Optimal Deterministic Massively Parallel Connectivity on Forests

We show fast deterministic algorithms for fundamental problems on forests in the challenging low-space regime of the well-known Massive Parallel Computation (MPC) model. A recent breakthrough result by Coy and Czumaj [STOC'22] shows that, in this setting, it is possible to deterministically identify connected components on graphs in $O(\log D + \log\log n)$ rounds, where $D$ is the diameter of the graph and $n$ the number of nodes. The authors left open a major question: is it possible to get rid of the additive $\log\log n$ factor and deterministically identify connected components in a runtime that is completely independent of $n$? We answer the above question in the affirmative in the case of forests. We give an algorithm that identifies connected components in $O(\log D)$ deterministic rounds. The total memory required is $O(n+m)$ words, where $m$ is the number of edges in the input graph, which is optimal as it is only enough to store the input graph. We complement our upper bound results by showing that $\Omega(\log D)$ time is necessary even for component-unstable algorithms, conditioned on the widely believed 1 vs. 2 cycles conjecture. Our techniques also yield a deterministic forest-rooting algorithm with the same runtime and memory bounds. Furthermore, we consider Locally Checkable Labeling problems (LCLs), whose solution can be verified by checking the $O(1)$-radius neighborhood of each node. We show that any LCL problem on forests can be solved in $O(\log D)$ rounds with a canonical deterministic algorithm, improving over the $O(\log n)$ runtime of Brandt, Latypov and Uitto [DISC'21]. We also show that there is no algorithm that solves all LCL problems on trees asymptotically faster.Comment: ACM-SIAM Symposium on Discrete Algorithms (SODA) 202

A New Method for Slant Calculation in Off-Line Handwriting Analysis

In this paper, we propose a new method for estimating the slant of word in handwritten text. The method allows a researcher to analyze snippets of a picture containing a few words in different lines. The main goal of the research is to present a tool to observe small changes of slant in the text during work. Student check sheets were used as a database for the research. Some changes in slant depending on speed of writing are discovered

Exponential Speedup over Locality in MPC with Optimal Memory

Locally Checkable Labeling (LCL) problems are graph problems in which a solution is correct if it satisfies some given constraints in the local neighborhood of each node. Example problems in this class include maximal matching, maximal independent set, and coloring problems. A successful line of research has been studying the complexities of LCL problems on paths/cycles, trees, and general graphs, providing many interesting results for the LOCAL model of distributed computing. In this work, we initiate the study of LCL problems in the low-space Massively Parallel Computation (MPC) model. In particular, on forests, we provide a method that, given the complexity of an LCL problem in the LOCAL model, automatically provides an exponentially faster algorithm for the low-space MPC setting that uses optimal global memory, that is, truly linear. While restricting to forests may seem to weaken the result, we emphasize that all known (conditional) lower bounds for the MPC setting are obtained by lifting lower bounds obtained in the distributed setting in tree-like networks (either forests or high girth graphs), and hence the problems that we study are challenging already on forests. Moreover, the most important technical feature of our algorithms is that they use optimal global memory, that is, memory linear in the number of edges of the graph. In contrast, most of the state-of-the-art algorithms use more than linear global memory. Further, they typically start with a dense graph, sparsify it, and then solve the problem on the residual graph, exploiting the relative increase in global memory. On forests, this is not possible, because the given graph is already as sparse as it can be, and using optimal memory requires new solutions

Exponential Speedup Over Locality in MPC with Optimal Memory

Locally Checkable Labeling (LCL) problems are graph problems in which a solution is correct if it satisfies some given constraints in the local neighborhood of each node. Example problems in this class include maximal matching, maximal independent set, and coloring problems. A successful line of research has been studying the complexities of LCL problems on paths/cycles, trees, and general graphs, providing many interesting results for the LOCAL model of distributed computing. In this work, we initiate the study of LCL problems in the low-space Massively Parallel Computation (MPC) model. In particular, on forests, we provide a method that, given the complexity of an LCL problem in the LOCAL model, automatically provides an exponentially faster algorithm for the low-space MPC setting that uses optimal global memory, that is, truly linear. While restricting to forests may seem to weaken the result, we emphasize that all known (conditional) lower bounds for the MPC setting are obtained by lifting lower bounds obtained in the distributed setting in tree-like networks (either forests or high girth graphs), and hence the problems that we study are challenging already on forests. Moreover, the most important technical feature of our algorithms is that they use optimal global memory, that is, memory linear in the number of edges of the graph. In contrast, most of the state-of-the-art algorithms use more than linear global memory. Further, they typically start with a dense graph, sparsify it, and then solve the problem on the residual graph, exploiting the relative increase in global memory. On forests, this is not possible, because the given graph is already as sparse as it can be, and using optimal memory requires new solutions

