A worst-case bound for topology computation of algebraic curves
| Autor: | Michael Sagraloff, Michael Kerber |
|---|---|
| Rok vydání: | 2012 |
| Předmět: |
Computer Science - Symbolic Computation
FOS: Computer and information sciences Circular algebraic curve Discrete mathematics Polynomial Stable curve Algebra and Number Theory Plane curve Lower limit topology Symbolic Computation (cs.SC) Amortized analysis Combinatorics Topological combinatorics Computational Mathematics Algebraic curve ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION General topology Topology computation Complexity analysis MathematicsofComputing_DISCRETEMATHEMATICS Mathematics |
| Zdroj: | Journal of Symbolic Computation. 47(3):239-258 |
| ISSN: | 0747-7171 |
| DOI: | 10.1016/j.jsc.2011.11.001 |
| Popis: | Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with coefficients bounded by $2^\rho$, the topology of the induced curve can be computed with $\tilde{O}(n^8(n+\rho^2))$ bit operations deterministically, and with $\tilde{O}(n^8\rho^2)$ bit operations with a randomized algorithm in expectation. Our analysis improves 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 by the consequent amortized analysis of the critical fibers of the algebraic curve. |
| Databáze: | OpenAIRE |
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