Showing papers in "Discrete and Computational Geometry in 2009"
TL;DR: A parameterized notion of feature size is introduced that interpolates between the minimum of the local feature size and the recently introduced weak feature size to ensure the topological correctness of a reconstruction given by an offset of the sampling.
Abstract: We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points, and critical values of distance functions.
146 citations
TL;DR: The theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates are presented.
Abstract: This paper presents a new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes. The algorithm—particularly strong for low-dimensional sets embedded in high dimensions—runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates.
101 citations
TL;DR: An affirmative answer to the case k=4 is provided to the conjectured that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O(n).
Abstract: A topological graph is called k -quasi-planar if it does not contain k pairwise crossing edges. It is conjectured that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O(n). We provide an affirmative answer to the case k=4.
80 citations
TL;DR: Several colorful generalizations of classical theorems in discrete geometry are proved, including Kirchberger’s theorem, which gives a generalization of the theorem of Tverberg on non-separated partitions.
Abstract: We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger’s theorem gives a generalization of the theorem of Tverberg on non-separated partitions.
72 citations
TL;DR: The Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement is extended to affine and toric hyperplane arrangements and Zaslavsky’s fundamental results on the number of regions are generalized.
Abstract: We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky’s fundamental results on the number of regions.
57 citations
TL;DR: A self-contained proof from first principles accessible to a nonexpert is given that the combinatorial Alexander duality of X is isomorphic to the (|V|−i−3)th reduced cohomology group of X* (over a given commutative ring R).
Abstract: Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X *={σ⊆V∣V∖σ ∉ X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (|V|−i−3)th reduced cohomology group of X * (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.
57 citations
TL;DR: It is shown that given any metric with constant doubling dimension k and any 0<ε<1, one can find (1+ε)-spanner for the metric with nearly linear number of edges and constant hop diameter; and it is proved that such tradeoffs between the number of edge and the hop diameter are asymptotically optimal.
Abstract: Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse (1+e)-spanners with small hop diameter for metrics of low doubling dimension.
In this paper, we show that given any metric with constant doubling dimension k and any 0
53 citations
TL;DR: In this paper, it was shown that the Frechet distance is upper semi-computable, i.e., there is a non-halting Turing machine which produces a decreasing sequence of rationals converging to the Frecheng distance.
Abstract: A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Frechet distance. Whereas efficient algorithms are known for computing the Frechet distance of polygonal curves, the same problem for triangulated surfaces is NP-hard. Furthermore, it remained open whether it is computable at all.
Using a discrete approximation, we show that it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a decreasing sequence of rationals converging to the Frechet distance. It follows that the decision problem, whether the Frechet distance of two given surfaces lies below a specified value, is recursively enumerable.
Furthermore, we show that a relaxed version of the Frechet distance, the weak Frechet distance can be computed in polynomial time. For this, we give a computable characterization of the weak Frechet distance in a geometric data structure called the Free Space Diagram.
49 citations
TL;DR: A new way of looking at k-triangulations, namely as complexes of star polygons, is introduced, namely with this tool, which gives new, direct proofs of the fundamental properties of k-Triangulations.
Abstract: Maximal (k+1)-crossing-free graphs on a planar point set in convex position, that is, k-triangulations, have received attention in recent literature, motivated by several interpretations of them.
We introduce a new way of looking at k-triangulations, namely as complexes of star polygons. With this tool we give new, direct proofs of the fundamental properties of k-triangulations, as well as some new results. This interpretation also opens up new avenues of research that we briefly explore in the last section.
46 citations
TL;DR: For manifolds with boundary, the authors derived an extension of Klee's Dehn-Sommerville relations and strengthened Kalai's result on the number of edges for manifolds without boundary.
Abstract: We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kuhnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kuhnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges.
44 citations
TL;DR: It is shown that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large depending on d.
Abstract: We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large depending on d. As a corollary, we determine the exact maximum number of unit distances for all even d≥6 and the exact maximum number of diameters for all d≥4 and all n sufficiently large depending on d.
TL;DR: In this article, a Delaunay refinement algorithm for meshing a piecewise smooth complex in 3D is presented, which protects edges with weighted points to avoid the difficulty posed by small angles between adjacent input elements.
Abstract: We present a Delaunay refinement algorithm for meshing a piecewise smooth complex in three dimensions. The algorithm protects edges with weighted points to avoid the difficulty posed by small angles between adjacent input elements. These weights are chosen to mimic the local feature size and to satisfy a Lipschitz-like property. A Delaunay refinement algorithm using the weighted Voronoi diagram is shown to terminate with the recovery of the topology of the input. Guaranteed bounds on the aspect ratios, normal variation, and dihedral angles are also provided. To this end, we present new concepts and results including a new definition of local feature size and a proof for a generalized topological ball property.
