Showing papers in "Discrete and Computational Geometry in 2015"

A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

Abstract: We establish that the extension complexity of the $$n\times n$$n×n correlation polytope is at least $$1.5\,^n$$1.5n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least $$1.5^n$$1.5n, and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least $$.58n$$.58n. We thereby slightly improve on the previously best known lower bounds $$1.24^n$$1.24n and $$.31n$$.31n, respectively.

Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations

Abstract: Differential privacy is a definition giving a strong privacy guarantee even in the presence of auxiliary information. In this work, we pursue the application of geometric techniques for achieving differential privacy, a highly promising line of work initiated by Hardt and Talwar (Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC'10, pp 705---714. ACM Press, New York, 2010). We apply these techniques to the problem of marginal release. Here, a database refers to a collection of the data of $$n$$n individuals, each characterized by $$d$$d binary attributes. A $$k$$k-way marginal query is specified by a subset $$S$$S of $$k$$k attributes, together with a $$|S|$$|S|-dimensional binary vector $$\beta $$β specifying their values. The true answer to this query is a count of the number of people in the database whose attribute vector restricted to $$S$$S agrees with $$\beta $$β. Information theoretically, the error complexity of marginal queries--how "wrong" do the answers have to be in order to preserve differential privacy--is well understood: the per-query additive error is known to be at least $$\varOmega (\min \{ \sqrt{n},d^{k/2}\})$$Ω(min{n,dk/2}) and at most $$\tilde{O}(\sqrt{n}d^{{\lceil k/2\rceil /4}})$$O~(nd?k/2?/4). However, no polynomial time algorithm with error complexity as low as the information-theoretic upper bound is known for small $$n$$n. We present a polynomial time algorithm that matches the best known information-theoretic bounds when $$k=2$$k=2; more generally, by reducing to the case $$k=2$$k=2, for any distribution on marginal queries, our algorithm achieves average error at most $$\tilde{O}(\sqrt{n}d^{{\lceil k/2\rceil /4}})$$O~(nd?k/2?/4), an improvement over previous work when $$k$$k is small and when error $$o(n)$$o(n) is desirable. Using private boosting, we are also able to give nearly matching worst-case error bounds. Our algorithms are based on the geometric techniques of Nikolov et al. (Proceedings of the 45th Annual ACM Symposium on Theory of Computing, STOC'13, pp 351---360. ACM Press, New York, 2013), wherein a vector of "sufficiently noisy" answers is projected onto a particular convex body. We reduce the projection step, which is expensive, to a simple geometric question: given (a succinct representation of) a convex body $$K$$K, find a containing convex body $$L$$L that one can efficiently optimize over, while keeping the Gaussian width of $$L$$L small. This reduction is achieved by a careful use of the Frank---Wolfe algorithm.

Crossing Numbers and Combinatorial Characterization of Monotone Drawings of $$K_n$$Kn

Abstract: In 1958, Hill conjectured that the minimum number of crossings in a drawing of $$K_n$$Kn is exactly $$Z(n) = \frac{1}{4} \big \lfloor \frac{n}{2}\big \rfloor \big \lfloor \frac{n-1}{2}\big \rfloor \big \lfloor \frac{n-2}{2}\big \rfloor \big \lfloor \frac{n-3}{2}\big \rfloor $$Z(n)=14?n2??n-12??n-22??n-32?. Generalizing the result by Abrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $$x$$x-monotone curves. In fact, our proof shows that the conjecture remains true for $$x$$x-monotone drawings of $$K_n$$Kn in which adjacent edges may cross an even number of times, and instead of the crossing number we count the pairs of edges which cross an odd number of times. We further discuss a generalization of this result to shellable drawings, a notion introduced by Abrego et al. We also give a combinatorial characterization of several classes of $$x$$x-monotone drawings of complete graphs using a small set of forbidden configurations. For a similar local characterization of shellable drawings, we generalize Caratheodory's theorem to simple drawings of complete graphs.

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Abstract: Let T be a triangulation of a simple polygon. A flip in T is the operation of replacing one diagonal of T by a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-hard. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.

