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Selection and sorting the Cartesian sum, X + Y, are classic and important problems. Here, a new algorithm is presented, which generates the top k values of the form Xi+Yj. The algorithm relies on layer-ordered heaps, partial orderings of exponentially sized layers. The algorithm relies only on median-of-medians and is simple to implement. Furthermore, it uses data structures contiguous in memory, cache efficient, and fast in practice. The presented algorithm is demonstrated to be theoretically optimal. 

Selection on the Cartesian product is a classic problem in computer science. Recently, an optimal algorithm for selection on A + B, based on soft heaps, was introduced. By combining this approach with layer-ordered heaps (LOHs), an algorithm using a balanced binary tree of A + B selections was proposed to perform selection on X1 + X2 + ⋯ + Xm in o(n⋅m + k⋅m), where Xi have length n. Here, that o(n⋅m + k⋅m) algorithm is combined with a novel, optimal LOH-based algorithm for selection on A + B (without a soft heap). Performance of algorithms for selection on X1 + X2 + ⋯ + Xm are compared empirically, demonstrating the benefit of the algorithm proposed here. 

Nonparametric statistical tests are an integral part of scientific experiments in a diverse range of fields. When performing such tests, it is standard to sort values; however, this requires Ω(n log(n)) time to sort n values. Thus given enough data, sorting becomes the computational bottleneck, even with very optimized implementations such as the C++ standard library routine, std::sort. Frequently, a nonparametric statistical test is only used to partition values above and below a threshold in the sorted ordering, where the threshold corresponds to a significant statistical result. Linear-time selection and partitioning algorithms cannot be directly used because the selection and partitioning are performed on the transformed statistical significance values rather than on the sorted statistics. Usually, those transformed statistical significance values (e.g., the p value when investigating the family-wise error rate and q values when investigating the false discovery rate (FDR)) can only be computed at a threshold. Because this threshold is unknown, this leads to sorting the data. Layer-ordered heaps, which can be constructed in O(n), only partially sort values and thus can be used to get around the slow runtime required to fully sort. Here we introduce a layer-ordering-based method for selection and partitioning on the transformed values (e.g., p values or q values). We demonstrate the use of this method to partition peptides using an FDR threshold. This approach is applied to speed up Percolator, a postprocessing algorithm used in mass-spectrometry-based proteomics to evaluate the quality of peptide-spectrum matches (PSMs), by >70% on data sets with 100 million PSMs. 

Computation of the isotopic distribution of compounds is crucial to applications of mass spectrometry, particularly as machine precision continues to improve. In the past decade, several tools have been created for doing so. In this paper we present a novel algorithm for calculating either the most abundant k isotopologue peaks of a compound or the minimal set of isotopologue peaks which have a combined total abundance of at least p. The algorithm uses Serang’s optimal method of selection on Cartesian products. The method is significantly faster than the state-of-the-art on large compounds (e.g., Titin protein) and on compounds whose elements have many isotopes (e.g., palladium alloys). 

Accurate protein inference in the presence of shared peptides is still one of the key problems in bottom-up proteomics. Most protein inference tools employing simple heuristic inference strategies are efficient but exhibit reduced accuracy. More advanced probabilistic methods often exhibit better inference quality but tend to be too slow for large data sets. Here, we present a novel protein inference method, EPIFANY, combining a loopy belief propagation algorithm with convolution trees for efficient processing of Bayesian networks. We demonstrate that EPIFANY combines the reliable protein inference of Bayesian methods with significantly shorter runtimes. On the 2016 iPRG protein inference benchmark data, EPIFANY is the only tested method that finds all true-positive proteins at a 5% protein false discovery rate (FDR) without strict prefiltering on the peptide-spectrum match (PSM) level, yielding an increase in identification performance (+10% in the number of true positives and +14% in partial AUC) compared to previous approaches. Even very large data sets with hundreds of thousands of spectra (which are intractable with other Bayesian and some non-Bayesian tools) can be processed with EPIFANY within minutes. The increased inference quality including shared peptides results in better protein inference results and thus increased robustness of the biological hypotheses generated. EPIFANY is available as open-source software for all major platforms at https://OpenMS.de/epifany. 

In the metabolomics, glycomics, and mass spectrometry of structured small molecules, the combinatoric nature of the problem renders a database impossibly large, and thus de novo analysis is necessary. De novo analysis requires an alphabet of mass difference values used to link peaks in fragmentation spectra when they are different by a mass in the alphabet divided by a charge. Often, this alphabet is not known, prohibiting de novo analysis. A method is proposed that, given fragmentation mass spectra, identifies an alphabet of m/z differences that can build large connected graphs from many intense peaks in each spectrum from a collection. We then introduce a novel approach to efficiently find recurring substructures in the de novo graph results.