A path formula for the sock sorting problem

Simon Korbel; Simon Korbel; Peter Mörters; Peter Mörters

doi:10.4213/tvp5506

A path formula for the sock sorting problem

Теория вероятностей и ее применения ◽

10.4213/tvp5506 ◽

2021 ◽

Vol 66 (4) ◽

pp. 889-894

Author(s):

Simon Korbel ◽

Peter Mörters ◽

Peter Mörters

Keyword(s):

Sorting Problem

Допустим, что в сушильном барабане находятся $n$ различных пар носков. По окончании сушки носки выкладываются на стол один за другим. Если очередной вынутый носок оказывается из той же пары, что и один из лежащих на столе, то пара убирается, если нет, то носок остается на столе до тех пор, пока из сушки не появится носок из его пары. Каждый раз, когда один из $2n$ носков выкладывается на стол, мы записываем число носков, остающихся на столе. В работе получена явная формула для вероятности события, состоящего в том, что полученная последовательность совпадает с заданной последовательностью длины $2n$.

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Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax

Natural Computing ◽

10.1007/s11047-021-09856-0 ◽

2021 ◽

Author(s):

Moritz Mühlenthaler ◽

Alexander Raß ◽

Manuel Schmitt ◽

Rolf Wanka

Keyword(s):

Particle Swarm Optimization ◽

Markov Chain ◽

Markov Model ◽

Optimization Problems ◽

Formal Analysis ◽

Particle Swarm ◽

Upper And Lower Bounds ◽

Swarm Optimization ◽

Local Optima ◽

Sorting Problem

AbstractMeta-heuristics are powerful tools for solving optimization problems whose structural properties are unknown or cannot be exploited algorithmically. We propose such a meta-heuristic for a large class of optimization problems over discrete domains based on the particle swarm optimization (PSO) paradigm. We provide a comprehensive formal analysis of the performance of this algorithm on certain “easy” reference problems in a black-box setting, namely the sorting problem and the problem OneMax. In our analysis we use a Markov model of the proposed algorithm to obtain upper and lower bounds on its expected optimization time. Our bounds are essentially tight with respect to the Markov model. We show that for a suitable choice of algorithm parameters the expected optimization time is comparable to that of known algorithms and, furthermore, for other parameter regimes, the algorithm behaves less greedy and more explorative, which can be desirable in practice in order to escape local optima. Our analysis provides a precise insight on the tradeoff between optimization time and exploration. To obtain our results we introduce the notion of indistinguishability of states of a Markov chain and provide bounds on the solution of a recurrence equation with non-constant coefficients by integration.

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T-SNE visualization of large-scale neural recordings

10.1101/087395 ◽

2016 ◽

Cited By ~ 5

Author(s):

George Dimitriadis ◽

Joana Neto ◽

Adam R. Kampff

Keyword(s):

Large Scale ◽

New Technologies ◽

Dimensional Space ◽

Clustering Algorithms ◽

Brain Regions ◽

Data Sets ◽

Neural Recordings ◽

Sorting Problem ◽

Feature Spaces ◽

High Degree

AbstractElectrophysiology is entering the era of ‘Big Data’. Multiple probes, each with hundreds to thousands of individual electrodes, are now capable of simultaneously recording from many brain regions. The major challenge confronting these new technologies is transforming the raw data into physiologically meaningful signals, i.e. single unit spikes. Sorting the spike events of individual neurons from a spatiotemporally dense sampling of the extracellular electric field is a problem that has attracted much attention [22, 23], but is still far from solved. Current methods still rely on human input and thus become unfeasible as the size of the data sets grow exponentially.Here we introduce the t-student stochastic neighbor embedding (t-sne) dimensionality reduction method [27] as a visualization tool in the spike sorting process. T-sne embeds the n-dimensional extracellular spikes (n = number of features by which each spike is decomposed) into a low (usually two) dimensional space. We show that such embeddings, even starting from different feature spaces, form obvious clusters of spikes that can be easily visualized and manually delineated with a high degree of precision. We propose that these clusters represent single units and test this assertion by applying our algorithm on labeled data sets both from hybrid [23] and paired juxtacellular/extracellular recordings [15]. We have released a graphical user interface (gui) written in python as a tool for the manual clustering of the t-sne embedded spikes and as a tool for an informed overview and fast manual curration of results from other clustering algorithms. Furthermore, the generated visualizations offer evidence in favor of the use of probes with higher density and smaller electrodes. They also graphically demonstrate the diverse nature of the sorting problem when spikes are recorded with different methods and arise from regions with different background spiking statistics.

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