A Szemerédi---Trotter Type Theorem in $$\mathbb {R}^4$$R4

Abstract: We show that m points and n two-dimensional algebraic surfaces in $${\mathbb {R}}^4$$R4 can have at most $$O(m^{{k}/({2k-1})}n^{({2k-2})/({2k-1})}+m+n)$$O(mk/(2k-1)n(2k-2)/(2k-1)+m+n) incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that $$m\le n^{(2k+2)/3k}$$m≤n(2k+2)/3k. As a special case, we obtain a Szemeredi---Trotter type theorem for 2-planes in $${\mathbb {R}}^4$$R4, provided $$m\le n$$m≤n and the planes intersect transversely. As a further special case, we obtain a Szemeredi---Trotter type theorem for complex lines in $${\mathbb {C}}^2$$C2 with no restrictions on m and n (this theorem was originally proved by Toth using a different method). As a third special case, we obtain a Szemeredi---Trotter type theorem for complex unit circles in $${\mathbb {C}}^2$$C2. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.

Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line: Algorithms and Complexity

Abstract: Given a set of $$n$$n axis-parallel rectangles in the plane, finding a maximum independent set ($$\mathrm {MIS}$$MIS), a maximum weighted independent set ($$\mathrm {WMIS}$$WMIS), and a minimum hitting set ($$\mathrm {MHS}$$MHS), are basic problems in computational geometry and combinatorics. They have attracted significant attention since the sixties, when Wegner conjectured that the duality gap, equal to the ratio between the size of $$\mathrm {MIS}$$MIS and the size of $$\mathrm {MHS}$$MHS, is always bounded by a universal constant. An interesting case is when there exists a diagonal line that intersects each of the given rectangles. Indeed, Chepoi and Felsner recently gave a 6-approximation algorithm for $$\mathrm {MHS}$$MHS in this setting, and showed that the duality gap is between 3/2 and 6. We consider the same setting and improve upon these results. First, we derive an $$O(n^2)$$O(n2)-time algorithm for the $$\mathrm {WMIS}$$WMIS when, in addition, every pair of intersecting rectangles have a common point below the diagonal. This improves and extends a classic result of Lubiw, and gives a 2-approximation algorithm for $$\mathrm {WMIS}$$WMIS. Second, we show that $$\mathrm {MIS}$$MIS is NP-hard. Finally, we prove that the duality gap is between 2 and 4. The upper bound, which implies a 4-approximation algorithm for $$\mathrm {MHS}$$MHS, follows from simple combinatorial arguments, whereas the lower bound represents the best known lower bound on the duality gap, even in the general setting of the rectangles.

On Pattern Entropy of Weak Model Sets

Abstract: We study point sets arising from cut-and-project constructions. An important class is that of weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern entropy in terms of the volume of the window boundary in internal space. This proves a conjecture by R.V. Moody.

Distinct Distance Estimates and Low Degree Polynomial Partitioning

Abstract: We give a shorter proof of a slightly weaker version of a theorem from Guth and Katz (Ann Math 181:155---190, 2015): we prove that if $$\mathfrak {L}$$L is a set of $$L$$L lines in $$\mathbb {R}^3$$R3 with at most $$L^{1/2}$$L1/2 lines in any low degree algebraic surface, then the number of $$r$$r-rich points of $$\mathfrak {L}$$L is $$\lesssim L^{(3/2) + \varepsilon } r^{-2}$$?L(3/2)+?r-2. This result is one of the main ingredients in the proof of the distinct distance estimate in Guth and Katz (2015). With our slightly weaker theorem, we get a slightly weaker distinct distance estimate: any set of $$N$$N points in $$\mathbb {R}^2$$R2 determines at least $$c_{\varepsilon } N^{1 -{\varepsilon }}$$c?N1-? distinct distances.