Harvojen verkkojen värittäminen kolmella värillä Massiivisen rinnakkaislaskennan (MPC) mallissa käyttäen vahvasti alilineaarista muistia

The question of what problems can be solved, and how efficiently, has always been at the core of theoretical computer science. One such fundamental problem is graph coloring; it is well researched and has numerous applications in areas of computer science such as scheduling and pattern matching. The challenges faced when designing graph coloring algorithms are dictated by the underlying graph family, the number of colors allowed, and the model of computation. In this work we consider the graph family of trees and the distributed Massively Parallel Computation (MPC) model, introduced by Karloff et al. \cite{karloff}. Our contribution to the field of distributed computing is a deterministic strongly sublinear MPC algorithm for 3-coloring unbounded degree trees with $n$ nodes in $O(\log \log n)$ time. To the best of our knowledge, this is the current state-of-the-art algorithm, improving on the work of Ghaffari et al. \cite{ghaffari}. It is loosely based on two previous works by Brandt et al. \cite{sparce,sirocco}. Before computing a 3-coloring, our algorithm partitions the input tree into disjoint node sets $H_1,H_2,\dots,H_l$, in $O(\log \log n)$ time. For each node $v \in H_i$, it holds that $v$ has at most two neighbors in the set $\bigcup_{j=i}^l H_j$. We consider this partitioning in and of itself an important contribution, since it has the potential of being a useful subroutine in future algorithms. For example, a similar technique was used by Chang et al. \cite{chang} in their seminal paper to establish an important time hierarchy theorem for the distributed LOCAL model on trees.Teoreettisen tietojenkäsittelytieteen tärkeimpiä tavoitteita on tutkia, mitkä ongelmat ovat ratkaistavissa, ja kuinka tehokkaasti nämä ratkeavat. Eräs sellainen ongelma on verkkojen väritysongelma, jolla on lukuisia sovelluksia tietojenkäsittelytieteen ongelmissa, kuten aikataulutusongelmissa ja hahmontunnistuksessa. Verkkojen värittämiseen liittyvät haasteet riippuvat verkkoperheestä, sallitusta värien määrästä ja valitusta mallista. Tässä työssä keskitytään puiden verkkoperheeseen ja hajautettuun Massiivisen rinnakkaislaskennan (MPC) malliin, jonka esittivät Karloff ym. \cite{karloff}. Työn tulos on MPC-mallissa vahvasti alilineaarista muistia käyttävä deterministinen 3-väritysalgoritmi rajoittamattoman asteen $n$-solmuisissa puissa, jonka ajoaika on $O(\log \log n)$. Kyseessä on läpimurtoalgoritmi, joka parantaa Ghaffari ym. \cite{ghaffari} esittämiä tuloksia ja pohjautuu löyhästi Brandt ym. \cite{sparce,sirocco} esittämiin algoritmeihin. Ratkaistaessa 3-väritysongelmaa algoritmi ensin osittaa puun solmut erillisiin joukkoihin $H_1,H_2,\dots,H_l$, ajassa $O(\log \log n)$. Osituksessa pätee, että jokaisella solmulla $v \in H_i$ on enintään kaksi naapuria joukossa $\bigcup_{j=i}^l H_j$. Kyseinen ositus on esille nostamisen arvoinen ja on myös sellaisenaan mittava tulos, koska osituksella on paljon potentiaalista käyttöä myös muissa algoritmeissa. Esimerkiksi Chang ym. \cite{chang} hyödynsivät vastaavanlaista ositusta mullistavassa artikkelissaan, missä he johtivat tärkeän hajautetun LOCAL-mallin aikahierarkian puissa

Mobile e-learning systems

Education is a key element in the life of every person. In the recent years we have seen strong growth and development of information technologies, and as a consequence - informatization process in all areas of society. Particular attention is now being paid to mobile technologies, based on the analysis of the market it has been revealed that tablets and smartphones are one of the most promising areas of development in the near future. The paper presents an approach to design of the e-learning systems in terms of diversity of implementation on the side of the client/end user (mobile application, web application and a classic application for PC). To improve the performance of teachers and students the implementation of special units is suggested, such as the module of the analytical statistics for teachers, the module of additional advanced checking of home work and feedback module