Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails

The Frechet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original algorithm to compute the Frechet distance of two polygonal curves with n vertices has a runtime of O(n^2 log n). More than 20 years later, the state of the art algorithms for most variants still take time more than O(n^2 / log n), but no matching lower bounds are known, not even under reasonable complexity theoretic assumptions. To obtain a conditional lower bound, in this paper we assume the Strong Exponential Time Hypothesis or, more precisely, that there is no O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption we show that the Frechet distance cannot be computed in strongly subquadratic time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subquadratic algorithm can be considered unlikely. Our result holds for both the continuous and the discrete Frechet distance. We extend the main result in various directions. Based on the same assumption we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2) present tight lower bounds in case the numbers of vertices of the two curves are imbalanced, and (3) examine realistic input assumptions (c-packed curves)

Applications of Bee Colony Optimization

Many computationally difficult problems are attacked using non-exact algorithms, such as approximation algorithms and heuristics. This thesis investigates an ex- ample of the latter, Bee Colony Optimization, on both an established optimization problem in the form of the Quadratic Assignment Problem and the FireFighting problem, which has not been studied before as an optimization problem. Bee Colony Optimization is a swarm intelligence algorithm, a paradigm that has increased in popularity in recent years, and many of these algorithms are based on natural pro- cesses. We tested the Bee Colony Optimization algorithm on the QAPLIB library of Quadratic Assignment Problem instances, which have either optimal or best known solutions readily available, and enabled us to compare the quality of solutions found by the algorithm. In addition, we implemented a couple of other well known algorithms for the Quadratic Assignment Problem and consequently we could analyse the runtime of our algorithm. We introduce the Bee Colony Optimization algorithm for the FireFighting problem. We also implement some greedy algorithms and an Ant Colony Optimization al- gorithm for the FireFighting problem, and compare the results obtained on some randomly generated instances. We conclude that Bee Colony Optimization finds good solutions for the Quadratic Assignment Problem, however further investigation on speedup methods is needed to improve its performance to that of other algorithms. In addition, Bee Colony Optimization is effective on small instances of the FireFighting problem, however as instance size increases the results worsen in comparison to the greedy algorithms, and more work is needed to improve the decisions made on these instances

A domain-specific language and matrix-free stencil code for investigating electronic properties of Dirac and topological materials

We introduce PVSC-DTM (Parallel Vectorized Stencil Code for Dirac and Topological Materials), a library and code generator based on a domain-specific language tailored to implement the specific stencil-like algorithms that can describe Dirac and topological materials such as graphene and topological insulators in a matrix-free way. The generated hybrid-parallel (MPI+OpenMP) code is fully vectorized using Single Instruction Multiple Data (SIMD) extensions. It is significantly faster than matrix-based approaches on the node level and performs in accordance with the roofline model. We demonstrate the chip-level performance and distributed-memory scalability of basic building blocks such as sparse matrix-(multiple-) vector multiplication on modern multicore CPUs. As an application example, we use the PVSC-DTM scheme to (i) explore the scattering of a Dirac wave on an array of gate-defined quantum dots, to (ii) calculate a bunch of interior eigenvalues for strong topological insulators, and to (iii) discuss the photoemission spectra of a disordered Weyl semimetal.Comment: 16 pages, 2 tables, 11 figure

Learning Tuple Probabilities

Learning the parameters of complex probabilistic-relational models from labeled training data is a standard technique in machine learning, which has been intensively studied in the subfield of Statistical Relational Learning (SRL), but---so far---this is still an under-investigated topic in the context of Probabilistic Databases (PDBs). In this paper, we focus on learning the probability values of base tuples in a PDB from labeled lineage formulas. The resulting learning problem can be viewed as the inverse problem to confidence computations in PDBs: given a set of labeled query answers, learn the probability values of the base tuples, such that the marginal probabilities of the query answers again yield in the assigned probability labels. We analyze the learning problem from a theoretical perspective, cast it into an optimization problem, and provide an algorithm based on stochastic gradient descent. Finally, we conclude by an experimental evaluation on three real-world and one synthetic dataset, thus comparing our approach to various techniques from SRL, reasoning in information extraction, and optimization