Multilevel Polynomial Partitions and Simplified Range Searching

Abstract: The polynomial partitioning method of Guth and Katz (arXiv:1011.4105) has numerous applications in discrete and computational geometry. It partitions a given n-point set $$P\subset {\mathbb {R}}^d$$P?Rd using the zero set Z(f) of a suitable d-variate polynomial f. Applications of this result are often complicated by the problem, "What should be done with the points of P lying within Z(f)?" A natural approach is to partition these points with another polynomial and continue further in a similar manner. So far this has been pursued with limited success--several authors managed to construct and apply a second partitioning polynomial, but further progress has been prevented by technical obstacles. We provide a polynomial partitioning method with up to d polynomials in dimension d, which allows for a complete decomposition of the given point set. We apply it to obtain a new algorithm for the semialgebraic range searching problem. Our algorithm has running time bounds similar to a recent algorithm by Agarwal et al. (SIAM J Comput 42:2039---2062, 2013), but it is simpler both conceptually and technically. While this paper has been in preparation, Basu and Sombra, as well as Fox, Pach, Sheffer, Suk, and Zahl, obtained results concerning polynomial partitions which overlap with ours to some extent.

Stress Matrices and Global Rigidity of Frameworks on Surfaces

Abstract: In 2005, Bob Connelly showed that a generic framework in $${\mathbb {R}}^d$$Rd is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in $${\mathbb {R}}^3$$R3. For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum-rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.

Fan Realizations of Type $$A$$A Subword Complexes and Multi-associahedra of Rank 3

Abstract: We present complete simplicial fan realizations of any spherical subword complex of type $$A_n$$An for $$n\le 3$$n≤3. This provides complete simplicial fan realizations of simplicial multi-associahedra $$\varDelta _{2k+4,k}$$Δ2k+4,k, whose facets are in correspondence with $$k$$k-triangulations of a convex $$(2k+4)$$(2k+4)-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. We also present fan realizations of two previously unknown cases of subword complexes of type $$A_4$$A4, namely the multi-associahedra $$\varDelta _{9,2}$$Δ9,2 and $$\varDelta _{11,3}$$Δ11,3.

Algebraic---exponential Data Recovery from Moments

Abstract: Let $$\mathbf {G}$$G be a bounded open subset of Euclidean space whose boundary $$\Gamma $$Γ is algebraic, i.e., contained in the real zero set of finitely many polynomials. Under the assumption that the degree d of this variety is given, and the power moments of the Lebesgue measure on $$\mathbf {G}$$G are known up to order 3d, we describe an algorithmic procedure for obtaining a polynomial vanishing on $$\Gamma $$Γ. The particular case of semi-algebraic sets defined by a single polynomial inequality raises an intriguing question related to the finite determinateness of the full moment sequence. The more general case of a measure with density equal to the exponential of a polynomial is treated in parallel. Our approach relies on Stokes' Theorem on spaces with singularities and simple Hankel-type matrix identities.

The Density of Sets Avoiding Distance 1 in Euclidean Space

Abstract: We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analog of the Lovasz theta number and of a combinatorial argument involving finite subgraphs of the unit distance graph. In turn, we straightforwardly obtain an asymptotic improvement for the measurable chromatic number of Euclidean space. We also tighten previous results for dimensions between 4 and 24.

Frameworks with Forced Symmetry I: Reflections and Rotations

Abstract: We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection To establish these results, we develop a new technique for deriving linear representations of sparsity matroids on colored graphs and extend the direction network method of proving rigidity characterizations to handle reflections

Gromov---Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs

Abstract: In many real-world applications, data appear to be sampled around 1-dimensional filamentary structures that can be seen as topological metric graphs. In this paper, we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated with respect to the Gromov---Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and provide an efficient and easy-to-implement algorithm to compute such approximations in almost linear time. We illustrate the performance of our algorithm on a few datasets.

Finite and Infinitesimal Rigidity with Polyhedral Norms

Abstract: We characterise finite and infinitesimal rigidity for bar-joint frameworks in $${\mathbb {R}}^d$$Rd with respect to polyhedral norms (i.e. norms with closed unit ball $${\mathcal {P}}$$P, a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in $${\mathbb {R}}^d$$Rd which are well-positioned with respect to $${\mathcal {P}}$$P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in $${\mathbb {R}}^d$$Rd in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on $${\mathbb {R}}^2$$R2.

