Fast Deterministic Selection

The Median of Medians (also known as BFPRT) algorithm, although a landmark theoretical achievement, is seldom used in practice because it and its variants are slower than simple approaches based on sampling. The main contribution of this paper is a fast linear-time deterministic selection algorithm QuickselectAdaptive based on a refined definition of MedianOfMedians. The algorithm's performance brings deterministic selection---along with its desirable properties of reproducible runs, predictable run times, and immunity to pathological inputs---in the range of practicality. We demonstrate results on independent and identically distributed random inputs and on normally-distributed inputs. Measurements show that QuickselectAdaptive is faster than state-of-the-art baselines.Comment: Pre-publication draf

QuickXsort: Efficient Sorting with n log n - 1.399n +o(n) Comparisons on Average

In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. With QuickWeakHeapsort and QuickMergesort we present two examples for the QuickXsort-construction. Both are efficient algorithms that incur approximately n log n - 1.26n +o(n) comparisons on the average. A worst case of n log n + O(n) comparisons can be achieved without significantly affecting the average case. Furthermore, we describe an implementation of MergeInsertion for small n. Taking MergeInsertion as a base case for QuickMergesort, we establish a worst-case efficient sorting algorithm calling for n log n - 1.3999n + o(n) comparisons on average. QuickMergesort with constant size base cases shows the best performance on practical inputs: when sorting integers it is slower by only 15% to STL-Introsort

Path Similarity Analysis: a Method for Quantifying Macromolecular Pathways

Diverse classes of proteins function through large-scale conformational changes; sophisticated enhanced sampling methods have been proposed to generate these macromolecular transition paths. As such paths are curves in a high-dimensional space, they have been difficult to compare quantitatively, a prerequisite to, for instance, assess the quality of different sampling algorithms. The Path Similarity Analysis (PSA) approach alleviates these difficulties by utilizing the full information in 3N-dimensional trajectories in configuration space. PSA employs the Hausdorff or Fr\'echet path metrics---adopted from computational geometry---enabling us to quantify path (dis)similarity, while the new concept of a Hausdorff-pair map permits the extraction of atomic-scale determinants responsible for path differences. Combined with clustering techniques, PSA facilitates the comparison of many paths, including collections of transition ensembles. We use the closed-to-open transition of the enzyme adenylate kinase (AdK)---a commonly used testbed for the assessment enhanced sampling algorithms---to examine multiple microsecond equilibrium molecular dynamics (MD) transitions of AdK in its substrate-free form alongside transition ensembles from the MD-based dynamic importance sampling (DIMS-MD) and targeted MD (TMD) methods, and a geometrical targeting algorithm (FRODA). A Hausdorff pairs analysis of these ensembles revealed, for instance, that differences in DIMS-MD and FRODA paths were mediated by a set of conserved salt bridges whose charge-charge interactions are fully modeled in DIMS-MD but not in FRODA. We also demonstrate how existing trajectory analysis methods relying on pre-defined collective variables, such as native contacts or geometric quantities, can be used synergistically with PSA, as well as the application of PSA to more complex systems such as membrane transporter proteins.Comment: 9 figures, 3 tables in the main manuscript; supplementary information includes 7 texts (S1 Text - S7 Text) and 11 figures (S1 Fig - S11 Fig) (also available from journal site

Hyperspherical Description of the Degenerate Fermi Gas: S-wave Interactions

We present a unique theoretical description of the physics of the spherically trapped $N$-atom degenerate Fermi gas (DFG) at zero temperature based on an ordinary Schr\"{o}dinger equation with a microscopic, two body interaction potential. With a careful choice of coordinates and a variational wavefunction, the many body Schr\"{o}dinger equation can be accurately described by a \emph{linear}, one dimensional effective Schr\"{o}dinger equation in a single collective coordinate, the rms radius of the gas. Comparisons of the energy, rms radius and peak density of ground state energy are made to those predicted by Hartree-Fock (HF). Also the lowest radial excitation frequency (the breathing mode frequency) agrees with a sum rule calculation, but deviates from a HF prediction