TL;DR: In this article, it was shown that the problem of finding the minimum number of vertices that need to be moved to untangle a straight-line drawing of a planar graph is NP-hard.
Abstract: A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem.
Further we define fix(G,δ)=n−shift(G,δ) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling δ. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log\log n}$vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing δ G of G with $\ensuremath {\mathrm {fix}}(G,\delta_{G})\leq \sqrt{n-2}+1$and an n-vertex outerplanar graph H and a drawing δ H of H with $\ensuremath {\mathrm {fix}}(H,\delta_{H})\leq2\sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.
TL;DR: It is shown that in dimension 4 only conjectures A and B are valid, while conjecture C fails, and that both conjectures B and C fail in all dimensions d≥5.
Abstract: In 1989, Kalai stated three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the “3d -conjecture.” It is well known that the three conjectures hold in dimensions d≤3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d≥5.
TL;DR: It is shown that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than $3(\sqrt{n}-1)$ vertices fixed.
Abstract: To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002) asked if every n-vertex geometric planar graph can be untangled while keeping at least n e vertices fixed. We answer this question in the affirmative with e=1/4. The previous best known bound was $\Omega(\sqrt{\log n/\log\log n})$. We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least $\Omega(\sqrt{n})$vertices fixed, while the best upper bound was $\mathcal{O}((n\log n)^{2/3})$. We answer a question of Spillner and Wolff (http://arxiv.org/abs/0709.0170) by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than $3(\sqrt{n}-1)$vertices fixed.
TL;DR: In this paper, the authors introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of fault-tolerant spanners of small size.
Abstract: We introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of region-fault tolerant spanners of small size. For a geometric graph $\mathcal{G}$on a point set P and a region F, we define $\mathcal{G}\ominus F$to be what remains of $\mathcal{G}$after the vertices and edges of $\mathcal{G}$intersecting F have been removed. A $\mathcal{C}$-fault tolerant t-spanner is a geometric graph $\mathcal{G}$on P such that for any convex region F, the graph $\mathcal{G}\ominus F$is a t-spanner for $\mathcal{G}_{c}(P)\ominus F$, where $\mathcal{G}_{c}(P)$is the complete geometric graph on P. We prove that any set P of n points admits a $\mathcal{C}$-fault tolerant (1+e)-spanner of size $\mathcal{O}(n\log n)$for any constant e>0; if adding Steiner points is allowed, then the size of the spanner reduces to $\mathcal{O}(n)$, and for several special cases, we show how to obtain region-fault tolerant spanners of $\mathcal{O}(n)$size without using Steiner points. We also consider fault-tolerant geodesic t -spanners: this is a variant where, for any disk D, the distance in $\mathcal{G}\ominus D$between any two points u,v∈P∖D is at most t times the geodesic distance between u and v in ℝ2∖D. We prove that for any P, we can add $\mathcal{O}(n)$Steiner points to obtain a fault-tolerant geodesic (1+e)-spanner of size $\mathcal{O}(n)$.
TL;DR: It is proved that, for fixed rank, Ehrhart polynomials of matroid polytopes and polymatroids are computable in polynomial time.
Abstract: We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that, for fixed rank, Ehrhart polynomials of matroid polytopes and polymatroids are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h *-vector and the coefficients of Ehrhart polynomials of matroid polytopes; we provide theoretical and computational evidence for their validity.
TL;DR: The mixed volume analogues of the Van der Waerden and Schrijver–Valiant conjectures on the permanent are proved and justify the abovementioned approximation by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm.
Abstract: Let K=(K 1,…,K n ) be an n-tuple of convex compact subsets in the Euclidean space R n , and let V(⋅) be the Euclidean volume in R n . The Minkowski polynomial V K is defined as V K (λ 1,…,λ n )=V(λ 1 K 1+⋅⋅⋅+λ n K n ) and the mixed volume V(K 1,…,K n ) as $$V(K_{1},\ldots,K_{n})=\frac{\partial^{n}}{\partial\lambda_{1}\cdots\partial \lambda_{n}}V_{\mathbf{K}}(\lambda_{1},\ldots,\lambda_{n}).$$ Our main result is a poly-time algorithm which approximates V(K 1,…,K n ) with multiplicative error e n and with better rates if the affine dimensions of most of the sets K i are small. Our approach is based on a particular approximation of log (V(K 1,…,K n )) by a solution of some convex minimization problem. We prove the mixed volume analogues of the Van der Waerden and Schrijver–Valiant conjectures on the permanent. These results, interesting on their own, allow us to justify the abovementioned approximation by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.
