Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs

A graph G is called generically minimally rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining tight bounds on the number of such embeddings, as a function of the number of vertices. The study of rigid graphs is motivated by numerous applications, mostly in robotics, bioinformatics, sensor networks and architecture. We capture embeddability by polynomial systems with suitable structure, so that their mixed volume, which bounds the number of common roots, yields interesting upper bounds on the number of embeddings. We explore different polynomial formulations so as to reduce the corresponding mixed volume, namely by introducing new variables that remove certain spurious roots, and by applying the theory of distance geometry. We focus on R2 and R3, where Laman graphs and 1-skeleta (or edge graphs) of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. Our implementation yields upper bounds for n ≤ 10 in R2 and R3, which reduce the existing gaps and lead to tight bounds for n ≤ 7 in both R2 and R3; in particular, we describe the recent settlement of the case of Laman graphs with 7 vertices. Our approach also yields a new upper bound for Laman graphs with 8 vertices, which is conjectured to be tight. We also establish the first lower bound in R3 of about 2.52n, where n denotes the number of vertices

On the maximal number of real embeddings of spatial minimally rigid graphs

The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in $\mathbb{R}^3$. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the {\em a priori} number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in $\mathbb{R}^3$, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in $\mathbb{R}^3$

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding

Recursive Polynomial Remainder Sequence and the Nested Subresultants

We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for investigation of the recursive PRS. Finally, we discuss usage of the reduced nested subresultant in approximate algebraic computation, which motivates the present work.Comment: 12 pages. Presented at CASC 2005 (Kalamata, Greece, Septermber 12-16, 2005

Multihomogeneous resultant formulae for systems with scaled support

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bézout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bézout-type matrices, known as partial Bezoutians. Our public-domain Mapleimplementation includes efficient storage of complexes in memory, and construction of resultant matrices. © 2011 Elsevier Ltd

Improved algorithms for computing determinants and resultants

Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via computing the resultant. This is achieved by combining our techniques with the primitive-element method, which leads to an effective implicit representation of the roots. We systematically examine quotient formulae of Sylvester-type resultant matrices, including matrix polynomials and the u-resultant. We reduce the known bit operation complexity bounds by almost an order of magnitude, in terms of the resultant matrix dimension. Our theoretical and practical improvements cover the highly important cases of sparse and degenerate systems. © 2004 Elsevier Inc. All rights reserved

Exact and efficient evaluation of the InCircle predicate for parametric ellipses and smooth convex objects

We study the Voronoi diagram, under the Euclidean metric, of a set of ellipses, given in parametric representation. The article concentrates on the InCircle predicate, which is the hardest to compute, and describes an exact and complete solution. It consists of a customized subdivision-based method that achieves quadratic convergence, leading to a real-time implementation for non-degenerate inputs. Degenerate cases are handled using exact algebraic computation. We conclude with experiments showing that most instances run in less than 0.1 s, on a 2.6 GHz Pentium-4, whereas degenerate cases may take up to 13 s. Our approach readily generalizes to smooth convex objects. © 2008 Elsevier Ltd. All rights reserved

Protein structure prediction using residual dipolar couplings

NMR is important for the determination of protein structures, but the usual NOE distance constraints cannot capture large structures. However, RDC experiments offer global orientation constraints for the H-N backbone vectors. Our first application validates local structure from 3 RDC values, by solving an elliptical equation. Second, we model the protein backbone by drawing upon robot kinematics, and compute the relative orientation of consecutive pairs of peptide planes; we obtain a unique orientation by considering also NOE distances. Third, we present a novel algebraic method for determining the relative orientation of secondary structures, a crucial question in fold classification. The orientation of the magnetic vector relative to the secondary structures is determined using two media, leading to a rotation matrix mapping one molecular frame to the other. A unique solution is obtained from RDC data, with no NOE constraints. Our algorithms use robust algebraic operations and are implemented in MAPLE. © Springer-Verlag Berlin Heidelberg 2007