DA14 Abstracts 27

DA14 Abstracts 27 IP1 [email protected] Shape, Homology, Persistence, and Stability CP0 My personal journey to the fascinating world of geomet- Distributed Computation of Persistent Homology ric forms started 30 years ago with the invention of al- pha shapes in the plane. It took about 10 years before Advances in algorithms for computing persistent homol- we generalized the concept to higher dimensions, we pro- ogy have reduced the computation time drastically – as duced working software with a graphics interface for the long as the algorithm does not exhaust the available mem- 3-dimensional case, and we added homology to the com- ory. Following up on a recently presented parallel method putations. Needless to say that this foreshadowed the in- for persistence computation on shared memory systems, we ception of persistent homology, because it suggested the demonstrate that a simple adaption of the standard reduc- study of filtrations to capture the scale of a shape or data tion algorithm leads to a variant for distributed systems. set. Importantly, this method has fast algorithms. The Our algorithmic design ensures that the data is distributed arguably most useful result on persistent homology is the over the nodes without redundancy; this permits the com- stability of its diagrams under perturbations. putation of much larger instances than on a single machine. The parallelism often speeds up the computation compared Herbert Edelsbrunner to sequential and even parallel shared memory algorithms. Institute of Science and Technology Austria In our experiments, we were able to compute the persis- [email protected] tent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 10GB of memory per node. IP2 Ulrich Bauer Recent Developments in the Sparse Fourier Trans- Institute of Science and Technology (Austria) form [email protected] The Fast Fourier Transform (FFT) is a widely used numer- Michael Kerber ical algorithm. It computes the Discrete Fourier Transform Stanford University (DFT) of an n-dimensional signal in O(n log n) time. It [email protected] is not known whether its running time can be improved. However, in many applications, most of the Fourier coef- Jan Reininghaus ficients of a signal are ”small” or equal to zero, i.e., the IST Austria output of the transform is (approximately) sparse. In this [email protected] case, it is known that one can compute the set of non-zero coefficients much faster, even in sub-linear time. In this talk I will give an overview of recent highly efficient algo- CP0 rithms for computing the Sparse Fourier Transform. I will On the Asymptotic Number of BCK(2)-Terms also give a few examples of applications impacted by these developments. We investigate the asymptotic number of a particular class of closed lambda-terms. This class is a generalization of a Piotr Indyk class of terms related to the axiom system BCK which is Massachusetts Institute of Technology well known in combinatory logic. We determine the asymp- [email protected] totic number of terms, when their size tends to infinity, up to a constant multiplicative factor and discover a surprising asymptotic behaviour involving an exponential with frac- IP3 tional powers in the exponent. Interlacing Families: Mixed Characteristic Polyno- Olivier Bodini mials and the Kadison-Singer Problem LIPN, Université Paris 13 [email protected] We introduce a new technique for demonstrating the ex- istence of combinatorial objects that we call the “Method Bernhard Gittenberger of Interlacing Polynomials’. We then use this technique Institute for Discrete Mathematics and Geometry, TU to prove a conjecture in discrepancy theory posed by Nik Wien Weaver. Weaver’s conjecture implies the truth of the [email protected] Paving Conjecture of Akemann and Anderson, which in turn provides a solution to the fifty year old problem of Kadison and Singer. Our main technical result is an upper CP0 bound on the largest root of the expected characteristic On the Scalability of Computing Triplet and Quar- polynomial of a sum of random symmetric rank-1 matri- tet Distances ces. We use this bound to prove that there exists a balanced partition of certain sets of vectors. This result can be We present an experimental evaluation of the algorithms viewed as a strengthening of the ”matrix Chernoff bounds” by Brodal et al. [SODA 2013] for computing the triplet and of Ahlswede and Winter, Rudelson and Vershynin, and quartet distance measures between two leaf labelled rooted Tropp. This is joint work with Adam Marcus and Nikhil and unrooted trees of arbitrary degree, respectively. In our Srivastava. experimental evaluation of the algorithms the typical over- head is about 2 KB and 10 KB per node in the input trees Daniel Spielman for triplet and quartet computations, respectively. This al- Yale University lows us to compute the distance measures for trees with up 28 DA14 Abstracts to millions of