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Chee Yap

New York University, Computer Science, Faculty Member

Massachusetts Institute of Technology (MIT), EECS, Undergraduate

Yale University, Computer Science, Graduate Student

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Professor of Computer Science, Courant Institute of Mathematical Sciences, NYU. I work in theoretical computer science, especially on algorithms and complexity analysis, computer algebra and certified numerical algorithms, computational geometry, algorithmic robotics.
Supervisors: Richard L. Lipton
Phone: 212-9983115
Address: 251 Mercer Street, New York, NY 10012

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Papers by Chee Yap

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from... more This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows to transform the considered differential systems to the normal formal of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms that determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method. • Computing methodologies → Symbolic and algebraic manipulation; • Symbolic and algebraic algorithms → Symbolic calculus algorithms.

Novel Range Functions via Taylor Expansions and Recursive Lagrange Interpolation with Application to Real Root Isolation

Range functions are an important tool for interval computations, and they can be employed for the... more Range functions are an important tool for interval computations, and they can be employed for the problem of root isolation. In this paper, we first introduce two new classes of range functions for real functions. They are based on the remainder form by Cornelius and Lohner and provide different improvements for the remainder part of this form. On the one hand, we use centered Taylor expansions to derive a generalization of the classical Taylor form with higher than quadratic convergence. On the other hand, we propose a recursive interpolation procedure, in particular based on quadratic Lagrange interpolation, leading to recursive Lagrange forms with cubic and quartic convergence. We then use these forms for isolating the real roots of square-free polynomials with the algorithm Eval, a relatively recent algorithm that has been shown to be effective and practical. Finally, we compare the performance of our new range functions against the standard Taylor form. Range functions are often compared in isolation; in contrast, our holistic comparison is based on their performance in an application. Specifically, Eval can exploit features of our recursive Lagrange forms which are not found in range functions based on Taylor expansion. Experimentally, this yields at least a twofold speedup in Eval.

The D-plus Discriminant and Complexity of Root Clustering

ArXiv, 2021

Let $p(x)$ be an integer polynomial with $m\ge 2$ distinct roots $\rho_1,\ldots,\rho_m$ whose mul... more Let $p(x)$ be an integer polynomial with $m\ge 2$ distinct roots $\rho_1,\ldots,\rho_m$ whose multiplicities are $\boldsymbol{\mu}=(\mu_1,\ldots,\mu_m)$. We define the D-plus discriminant of $p(x)$ to be $D^+(p):= \prod_{1\le i<j\le m}(\rho_i-\rho_j)^{\mu_i+\mu_j}$. We first prove a conjecture that $D^+(p)$ is a $\boldsymbol{\mu}$-symmetric function of its roots $\rho_1,\ldots,\rho_m$. Our main result gives an explicit formula for $D^+(p)$, as a rational function of its coefficients. Our proof is ideal-theoretic, based on re-casting the classic Poisson resultant as the"symbolic Poisson formula". The D-plus discriminant first arose in the complexity analysis of a root clustering algorithm from Becker et al. (ISSAC 2016). The bit-complexity of this algorithm is proportional to a quantity $\log(|D^+(p)|^{-1})$. As an application of our main result, we give an explicit upper bound on this quantity in terms of the degree of $p$ and its leading coefficient.

Soft Subdivision Search in Motion Planning, II: Axiomatics

Frontiers in Algorithmics, 2015

We propose to design motion planning algorithms with a strong form of resolution completeness, ca... more We propose to design motion planning algorithms with a strong form of resolution completeness, called resolution-exactness. Such planners can be implemented using soft predicates within the subdivision paradigm. The advantage of softness is that we avoid the Zero problem and other issues of exact computation. Soft Subdivision Search (SSS) is an algorithmic framework for such planners. There are many parallels between our framework and the well-known Probabilistic Road Map (PRM) framework. Both frameworks lead to algorithms that are practical, flexible, extensible, with adaptive and local complexity. Our several recent papers have demonstrated these favorable properties on various non-trivial motion planning problems. In this paper, we provide a general axiomatic theory underlying these results. We also address the issue of subdivision in non-Euclidean configuration spaces, and how exact algorithms can be recovered using soft methods.