TL;DR: A consequence of the results is a proof that the Delaunay triangulations for the L∞ metric and the L1 metric always admit a Hamiltonian path.
Abstract: Given a class $\mathcal{C}$of geometric objects and a point set P, a $\mathcal{C}$-matching of P is a set $M=\{C_{1},\dots,C_{k}\}\subseteq \mathcal{C}$of elements of $\mathcal{C}$such that each C i contains exactly two elements of P and each element of P lies in at most one C i . If all of the elements of P belong to some C i , M is called a perfect matching. If, in addition, all of the elements of M are pairwise disjoint, we say that this matching M is strong. In this paper we study the existence and characteristics of $\mathcal{C}$-matchings for point sets in the plane when $\mathcal{C}$is the set of isothetic squares in the plane. A consequence of our results is a proof that the Delaunay triangulations for the L ∞ metric and the L 1 metric always admit a Hamiltonian path.
TL;DR: It is proved that the maximum number of facets of the Minkowski sum of k polytopes with m1,m2,…,mk facets, respectively, is bounded from above by $\sum_{1\leq i
Abstract: We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in ℝ3. In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m 1,m 2,…,m k facets, respectively, is bounded from above by $\sum_{1\leq i
TL;DR: The problem of estimating b(n), the minimum of b(P) over all n-point sets P with no three points collinear, is raised.
Abstract: For a finite set P in the plane, let b(P) be the smallest possible size of a set Q, Q∩P=∅, such that every segment with both endpoints in P contains at least one point of Q. We raise the problem of estimating b(n), the minimum of b(P) over all n-point sets P with no three points collinear. We review results providing bounds on b(n) and mention some additional observations.
TL;DR: For arbitrary point sets, the general problem of traversing an arbitrary set of points in ℝd with an axis-aligned spanning path having a minimum number of links with a constant ratio (depending on the dimension d) approximation algorithm is presented.
Abstract: Given a finite set of points S in źd, consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let Gnd be an nן×n grid in źd. Kranakis et al. (Ars Comb. 38:177---192, 1994) showed that L(Gn2)=2nź1 and $\frac{4}{3}n^{2}-O(n)\le L(G^{3}_{n})\le \frac{3}{2}n^{2}+O(n)$ and conjectured that, for all dź3, $L(G^{d}_{n})=\frac{d}{d-1}n^{d-1}\pm O(n^{d-2}).$ We prove the conjecture for d=3 by showing the lower bound for L(Gn3). For d=4, we prove that $L(G^{4}_{n})=\frac{4}{3}n^{3}\pm O(n^{5/2}).$
For general d, we give new estimates on L(Gnd) that are very close to the conjectured value. The new lower bound of $(1+\frac{1}{d})n^{d-1}-O(n^{d-2})$ improves previous result by Collins and Moret (Inf. Process. Lett. 68:317---319, 1998), while the new upper bound of $(1+\frac{1}{d-1})n^{d-1}+O(n^{d-3/2})$ differs from the conjectured value only in the lower order terms.
For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in źd with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm.
TL;DR: It is shown that if the order of the curvature is less than the dimension d for all polytopes defined by 2d inequalities and for all d, then the orders are less that the number of inequalities for allpolytopes.
Abstract: The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as a continuous analogue of its diameter. We prove an analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytopes defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.
TL;DR: It is shown that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1.
Abstract: A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.
TL;DR: It is proved that the chromatic number of this graph is at least five and that the odd-distance graph in ℝ2 is countably choosable, while such a graph inℝ3 is not.
Abstract: The vertices of the odd-distance graph are the points of the plane ℝ2. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ2 is countably choosable, while such a graph in ℝ3 is not.
TL;DR: “dimension gaps” are investigated among these notions and construct, for all d≥1, a 2d-Leray complex that is not (3d−1)-collapsible and a d-collapsable complex that are not (2d−2)-representable.
Abstract: A simplicial complex $\mathsf{K}$is called d -representable if it is the nerve of a collection of convex sets in ℝd ; $\mathsf{K}$is d -collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d−1 that is contained in a unique maximal face; and $\mathsf{K}$is d -Leray if every induced subcomplex of $\mathsf{K}$has vanishing homology of dimension d and larger
It is known that d-representable implies d-collapsible implies d-Leray, and no two of these notions coincide for d≥2 The famous Helly theorem and other important results in discrete geometry can be regarded as results about d-representable complexes, and in many of these results, “d-representable” in the assumption can be replaced by “d-collapsible” or even “d-Leray”
We investigate “dimension gaps” among these notions and construct, for all d≥1, a 2d-Leray complex that is not (3d−1)-collapsible and a d-collapsible complex that is not (2d−2)-representable In the proofs, we obtain two results of independent interest: (i) The nerve of every finite family of sets, each of size at most d, is d-collapsible; (ii) If the nerve of a simplicial complex $\mathsf{K}$is d-representable, then $\mathsf{K}$ embeds in ℝd
TL;DR: In this paper, the authors recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument.
