Plantinga-Vegter algorithm takes average polynomial time Felipe Cucker Alperen A. Ergur¨ Josue Tonelli-Cueto City University of Hong Kong Technische Universitat¨ Berlin Technische Universitat¨ Berlin Dept. of Mathematics Institut fur¨ Mathematik Institut fur¨ Mathematik Tat Chee Avenue Str. des 17. Juni 136 Str. des 17. Juni 136 Kowloon Tong, Hong Kong Berlin 10623, Germany Berlin 10623, Germany [email protected] [email protected] [email protected] ABSTRACT and dominate, the cost of the computation. e paper how- We exhibit a condition-based analysis of the adaptive subdi- ever, contained no complexity analysis and not even a formal vision algorithm due to Plantinga and Vegter. e first com- seing fixing either the kind of functions implicitly defining plexity analysis of the PV Algorithm is due to Burr, Gao and the considered curves and surfaces or the arithmetic used. 4 Tsigaridas who proved a 2τd log d worst-case cost bound An article doing so was published in 2017 by Burr, Gao and O for degree d plane curves with maximum coefficient bit-size τ. Tsigaridas [6]. e functions this article deals with are poly- is exponential bound, it was observed, is in stark contrast nomials with integer coefficients and smooth zero set. Consis- with the good performance of the algorithm in practice. More tently with this choice of data, the arithmetic is infinite preci- in line with this performance, we show that, with respect to sion. e main result in the paper is a worst-case complexity a broad family of measures, the expected time complexity of analysis for the number of cubes in the description of the ap- the PV Algorithm is bounded by O d7 for real, degree d, plane proximation which, as we just mentioned, dominates the cost ( ) curves. We also exhibit a smoothed analysis of the PV Algo- of the computation. e bounds proved for this quantity are rithm that yields similar complexity estimates. To obtain these shown to be optimal. Yet, these bounds are exponential (both results we combine robust probabilistic techniques coming from in the degree of the input polynomial and in its logarithmic geometric functional analysis with condition numbers and the height), a fact that motivates the following comment at the continuous amortization paradigm introduced by Burr, Krah- end of the paper mer and Yap. We hope this will motivate a fruitful exchange Even though our bounds are optimal, in prac- of ideas between the different approaches to numerical com- tice, these are quite pessimistic [...] putation. e authors further observe that, following from their Propo- sition 5.2 (see eorem 6.3 below) an instance-based analysis CCS CONCEPTS of the algorithm (i.e., one yielding a cost that depends on the •Mathematics of computing Computations on polyno- input at hand) could be derived from the evaluation of a cer- → mials; Interval arithmetic; •eory of computation De- tain integral. And they conclude their paper by writing → sign and analysis of algorithms; Computational geometry; Since the complexity of the algorithm can be exponential in the inputs [size], the integral KEYWORDS must be described in terms of additional geo- computational algebraic geometry, numerical methods, adap- metric and intrinsic parameters. tive subdivision methods, isotopy of curves, complexity A number of features in this state of affairs suggest that a condition-based approach to the analysis of our quantity of ACKNOWLEDGMENTS interest could be useful. To begin with, the fact that a condi- arXiv:1901.09234v2 [cs.CG] 18 Apr 2019 tion number is a perfect fit for the notion of an “additional We thank Michael Burr for useful discussions. is work was geometric and intrinsic parameter.” To which we may add the supported by the Einstein Foundation Berlin. FC was partially fact that the obvious set of ill-posed inputs, the set of poly- supported by a GRF grant from the Research Grants Council nomials having a non-smooth zero set, is precisely the set of of the Hong Kong SAR (project number CityU 11202017). data which are not allowed as inputs in [6]. Of course, such a condition-based analysis would drop the assumption of inte- 1 INTRODUCTION ger coefficients and replace it by that of real coefficients but In 2004 Plantinga and Vegter proposed an algorithm for com- this is a common practice for numerical algorithms and, as we puting a regularly isotopic piecewise linear approximation of will see, it pays off in our case as it yields small (i.e., polyno- a curve or surface [16]. eir