Neighborly inscribed polytopes and Delaunay triangulations

We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there are superexponentially many combinatorially distinct neighborly polytopes that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations.Comment: 15 pages, 2 figures. We extended our results to arbitrary smooth strictly convex bodie

A Poisson sample of a smooth surface is a good sample

International audienceThe complexity of the 3D-Delaunay triangulation (tetrahedralization) of n points distributed on a surface ranges from linear to quadratic. When the points are a deterministic good sample of a smooth compact generic surface, the size of the Delaunay triangulation is O(n log n). Using this result, we prove that when points are Poisson distributed on a surface under the same hypothesis, whose expected number of vertices is λ, the expected size is O(λ log^2 λ)

A Framework for Algorithm Stability

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees

Constructing Delaunay triangulations along space-filling curves

Incremental construction con BRIO using a space-filling curve order for insertion is a popular algorithm for constructing Delaunay triangulations. So far, it has only been analyzed for the case that a worst-case optimal point location data structure is used which is often avoided in implementations. In this paper, we analyze its running time for the more typical case that points are located by walking. We show that in the worst-case the algorithm needs quadratic time, but that this can only happen in degenerate cases. We show that the algorithm runs in O(n logn) time under realistic assumptions. Furthermore, we show that it runs in expected linear time for many random point distributions. This research was supported by the Deutsche Forschungsgemeinschaft within the European graduate program ’Combinatorics, Geometry, and Computation’ (No. GRK 588/2) and by the Netherlands’ Organisation for Scientific Research (NWO) under BRICKS/FOCUS grant number 642.065.503 and project no. 639.022.707

Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm

Light Euclidean Spanners with Steiner Points

The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy $(1+\epsilon)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(\epsilon^{-d})$ for any $d = O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$ (where $\tilde{O}$ hides polylogarithmic factors of $\frac{1}{\epsilon}$), and also shows the existence of point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have lightness $\Omega(\epsilon^{-d})$. Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in $\mathbb{R}^2$ with lightness $O(\epsilon^{-1} \log \Delta)$, where $\Delta$ is the spread of the point set. In the regime of $\Delta \ll 2^{1/\epsilon}$, this provides an improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications $\epsilon$ often controls the precision, and it sometimes needs to be much smaller than $O(1/\log n)$. Moreover, for spread polynomially bounded in $1/\epsilon$, this upper bound provides a quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019], We then demonstrate that such a light spanner can be constructed in $O_{\epsilon}(n)$ time for polynomially bounded spread, where $O_{\epsilon}$ hides a factor of $\mathrm{poly}(\frac{1}{\epsilon})$. Finally, we extend the construction to higher dimensions, proving a lightness upper bound of $\tilde{O}(\epsilon^{-(d+1)/2} + \epsilon^{-2}\log \Delta)$ for any $3\leq d = O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$.Comment: 23 pages, 2 figures, to appear in ESA 2

The impact of heterogeneity and geometry on the proof complexity of random satisfiability

Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real-world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random -SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random -SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time