A simple but exact and efficient algorithm for complex root isolation

Proceedings of the 36th international symposium on Symbolic and algebraic computation, 2011

We present a new exact subdivision algorithm Ceval for isolating the complex roots of a square-fr... more We present a new exact subdivision algorithm Ceval for isolating the complex roots of a square-free polynomial in any given box. It is a generalization of a previous real root isolation algorithm called Eval. Under suitable conditions, our approach is applicable for general analytic functions. Ceval is based on the simple Bolzano Principle and is easy to implement exactly. Preliminary experiments have shown its competitiveness. We further show that, for the "benchmark problem" of isolating all roots of a square-free polynomial with integer coefficients, the asymptotic complexity of both algorithms Eval and Ceval matches (up a logarithmic term) that of more sophisticated real root isolation methods which are based on Descartes' Rule of Signs, Continued Fraction or Sturm sequences. In particular, we show that the tree size of Eval matches that of other algorithms. Our analysis is based on a novel technique called δ-clusters from which we expect to see further applications.

D1-OPTIMAL Motion for a Rod (Extended Abstract)

Proceedings of the twelfth annual symposium on Computational geometry - SCG '96, 1996

ABSTRACT

Complete subdivision algorithms, I

Proceedings of the twenty-second annual symposium on Computational geometry, 2006

We give the first complete subdivision algorithm for the intersection of two Bezier curves F, G, ... more We give the first complete subdivision algorithm for the intersection of two Bezier curves F, G, possibly with tangential intersections. Our approach to robust subdivision algorithms is based on geometric separation bounds, and using a criterion for detecting non-crossing intersection of curves. Our algorithm is adaptive, being based only on exact bigfloat computations. In particular, we avoid manipulation of algebraic numbers and resultant computations. It is designed to be competitive with current algorithms on "nice" inputs. All standard algorithms assume F, G to be relatively prime -our algorithm needs a generalization of this.

k-Link Shortest Paths in Weighted Subdivisions

Lecture Notes in Computer Science, 2005

We study the shortest path problem in weighted polygonal subdivisions of the plane, with the addi... more We study the shortest path problem in weighted polygonal subdivisions of the plane, with the additional constraint of an upper bound, k, on the number of links (segments) in the path. We prove structural properties of optimal paths and utilize these results to obtain approximation algorithms that yield a path having O(k) links and weighted length at most (1 + ) times the weighted length of an optimal k-link path, for any fixed > 0. Some of our results make use of a new solution for the 1-link case, based on computing optimal solutions for a special sum-of-fractionals (SOF) problem. We have implemented a system, based on the CORE library, for computing optimal 1-link paths; we experimentally compare our new solution with a previous method for 1-link optimal paths based on a prune-and-search scheme.

Almost tight recursion tree bounds for the Descartes method

Proceedings of the 2006 international symposium on Symbolic and algebraic computation, 2006

We give a unified ("basis free") framework for the Descartes method for real root isolation of sq... more We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiX i with integer coefficients |ai| < 2 L , this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case. F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems-Computations on polynomials; G.1.5 [Numerical Analysis]: Roots of Nonlinear Equations-Methods for polynomials

Robust Approximate Zeros

Lecture Notes in Computer Science, 2005

Smale’s notion of an approximate zero of an analytic function f: C →C is extended to take into ac... more Smale’s notion of an approximate zero of an analytic function f: C →C is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero.. We develop a corresponding robust point estimate for such zeros: we prove that if z 0 ∈ C satisfies α(f,z 0)z 0 is a

A Unified Approach to HGCD Algorithms for polynomials and integers

We present a unified framework for the asymptotically fast Half-GCD(HGCD) algorithms, based on pr... more We present a unified framework for the asymptotically fast Half-GCD(HGCD) algorithms, based on properties of the norm. Two other benefitsof our approach are (a) a simplified correctness proof of the polynomialHGCD algorithm and (b) the first explicit integer HGCD algorithm. Theinteger HGCD algorithm turns out to be rather intricate.Keywords: Integer GCD, Euclidean algorithm, Polynomial GCD, Half GCDalgorithm, efficient algorithm.On leave from the Courant Institute, New York University, ...