nodes. of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Gerth S. Brodal MADALGO, Aarhus University Gwendal Collet [email protected] LIX, Ecole Polytechnique [email protected] Jens Johansen, Morten Holt Aarhus University Olivier Bernardi [email protected], [email protected] Brandeis University, MA [email protected] CP0 Eric Fusy Enumerating Fundamental Normal Surfaces: Algo- LIX, Ecole Polytechnique, France rithms, Experiments and Invariants [email protected] Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the CP0 fact that many key algorithms have complexity bounds A Statistical View on Exchanges in Quickselect that are exponential or greater. In this setting, experimen- tation is essential for understanding the limits of practical- In this paper we study the number of key exchanges re- ity, as well as for gauging the relative merits of competing quired by Hoare’s FIND algorithm (also called Quickselect) algorithms. Here we focus on normal surface theory, a key when operating on a uniformly distributed random permu- tool in low-dimensional computational topology. In par- tation and selecting an independent uniformly distributed ticular, we aim to compute fundamental normal surfaces, rank. After normalization we give a limit theorem where which form an integral basis for a high-dimensional poly- the limit law is a perpetuity characterized by a recursive hedral cone. We develop, implement and experimentally distributional equation. To make the limit theorem usable compare a primal and a dual algorithm, both of which for statistical methods and statistical experiments we pro- combine domain-specific techniques with classical Hilbert vide an explicit rate of convergence in the Kolmogorov– basis algorithms. Our experiments indicate that we can Smirnov metric, a numerical table of the limit law’s dis- solve extremely large problems that were once though in- tribution function and an algorithm for exact simulation tractable. As a practical application, we fill gaps from the from the limit distribution. We also investigate the limit KnotInfo database by computing 398 previously-unknown law’s density. This case study provides a program applica- crosscap numbers of knots. ble to other cost measures, alternative models for the rank selected and more balanced choices of the pivot element Benjamin A. Burton such as median-of-2t + 1 versions of Quickselect as well as The University of Queensland further variations of the algorithm. [email protected] Benjamin Dadoun ´ CP0 Ecole Normale SupérieuredeCachan Département Informatique An Exact Approach To Upward Crossing Mini- [email protected] mization The upward crossing number problem asks for a drawing Ralph Neininger of the graph into the plane with the minimum number of J.W. Goethe University, Frankfurt edge crossings where the edges are drawn as monotonously [email protected] increasing curves w.r.t. the y-axis. While there is a large body of work on solving this central graph drawing problem heuristically, we present the first approach to solve the CP0 problem to proven optimality. Our approach is based on Small Superpatterns for Dominance Drawing a reformulation of the problem as a boolean formula that can be iteratively tightened and resolved. In our experi- We exploit the connection between dominance drawings of ments, we show the practical applicability and limits of our directed acyclic graphs and permutations, in both direc- approach. tions, to provide improved bounds on the size of universal point sets for certain types of dominance drawing and on Robert Zeranski superpatterns for certain natural classes of permutations. Friedrich-Schiller-Universität Jena In particular we show that there exist universal point sets [email protected] for dominance drawings of the Hasse diagrams of width-two partial orders of size O(n3/2), universal point sets for dom- Markus Chimani inance drawings of st-outerplanar graphs of size O(n log n), Osnabrueck University and universal point sets for dominance drawings of directed [email protected] trees of size O(n2). We show that 321-avoiding permutations have superpatterns of size O(n3/2), riffle permutations (321-, 2143-, and 2413-avoiding permutations) have CP0 superpatterns of size O(n), and the concatenations of se- A Bijection for Plane Graphs and Its Applications quences of riffles and their inverses have superpatterns of size O(n log n). Our analysis includes a calculation of the This paper is concerned with the counting and random leading constants in these bounds. sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the Michael J. Bannister, William Devanny, David Eppstein class of plane graphs with triangular outer face, and a class University of California, Irvine DA14 Abstracts 29 Department of Computer Science greatly simplify the classical enumeration of these texts [email protected], [email protected], [email protected] by inclusion-exclusion approaches.

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