An Efficient Representation for Filtrations of Simplicial Complexes

A filtration over a simplicial complex $K$ is an ordering of the simplices of $K$ such that all prefixes in the ordering are subcomplexes of $K$. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest. In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica '17]. Our data structure allows one to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Finally, we show that the CSD representation admits fast construction algorithms for Flag complexes and relaxed Delaunay complexes.Comment: A preliminary version appeared in SODA 201

Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

Structural Analysis: Shape Information via Points-To Computation

This paper introduces a new hybrid memory analysis, Structural Analysis, which combines an expressive shape analysis style abstract domain with efficient and simple points-to style transfer functions. Using data from empirical studies on the runtime heap structures and the programmatic idioms used in modern object-oriented languages we construct a heap analysis with the following characteristics: (1) it can express a rich set of structural, shape, and sharing properties which are not provided by a classic points-to analysis and that are useful for optimization and error detection applications (2) it uses efficient, weakly-updating, set-based transfer functions which enable the analysis to be more robust and scalable than a shape analysis and (3) it can be used as the basis for a scalable interprocedural analysis that produces precise results in practice. The analysis has been implemented for .Net bytecode and using this implementation we evaluate both the runtime cost and the precision of the results on a number of well known benchmarks and real world programs. Our experimental evaluations show that the domain defined in this paper is capable of precisely expressing the majority of the connectivity, shape, and sharing properties that occur in practice and, despite the use of weak updates, the static analysis is able to precisely approximate the ideal results. The analysis is capable of analyzing large real-world programs (over 30K bytecodes) in less than 65 seconds and using less than 130MB of memory. In summary this work presents a new type of memory analysis that advances the state of the art with respect to expressive power, precision, and scalability and represents a new area of study on the relationships between and combination of concepts from shape and points-to analyses

FLASH: Randomized Algorithms Accelerated over CPU-GPU for Ultra-High Dimensional Similarity Search

We present FLASH (\textbf{F}ast \textbf{L}SH \textbf{A}lgorithm for \textbf{S}imilarity search accelerated with \textbf{H}PC), a similarity search system for ultra-high dimensional datasets on a single machine, that does not require similarity computations and is tailored for high-performance computing platforms. By leveraging a LSH style randomized indexing procedure and combining it with several principled techniques, such as reservoir sampling, recent advances in one-pass minwise hashing, and count based estimations, we reduce the computational and parallelization costs of similarity search, while retaining sound theoretical guarantees. We evaluate FLASH on several real, high-dimensional datasets from different domains, including text, malicious URL, click-through prediction, social networks, etc. Our experiments shed new light on the difficulties associated with datasets having several million dimensions. Current state-of-the-art implementations either fail on the presented scale or are orders of magnitude slower than FLASH. FLASH is capable of computing an approximate k-NN graph, from scratch, over the full webspam dataset (1.3 billion nonzeros) in less than 10 seconds. Computing a full k-NN graph in less than 10 seconds on the webspam dataset, using brute-force ($n^2D$), will require at least 20 teraflops. We provide CPU and GPU implementations of FLASH for replicability of our results

The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website