algorithm relied on a subdivi- mial) average complexity bounds for a large class of probabil- sion method enhanced with interval arithmetic to certificate ity measures. the procedure (i.e., ensure its correctness) and in Section 7 of Although our approach follows the condition-based ideas their paper they provided some examples with their approx- of, e.g., [1, 8, 9, 12, 18], the complexity analysis in this paper imations and the record of how many cubes (squares in the would have been impossible without the continuous amortiza- case of plane curves) were in the description of these approx- tion technique developed in the exact numerical context [4, 5]. imations. is number of cubes appears to be proportional to, We hope that this merging of techniques will start a fruitful , , Felipe Cucker, Alperen A. Ergur,¨ and Josue Tonelli-Cueto exchange of ideas between different approaches to continuous 2 THE PV ALGORITHM computation. Given a real smooth hypersurface in Rn described implicitly by a map f : Rn R and a region a, a n, the PV Algo- → [− ] rithm constructs a piecewise-linear approximation of the in- tersection of its zero set VR f with a,a n isotopic to this ( ) [− ] 1.1 Notation intersection inside a,a n. [− ] m Let m be the set of m-cubes of R . Recall that an interval roughout the paper, we will assume some familiarity with I approximation of a function F : Rm Rm′ is a map F : the basics of differential geometry and with the sphere Sn as → [ ] m such that for all J m, F J F J (c.f. [17]). a Riemannian manifold. For scalar smooth maps f : Rm R, I → Im′ ∈ I ( ) ⊆ [ ]( ) m m → We notice that if we see J as error bounds for the midpoint we will write the tangent map at x R as ∂x f : R R ∈ → m J , then F J is nothing more than error bounds for F m J . when we want to emphasize it as a linear map and as ∂f : ( ) [ ]( ) ( ( )) Assume that we have interval approximations of both f Rm Rm, x ∂f x , when we want to emphasize it as a → 7→ ( ) and its tangent map ∂f or, more generally, of hf and h′∂f for smooth function. For general smooth maps F : , we n M → N some positive maps h,h′ : R 0, . e PV Algorithm will just write ∂x F :Tx Tx as the tangent map. → ( ∞) M → N on a,a n will subdivide this region into smaller and smaller In what follows, , will denote the set of real polyno- [− ] Pn d cubes until the condition mials in n variables with degree at most d, n,d the set of + H homogeneous real polynomials in n 1 variables of degree d, Cf J : either 0 < hf I or 0 < h′∂f J , h′∂f J and and , will denote the usual norm and inner prod- ( ) [ ]( ) h [ ]( ) [ ]( )i k k h i uct in Rm as well as the Weyl norm and inner product in m is satisfied in each of the n-cubes J of the obtained subdivi- Pn,d , n m h sion of a a . In Section 4, we will be more precise on the and . Given a polynomial f , , f , will be [− ] Hn,d ∈ Pn d ∈ Hn d assumptions on our interval approximations and the functions its homogenization and ∂f the polynomial map given by its h and h′ that we will use. partial derivatives. For details about the concrete definition of each of these notions, see Section 4. Additionally, VR f and ( ) VC f will be, respectively, the real and complex zero sets of Algorithm 2.1: Subdivision routine of PV Algorithm ( ) f . Input: a 0, and f : Rn R n ∈( ∞) → We will denote by n the set of n-cubes of R and, for a with interval approximations hf and h ∂f I [ ] [ ′ ] given J n, m J will be its middle point, w J its width, and ∈I ( ) ( ) vol J = w J n its volume. Starting with the trivial subdivision := a,a n , ( ) S n {[− ] } Also, P A will denote the probability of the event A, Ex Kg x repeatedly subdivide each J into 2 cubes until the ( ) ∈ ( ) ∈S the expectation of g x when x is sampled uniformly from K condition C J holds for all J ( ) f ( ) ∈S and Eyg x the expectation ofg y with respect to a previously ( ) ( ) n specified probability distribution of y. Output: Subdivision n of a,a S⊆I [− ] Regarding complexity parameters, n will be the number of such that for all J , Cf J is true = n+d ∈S ( ) variables, d the degree bound, and N n the dimension of , . Pn d Finally, ln will denote the natural logarithm and log the log- e procedure in Algorithm 2.1 is only the subdivision rou- arithm in base 2. tine of the PV Algorithm but it dominates its complexity in the sense that the remaining computations do not add to the final cost estimates in Landau notation. Moreover, these addi- tional computations have been implemented only for n 3.