Abstract: We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if a code has a large distance, then its dual has a small covering radius and, therefore, is large. This implies the original code to be small.
We also point out that this bound is a natural isoperimetric constant of the Hamming cube, related to its Faber–Krahn minima.
While our approach belongs to the general framework of Delsarte’s linear programming method, its main technical ingredient is Fourier duality for the Hamming cube. In particular, we do not deal directly with Delsarte’s linear program or orthogonal polynomial theory.
TL;DR: Lower bounds on the worst-case complexity of approximate range searching in the semigroup arithmetic model for ranges consisting of d-dimensional unit hypercubes under rigid motions are established and the improvements offered by idempotence do apply to smooth convex ranges.
Abstract: Given an n-element point set in ℝd , the range searching problem involves preprocessing these points so that the total weight, or for our purposes the semigroup sum, of the points lying within a given query range η can be determined quickly. In e-approximate range searching we assume that η is bounded, and the sum is required to include all the points that lie within η and may additionally include any of the points lying within distance e⋅diam(η) of η’s boundary.
In this paper we contrast the complexity of approximate range searching based on properties of the semigroup and range space. A semigroup (S,+) is idempotent if x+x=x for all x∈S, and it is integral if for all k≥2, the k-fold sum x+⋅⋅⋅+x is not equal to x. Recent research has shown that the computational complexity of approximate spherical range searching is significantly lower for idempotent semigroups than it is for integral semigroups in terms of the dependencies on e. In this paper we consider whether these results can be generalized to other sorts of ranges. We show that, as with integrality, allowing sharp corners on ranges has an adverse effect on the complexity of the problem. In particular, we establish lower bounds on the worst-case complexity of approximate range searching in the semigroup arithmetic model for ranges consisting of d-dimensional unit hypercubes under rigid motions. We show that for arbitrary (including idempotent) semigroups and linear space, the query time is at least $\varOmega(1/{\varepsilon }^{d-2\sqrt{d}})$. In the case of integral semigroups we prove a tighter lower bound of Ω(1/e d−2). These lower bounds nearly match existing upper bounds for arbitrary semigroups.
In contrast, we show that the improvements offered by idempotence do apply to smooth convex ranges. We say that a range is smooth if at every boundary point there is an incident Euclidean sphere that lies entirely within the range whose radius is proportional to the range’s diameter. We show that for smooth ranges and idempotent semigroups, e-approximate range queries can be answered in O(log n+(1/e)(d−1)/2log (1/e)) time using O(n/e) space. We show that this is nearly tight by presenting a lower bound of Ω(log n+(1/e)(d−1)/2). This bound is in the decision-tree model and holds irrespective of space.
TL;DR: It is shown that one can compute the combinatorial facets of a simple polytope from its graph in polynomial time, and that this allows one to characterize the face lattice of thepolytope, via a simple face recognition algorithm.
Abstract: We show that one can compute the combinatorial facets of a simple polytope from its graph in polynomial time. Our proof relies on a primal-dual characterization (by Joswig, Kaibel, and Korner in Israel J. Math. 129:109–118, 2002) and a linear program, with an exponential number of constraints, which can be used to construct the solution and can be solved in polynomial time. We show that this allows one to characterize the face lattice of the polytope, via a simple face recognition algorithm. In addition, we use this linear program to construct several interesting polynomial-time computable sets of graphs which may be of independent interest.
TL;DR: This paper describes an algorithm for generating a guaranteed intersection-free interpolation sequence between any pair of compatible polygons that builds on prior results from linkage unfolding, and allows additional control by accommodating a set of algebraic constraints that can be weakly enforced throughout the interpolation sequences.
Abstract: This paper describes an algorithm for generating a guaranteed intersection-free interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of the interpolation sequence. The computational machinery that ensures against self-intersection is independent from a distance metric that determines the overall character of the interpolation sequence. This decoupled approach provides a powerful control mechanism for determining how the interpolation should appear, while still assuring against intersection and guaranteeing termination of the algorithm. Our algorithm also allows additional control by accommodating a set of algebraic constraints that can be weakly enforced throughout the interpolation sequence.