Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

We present a deterministic distributed algorithm that computes a $(2\Delta-1)$-edge-coloring, or even list-edge-coloring, in any $n$-node graph with maximum degree $\Delta$, in $O(\log^7 \Delta \log n)$ rounds. This answers one of the long-standing open questions of \emph{distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were $2^{O(\sqrt{\log n})}$ by Panconesi and Srinivasan [STOC'92] and $\tilde{O}(\sqrt{\Delta}) + O(\log^* n)$ by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of $(2\Delta-1)$-edge-coloring to poly$(\log\log n)$ rounds. The key technical ingredient is a deterministic distributed algorithm for \emph{hypergraph maximal matching}, which we believe will be of interest beyond this result. In any hypergraph of rank $r$ --- where each hyperedge has at most $r$ vertices --- with $n$ nodes and maximum degree $\Delta$, this algorithm computes a maximal matching in $O(r^5 \log^{6+\log r } \Delta \log n)$ rounds. This hypergraph matching algorithm and its extensions lead to a number of other results. In particular, a polylogarithmic-time deterministic distributed maximal independent set algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a $((\log \Delta/\varepsilon)^{O(\log (1/\varepsilon))})$-round deterministic algorithm for $(1+\varepsilon)$-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting $\lambda$-arboricity graphs with out-degree at most $(1+\varepsilon)\lambda$, for any constant $\varepsilon>0$, hence partially answering Open Problem 10 of Barenboim and Elkin's book

Improved Deterministic Distributed Matching via Rounding

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows. - An O(log^2 Delta log n)-round deterministic distributed algorithm for computing a maximal matching, in n-node graphs with maximum degree Delta. This is the first improvement in about 20 years over the celebrated O(log^4 n)-round algorithm of Hanckowiak, Karonski, and Panconesi [SODA\u2798, PODC\u2799]. - A deterministic distributed algorithm for computing a (2+epsilon)-approximation of maximum matching in O(log^2 Delta log(1/epsilon) + log^* n) rounds. This is exponentially faster than the classic O(Delta + log^* n)-round 2-approximation of Panconesi and Rizzi [DIST\u2701]. With some modifications, the algorithm can also find an epsilon-maximal matching which leaves only an epsilon-fraction of the edges on unmatched nodes. - An O(log^2 Delta log(1/epsilon) + log^* n)-round deterministic distributed algorithm for computing a (2+epsilon)-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted b-matching. These improve over the O(log^4 n log_(1+epsilon) W)-round (6+epsilon)-approximation algorithm of Panconesi and Sozio [DIST\u2710], where W denotes the maximum normalized weight. - A deterministic local computation algorithm for a (2+epsilon)-approximation of maximum matching with 2^O(log^2 Delta) log^* n queries. This improves almost exponentially over the previous deterministic constant approximations which have query-complexity of 2^Omega(Delta log Delta) log^* n

A Simple Parallel and Distributed Sampling Technique: Local Glauber Dynamics

Sampling constitutes an important tool in a variety of areas: from machine learning and combinatorial optimization to computational physics and biology. A central class of sampling algorithms is the Markov Chain Monte Carlo method, based on the construction of a Markov chain with the desired sampling distribution as its stationary distribution. Many of the traditional Markov chains, such as the Glauber dynamics, do not scale well with increasing dimension. To address this shortcoming, we propose a simple local update rule based on the Glauber dynamics that leads to efficient parallel and distributed algorithms for sampling from Gibbs distributions. Concretely, we present a Markov chain that mixes in O(log n) rounds when Dobrushin\u27s condition for the Gibbs distribution is satisfied. This improves over the LubyGlauber algorithm by Feng, Sun, and Yin [PODC\u2717], which needs O(Delta log n) rounds, and their LocalMetropolis algorithm, which converges in O(log n) rounds but requires a considerably stronger mixing condition. Here, n denotes the number of nodes in the graphical model inducing the Gibbs distribution, and Delta its maximum degree. In particular, our method can sample a uniform proper coloring with alpha Delta colors in O(log n) rounds for any alpha >2, which almost matches the threshold of the sequential Glauber dynamics and improves on the alpha>2 + sqrt{2} threshold of Feng et al

Web-based Aptitude Tests at Universities in German-speaking Countries

Universities can select their students more and more independently. In order to support the selection process, web-based aptitude tests are a possibility to balance benefits und efforts of this task. Within this paper, we will point out how current web-based aptitude tests are designed, what competences are covered, and which methods for development are used. For this purpose, we developed a classification how web-based aptitude tests are implemented. Furthermore, competences as the basis of web-based aptitude tests are appraised. Four competence categories (professional, methodological, personal, and social competences) are selected as the most appropriate pattern. Thereafter, we analyse methods for developing competence specifications. Finally, we state lessons learned for the development of web-based aptitude tests at universities. These results are an important preparatory work and a basis for a systematically development of a web-based aptitude test for the University of Erlangen-Nuremberg

Evaluation of User Acceptance for Web-based Aptitude Tests

Web-based aptitude tests, which are a special category of aptitude tests, can be used for rather standardized test methods and for a large amount of users. The characteristics of web-based aptitude tests can have an impact on the test results and the user acceptance. The aim of our research is to develop a method for the evaluation of the user acceptance for web-based aptitude tests. Therefore, we used the DART-approach with the dimension (Perceived) Usefulness, (Perceived) Ease of Use, (Perceived) Network Effects and (Perceived) Costs as the theoretical basis, identified important acceptance indicators, developed a questionnaire and conducted a survey. Afterwards, we proved the reliability and conducted a factor analysis. The results point out that some of the defined acceptance indicators should be revised. Additionally, the factor analysis shows that a combination of two dimensions (Perceived) Usefulness and (Perceived) Network Effects is useful especially with regard to web-based aptitude tests. Finally, we conducted a univariate analysis to evaluate the user acceptance of a web-based aptitude test. The visualised result on the basis of a DART-chart clearly shows that the interviewees evaluated the indicators very differently. There are fields, where the aptitude test fulfils the expectations, and fields, which can be improved

Acoustic measurements in air: A model apparatus for education and testing

The speed of sound in air mainly depends on temperature and flow properties along the propagation path of acoustic signals. In turn, distributions of these quantities can be determined from sound speed measurements along different propagation paths. In this article, a modular measurement system is presented which is suited to demonstrate the effects of surrounding conditions on the speed of sound, e.g. for educational purposes, and which can be used for an easy testing of new hardware components or algorithms. Underlying mathematical relations are explained and uncertainties are discussed. A sample application of the system within a laboratory shows the effect of local heating on the sound speed along several propagation paths which differ in their spatial distribution with respect to the heating source