopic to C and thus represents its topology in R-2. We prove that, for a polynomial of degree n with integer coefficients bounded by 2(rho), the topology of the induced curve can be computed with O(n(8) rho(n + rho)) bit operations ((O) over tilde indicates that we omit logarithmic factors). Our analysis improves the previous best known complexity bounds by a factor of n(2). 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. (C) 2011 Elsevier Ltd. All rights reserved.
topology computation algebraic curve amortized analysis complexity analysis