There are Plane Spanners of Degree 4 and Moderate Stretch Factor

Abstract: Let $${\fancyscript{E}}$$E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant $$t \ge 1$$t?1, a spanning subgraph $$G$$G of $${\fancyscript{E}}$$E is said to be a $$t$$t-spanner, or simply a spanner, if for any pair of nodes $$u,v$$u,v in $${\fancyscript{E}}$$E the distance between $$u$$u and $$v$$v in $$G$$G is at most $$t$$t times their distance in $${\fancyscript{E}}$$E. The constant $$t$$t is referred to as the stretch factor. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of $${\fancyscript{E}}$$E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper, we show that the complete Euclidean graph always contains a plane spanner of maximum degree 4 and make a big step toward closing the question. The stretch factor of the spanner is bounded by $$156.82$$156.82. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's $$L_1$$L1-Delaunay triangulation.

Iterative Universal Rigidity

Abstract: A bar framework determined by a finite graph $$G$$G and a configuration $$\mathbf{p =(p_1,\ldots , p_n) }$$p=(p1,?,pn) in $$\mathbb {R}^d$$Rd is universally rigid if it is rigid in any $$\mathbb {R}^D \supset \mathbb {R}^d$$RD?Rd. We provide a characterization of universal rigidity for any graph $$G$$G and any configuration $$\mathbf{p}$$p in terms of a sequence of affine subsets of the space of configurations. This corresponds to a facial reduction process for closed finite-dimensional convex cones.

Inductive Constructions for Frameworks on a Two-Dimensional Fixed Torus

Abstract: An infinite periodic framework in the plane can be represented as a framework on a torus, using a $${\mathbb {Z}}^2$$Z2-labelled gain graph. We find necessary and sufficient conditions for the generic minimal rigidity of frameworks on the two-dimensional fixed torus $${\mathcal {T}}_0^2$$T02. It is also shown that every minimally rigid periodic orbit framework on $${\mathcal {T}}_0^2$$T02 can be constructed from smaller frameworks through a series of inductive constructions. These are fixed torus adapted versions of the results of Laman and Henneberg, respectively, for finite frameworks in the plane. The proofs involve the development of inductive constructions for $${\mathbb {Z}}^2$$Z2-labelled graphs.

Liftings and Stresses for Planar Periodic Frameworks

Abstract: We formulate and prove a periodic analog of Maxwell's theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.

A Quadratic Lower Bound for the Convergence Rate in the One-Dimensional Hegselmann---Krause Bounded Confidence Dynamics

Abstract: Let $$f_{k}(n)$$fk(n) be the maximum number of time steps taken to reach equilibrium by a system of $$n$$n agents obeying the $$k$$k-dimensional Hegselmann---Krause bounded confidence dynamics. Previously, it was known that $$\varOmega (n) = f_{1}(n) = O(n^3)$$Ω(n)=f1(n)=O(n3). Here we show that $$f_{1}(n) = \varOmega (n^2)$$f1(n)=Ω(n2), which matches the best-known lower bound in all dimensions $$k \ge 2$$k?2.

A Computer Search for Planar Substitution Tilings with $$n$$n-Fold Rotational Symmetry

Abstract: We describe a computer algorithm that searches for substitution rules on a set of triangles, the angles of which are all integer multiples of $$\pi /n$$?/n. We find new substitution rules admitting $$7$$7-fold rotational symmetry at many different inflation factors.

New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-Space

Abstract: In this paper we consider ball packings in $$4$$4-dimensional hyperbolic space. We show that it is possible to exceed the conjectured $$4$$4-dimensional realizable packing density upper bound due to L. Fejes-Toth (Regular Figures, Macmillian, New York, 1964). We give seven examples of horoball packing configurations that yield higher densities of $$0.71644896\dots $$0.71644896?, where horoballs are centered at ideal vertices of certain Coxeter simplices, and are invariant under the actions of their respective Coxeter groups.

