arXiv subjects

Jens Oehlschlägel

Publications and source records attributed to Jens Oehlschlägel.

5 recordsLinked to original sources

arXiv2024

Simple Symmetric Sustainable Sorting -- the greeNsort article

We explored an uncharted part of the solution space for sorting algorithms: the role of symmetry in divide&conquer algorithms. We found/designed novel simple binary Quicksort and Mergesort algorithms operating in contiguous space which achieve improved trade-offs between worst-case CPU-efficiency, best-case adaptivity and RAM-requirements. The 'greeNsort' algorithms need less hardware (RAM) and/or less energy (CPU) compared to the prior art. The new algorithms fit a theoretical framework: 'Footprint' KPIs allow to compare algorithms with different RAM-requirements, a new 'definition' of sorting API-targets simplifies construction of stable algorithms with mirrored scan directions, and our ordinal machine model encourages robust algorithms that minimize access 'distance'. Unlike earlier 'Quicksorts', our 'Zacksort', 'Zucksort' and 'Ducksort' algorithms optimally marry CPU-efficiency and tie-adaptivity. Unlike earlier 'Mergesorts' which required 100% distant buffer, our 'Frogsort' and 'Geckosort' algorithms achieve similar CPU-efficiency with 50% or less local buffer. Unlike natural Mergesorts such as 'Timsort' which are optimized for the best case of full-presorting, our 'Octosort' and 'Squidsort' algorithms achieve excellent bi-adaptivity to presorted best-cases without sacrificing worst-case efficiency in real sorting tasks. Our 'Walksort' and 'Jumpsort' have lower Footprint than the impressive low-memory 'Grailsort' and 'Sqrtsort' of Astrelin. Given the current climate-emergency, this is a call to action for all maintainers of sorting libraries, all software-engineers using custom sorting code, all professors teaching algorithms, all IT professionals designing programming languages, compilers and CPUs: check for better algorithms and consider symmetric code-mirroring.

cs.DS↗

arXiv2019

Reasoning about Primes (I)

We prove the twin prime conjecture and the generalized conjectures of Kronecker and Polignac. Key to the proofs is a new theoretical sieve that combines two concepts that go back to Eratosthenes: the 'sieve' filtering a finite set of numbers and the 'hydra' as a representation of infinity. Using functional programming notation (and a reference implementation in R) we define a data structure 'hydra' that partitions the infinite set of numbers into a finite set of partitions. On top of this data structure we define an algorithm 'split' which sub-partitions each partition using the modulus prime function. Hydra splitting along the natural sequence of primes is a recursive version of wheel factorization. Hydra recursion allows elementary proofs of some statements about primes. We consider one new and two classical proof structures. Using these proof methods we find that hydra recursion proves the infinity of twin primes. Then we show how to use specific selections of primes to create hydras that contain pairs of partitions with arbitrary even distance, which proves Maillet's conjecture and the stricter 'consecutive existence conjecture'. Together with our proof methodology we obtain elementary proofs of Kronecker's and Polignac's conjecture.

math.HO↗

arXiv2015

Reasoning about Primes (II)

We prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes <= p in their doubled primorial interval 0..p#..2p# where we relax a constraint that the primes usually follow: if the bound g(P)=2p-5 for maximizing the gap length applies with more degrees of freedom, it also applies in the more constrained prime game as g(P)<=2p-5, at least in the subregion $[1,{p'}^2]$ where no other primes have influence (p' notates the next other prime). From here proving the mentioned theorems is straightforward, for example Legendre's interval $]n^2,(n+1)^2[$ is located completely inside the valid subregion and is greater than the greatest possible gap. Another consequence is that there must be a prime within n+-(sqrt(n)-1) for all n>1. For small numbers the proofs are verified using the R statistical language.

math.GM↗

arXiv2007

Truecluster matching

Cluster matching by permuting cluster labels is important in many clustering contexts such as cluster validation and cluster ensemble techniques. The classic approach is to minimize the euclidean distance between two cluster solutions which induces inappropriate stability in certain settings. Therefore, we present the truematch algorithm that introduces two improvements best explained in the crisp case. First, instead of maximizing the trace of the cluster crosstable, we propose to maximize a chi-square transformation of this crosstable. Thus, the trace will not be dominated by the cells with the largest counts but by the cells with the most non-random observations, taking into account the marginals. Second, we suggest a probabilistic component in order to break ties and to make the matching algorithm truly random on random data. The truematch algorithm is designed as a building block of the truecluster framework and scales in polynomial time. First simulation results confirm that the truematch algorithm gives more consistent truecluster results for unequal cluster sizes. Free R software is available.

cs.AI↗

arXiv2007

Truecluster: robust scalable clustering with model selection

Data-based classification is fundamental to most branches of science. While recent years have brought enormous progress in various areas of statistical computing and clustering, some general challenges in clustering remain: model selection, robustness, and scalability to large datasets. We consider the important problem of deciding on the optimal number of clusters, given an arbitrary definition of space and clusteriness. We show how to construct a cluster information criterion that allows objective model selection. Differing from other approaches, our truecluster method does not require specific assumptions about underlying distributions, dissimilarity definitions or cluster models. Truecluster puts arbitrary clustering algorithms into a generic unified (sampling-based) statistical framework. It is scalable to big datasets and provides robust cluster assignments and case-wise diagnostics. Truecluster will make clustering more objective, allows for automation, and will save time and costs. Free R software is available.

cs.AI↗