Constructive root bound for k-ary rational input numbers

Theoretical Computer Science, 2006

Constructive root bounds is the fundamental technique needed to achieve guaranteed accuracy, the ... more Constructive root bounds is the fundamental technique needed to achieve guaranteed accuracy, the critical capability in Exact Geometric Computation. Known bounds are overly pessimistic in the presense of general rational input numbers. In this paper, we introduce a method which greatly improves the known bounds for k-ary rational input numbers. Since majority of input numbers in scientific and engineering applications are such numbers, this could lead to a significant speedup for a large class of applications. We apply our method to the BFMSS Bound. Implementation and experimental results based on the Core Library are reported.

Some consequences of non-uniform conditions on uniform classes

Theoretical Computer Science, 1983

We obtain some results of the form: If certain complexity classes satisfy a non-uniform condition... more We obtain some results of the form: If certain complexity classes satisfy a non-uniform condition, then some unlikely consequences follow. More precisely: (1) If the 'non-uniform polynomial-time hierarchy' collapses at level i >O. i.e., &/poly = Ui/pOly, then the Meyer-Stockmeyer hierarchy collapses at Ieke i+2, i.e., Ii+? = I7,, ?. This strengthens a generalization of a result of Karp and Lipton ( IWP). (2) If co-NP is conjunctively reducible to a sparse set, then P = NP. This generalizes a theorem of Fortune ( ). (3) If NP is conjunctively and disjunctively reducible to a sparse NP-cor,lplete set, then P = NP. This is a partial generalization of a result of . Conjuctive and disjunctive reducibility were introduced by Ladner, Lynch and Selman (1975). (4) If co-NP is y-reducible to a sparse set, then NP = co-NP. y-reducibility was introduced by Adleman and Manders (1977).

The Format Model

Journal of the ACM, 1984

A mathematical theory for the study of data representation in databases is introduced and develop... more A mathematical theory for the study of data representation in databases is introduced and developed. The theory focuses on three data constructs (collection, composition and classification). "Formats" with semantically rich yet tractable structure are built recursively using these constructs. Using formats, we obtain several nontrivial results concerning notions of relative information capacity and restructuring of data sets. As such, the format model provides a new approach for the formal study of the construction of "user views" and other data manipulations in databases.

The Habicht Approach to Subresultants

Journal of Symbolic Computation, 1996

The Habicht approach to the theory of subresultants is based on studying polynomial remainder seq... more The Habicht approach to the theory of subresultants is based on studying polynomial remainder sequences (PRS) with indeterminate coefficients, and predicting the effects of specializing these coefficients. This has advantages as noted by Loos. We give a complete treatment of this approach by introducing the concept of pseudo-subresultants.

Complete numerical isolation of real roots in zero-dimensional triangular systems

Journal of Symbolic Computation, 2009

We present a complete numerical algorithm of isolating all the real zeros of a zero-dimensional t... more We present a complete numerical algorithm of isolating all the real zeros of a zero-dimensional triangular polynomial system Fn ⊆ Z[x1, . . . , xn]. Our system Fn is general, with no further assumptions. In particular, our algorithm successfully treat multiple zeros directly in such systems. A key idea is to introduce evaluation bounds and sleeve bounds. We also present a much more efficient algorithm for zero-dimensional triangular systems without multiple roots. We implemented our algorithms and promising experimental results are shown.

Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves

Journal of Symbolic Computation, 2012

Given a real valued function f (X, Y ), a box region B 0 ⊆ R 2 and ε > 0, we want to compute an ε... more Given a real valued function f (X, Y ), a box region B 0 ⊆ R 2 and ε > 0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curve S = f -1 (0) = {p ∈ R 2 : f (p) = 0} to B 0 . We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities.

On formulating simultaneity for studying parallelism and synchronization

Journal of Computer and System Sciences, 1980

A “retraction” method for planning the motion of a disc

Journal of Algorithms, 1985

A new approach to certain motion-planning problems in robotics is introduced. This approach is ba... more A new approach to certain motion-planning problems in robotics is introduced. This approach is based on the use of a generalized Voronoi diagram, and reduces the search for a collision-free continuous motion to a search for a connected path along the edges of such a diagram. This approach yields an O(n log n) algorithm for planning an obstacle-avoiding motion of a single circular disc amid polygonal obstacles. Later papers will show that extensions of the approach can solve other motion-planning problems, including those of moving a straight line segment or several coordinated discs in the plane amid polygonal obstacles. Q 1985 Academic Press. Inc.

A hybrid algorithm for the shortest path between two nodes in the presence of few negative arcs

Information Processing Letters, 1983