Uli Wagner

Online Conflict-Free Coloring for Intervals

We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√n) colors in the worst case. We then modify this approach, to obtain an efficient deterministic algorithm that uses a maximum of Θ(log2 n) colors. Next, we present two randomized solutions. The first algorithm requires an expected number of at most O(log2 n) colors, and produces a coloring which is valid with high probability, and the second one, which is a variant of our efficient deterministic algorithm, requires an expected number of at most O(log n log log n) colors but always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case. The average-case behavior for disks, and cases involving other planar ranges, are still open.

Shape dimension and intrinsic metric from samples of manifolds with high co-dimension

We introduce the <i>adaptive neighborhood graph</i> as a data structure for modeling a smooth manifold <i>M</i> embedded in some (potentially very high-dimensional) Euclidean space <i>R^d</i>. We assume that <i>M</i> is known to us only through a finite sample <i>P? M</i>, as it is often the case in applications. The adaptive neighborhood graph is a geometric graph on <i>P</i>. Its complexity is at most <i>min[2^O(k)n, n²]</i>, where <i>n=|P|</i> and <i>k=dim M</i>, as opposed to the <i>n^[d/2]</i> complexity of the Delaunay triangulation, which is often used to model manifolds. We show that we can provably correctly infer the connectivity of <i>M</i> and the dimension of <i>M</i> from the adaptive neighborhood graph provided a certain standard sampling condition is fulfilled. The running time of the dimension detection algorithm is <i>d2^{O(k⁷log k)}</i> for each connected component of <i>M</i>. If the dimension is considered constant, this is a constant-time operation, and the adaptive neighborhood graph is of linear size. Moreover, the exponential dependence of the constants is only on the<i>intrinsic</i> dimension <i>k</i>, not on the ambient dimension <i>d</i>. This is of particular interest if the co-dimension is high, i.e., if <i>k</i> is much smaller than <i>d</i>, as is the case in many applications. The adaptive neighborhood graph also allows us to approximate the geodesic distances between the points in <i>P</i>.

Computing All Maps into a Sphere

Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X → Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of such maps. We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d−2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X=S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A → Y and ask whether it extends to a map X → Y, or computing the ℤ2-index—everything in the stable range. Outside the stable range, the extension problem is undecidable.

On laplacians of random complexes

Eigenvalues associated to graphs are a well-studied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n,p) are known quite precisely. We consider generalizations of these matrices to simplicial complexes of higher dimensions and study their eigenvalues for the Linial--Meshulam model Xk(n,p) of random k-dimensional simplicial complexes on n vertices. We show that for p=Ω(log n/n), the eigenvalues of both, the higher-dimensional adjacency matrix and the Laplacian, are a.a.s.~sharply concentrated around two values. In a second part of the paper, we discuss a possible higher-dimensional analogue of the Discrete Cheeger Inequality. This fundamental inequality expresses a close relationship between the eigenvalues of a graph and its combinatorial expansion properties; in particular, spectral expansion (a large eigenvalue gap) implies edge expansion. Recently, a higher-dimensional analogue of edge expansion for simplicial complexes was introduced by Gromov, and independently by Linial, Meshulam and Wallach and by Newman and Rabinovich. It is natural to ask whether there is a higher-dimensional version of Cheegers inequality. We show that the most straightforward version of a higher-dimensional Cheeger inequality fails: for every k>1, there is an infinite family of k-dimensional complexes that are spectrally expanding (there is a large eigenvalue gap for the Laplacian) but not combinatorially expanding.

New Constructions of Weak ε-Nets

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Minors in random and expanding hypergraphs

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Shape Dimension and Intrinsic Metric from Samples of Manifolds

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Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimension

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Embeddability in the 3-sphere is decidable

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Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

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