New Results for the Growth of Sets of Real Numbers

Abstract: We use the theory of cross ratios to construct a real-valued function f of only three variables with the property that for any finite set A of reals, the set $$f(A)=\{f(a,b,c):a,b,c \in A\}$$f(A)={f(a,b,c):a,b,c?A} has cardinality at least $$C|A|^2/\log |A|$$C|A|2/log|A|, for an absolute constant C. Previously-known functions with this property have all been of four variables. We also improve on the state of the art for functions of four variables by constructing a function g for which g(A) has cardinality at least $$C|A|^2$$C|A|2; the previously best-achieved bound was $$C|A|^2/\log |A|$$C|A|2/log|A|. Finally, we give an example of a five-variable function h for which h(A) has cardinality at least $$C|A|^4/\log |A|$$C|A|4/log|A|. Proving these results depends only on the Szemeredi---Trotter incidence theorem and an analoguous result for planes due to Edelsbrunner, Guibas and Sharir, each applied in the Erlangen-type framework of Elekes and Sharir. In particular the proofs do not employ the Guth---Katz polynomial partitioning technique or the theory of ruled surfaces. Although the growth exponents for f, g and h are stronger than those for previously-considered functions, it is not clear that they are necessarily sharp. So we pose a question as to whether the bounds on the cardinalities of f(A), g(A) and h(A) can be further strengthened.

Local Digital Algorithms for Estimating the Integrated Mean Curvature of r-Regular Sets

Abstract: Suppose an r-regular set is sampled on a random lattice A fast algorithm for estimating the integrated mean curvature is to use a weighted sum of $$2\times \cdots \times 2$$2×?×2 configuration counts We show that for a randomly translated lattice, no asymptotically unbiased estimator of this type exists in dimensions larger than two, while for stationary isotropic lattices, asymptotically unbiased estimators are plenty The basis for this is a formula for the asymptotic behavior of hit-or-miss transforms of r-regular sets

Geometry-driven Collapses for Converting a Čech Complex into a Triangulation of a Nicely Triangulable Shape

Abstract: Given a set of points that sample a shape, the Rips complex of the points is often used to provide an approximation of the shape easily-computed. It has been proved that the Rips complex captures the homotopy type of the shape, assuming that the vertices of the complex meet some mild sampling conditions. Unfortunately, the Rips complex is generally high-dimensional. To remedy this problem, it is tempting to simplify it through a sequence of collapses. Ideally, we would like to end up with a triangulation of the shape. Experiments suggest that, as we simplify the complex by iteratively collapsing faces, it should indeed be possible to avoid entering a dead end such as the famous Bing's house with two rooms. This paper provides a theoretical justification for this empirical observation. We demonstrate that the Rips complex of a point cloud (for a well-chosen scale parameter) can always be turned into a simplicial complex homeomorphic to the shape by a sequence of collapses, assuming that the shape is nicely triangulable and well-sampled (two concepts we will explain in the paper). To establish our result, we rely on a recent work which gives conditions under which the Rips complex can be converted into a Cech complex by a sequence of collapses. We proceed in two phases. Starting from the Cech complex, we first produce a sequence of collapses that arrives to the Cech complex, restricted by the shape. We then apply a sequence of collapses that transforms the result into the nerve of some covering of the shape. Along the way, we establish results which are of independent interest. First, we show that the reach of a shape cannot decrease when intersected with a (possibly infinite) collection of balls, assuming the balls are small enough. Under the same hypotheses, we show that the restriction of a shape with respect to an intersection of balls is either empty or contractible. We also provide conditions under which the nerve of a family of compact sets undergoes collapses as the compact sets evolve over time. We believe conditions are general enough to be useful in other contexts as well.

Polytopes with Preassigned Automorphism Groups

Abstract: We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.

Indecomposable Coverings with Homothetic Polygons

Abstract: We prove that for any convex polygon $$S$$S with at least four sides, or a concave one with no parallel sides, and any $$m>0$$m>0, there is an $$m$$m-fold covering of the plane with homothetic copies of $$S$$S that cannot be decomposed into two coverings.

Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

Abstract: We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Mobius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $${\mathbb {Q}}$$Q are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnev universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

On the Geometric Ramsey Number of Outerplanar Graphs

Abstract: We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-$$2$$2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on $$2n$$2n vertices are bounded by $$O(n^{3})$$O(n3) and $$O(n^{10})$$O(n10), in the convex and general case, respectively. We then apply similar methods to prove an $$n^{O(\log (n))}$$nO(log(n)) upper bound on the Ramsey number of a path with $$n$$n ordered vertices.