A Pseudorandom Sequence Generated over a Finite Field Using The Möbius Function

The aim of this paper is to generate and examine a pseudorandom sequence over a finite field using the Möbius function. In the main part of the paper, after generating a number of sequences using the Möbius function, we examine the sequences’ pseudorandomness using autocorrelation and prove that the second half of any sequence in $\mathbb{F}_{3^n}$ is the same as the first, but for the sign of the terms. I reach the conclusion, that it is preferable to generate sequences in fields of the form $\mathbb{F}_{3^n}$, thereby obtaining a sequence of the numbers $-1$,$0$,$1$, each of which appear in the same amounts. There is a variety of applications of the discussed pseudorandom generator and other generators such as cryptography or randomized algorithms.

New correlative randomized algorithms for statistical modelling of radiation transfer in stochastic medium

Correlative randomized algorithms are constructed by simple randomization of the algorithm of maximum cross-section (equalization, delta tracking) with the use of a one-dimensional distribution and the correlation function or only correlation length of a random medium. The value of the used correlation length can be adjusted using simple test studies. The calculations carried out confirmed the practical effectiveness of the new algorithms.

A Framework for Adversarially Robust Streaming Algorithms

We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems.

Concurrent disjoint set union

We develop and analyze concurrent algorithms for the disjoint set union (“union-find” ) problem in the shared memory, asynchronous multiprocessor model of computation, with CAS (compare and swap) or DCAS (double compare and swap) as the synchronization primitive. We give a deterministic bounded wait-free algorithm that uses DCAS and has a total work bound of $$O\biggl ( m \cdot \left( \log {\left( \frac{np}{m} + 1 \right) } + \alpha {\left( n, \frac{m}{np} \right) } \right) \biggr )$$ O ( m · log np m + 1 + α n , m np ) for a problem with n elements and m operations solved by p processes, where $$\alpha $$ α is a functional inverse of Ackermann’s function. We give two randomized algorithms that use only CAS and have the same work bound in expectation. The analysis of the second randomized algorithm is valid even if the scheduler is adversarial. Our DCAS and randomized algorithms take $$O(\log n)$$ O ( log n ) steps per operation, worst-case for the DCAS algorithm, high-probability for the randomized algorithms. Our work and step bounds grow only logarithmically with p, making our algorithms truly scalable. We prove that for a class of symmetric algorithms that includes ours, no better step or work bound is possible. Our work is theoretical, but Alistarh et al (In search of the fastest concurrent union-find algorithm, 2019), Dhulipala et al (A framework for static and incremental parallel graph connectivity algorithms, 2020) and Hong et al (Exploring the design space of static and incremental graph connectivity algorithms on gpus, 2020) have implemented some of our algorithms on CPUs and GPUs and experimented with them. On many realistic data sets, our algorithms run as fast or faster than all others.