Towards Optimjum Execution For Wien2k Using Parallel Computing Models (Comparative Study)_

In the present work, a comparison between two parallel methods have been done using message passing with OpenMPI API. Both Distribute k-pointmethod and Data distribution method have been used to run physical package which is used to study the physical and chemical properties of the materials which is called Wien2k package. Two data set size are used to be as a benchmark of this study, execution time of running with respect to RAM size either in shared or distributed case has been studied, two different size of RAM per CPU has been tested in Distribute k-point for the two benchmark, respectively. Network speed and its effect on calculation time has been studied, the network speed used in the study was 100 Mbps and 1 Gbps, the network effect has been tested on Distribute k-point method for two classes of CPUs and different RAM size per CPUs . The effect of CPUs speed on execution time has been studied by distribute k-point and Data distribution methods; different speed of CPUs has been used to study homogenous and heterogeneous effects on execution time. In all tests, RAM size considerably affect the time of calculation effectively, it’s found that increasing size of RAM available per CPU will cause considerable decrease in the calculation time, the study showed a small effect for the network speed on the calculation time, the effect of network can be neglected with respect to RAM size effect, The execution time showed that Data distribution gives better reduction in the time of calculation and higher speed up factor with increasing number of CPU’s, two scalable quantity has been used to compare and analyze the results, speed up factor and the power factor of decaying formula for time of execution

Parallel Model Counting with CUDA: Algorithm Engineering for Efficient Hardware Utilization

Propositional model counting (MC) and its extensions as well as applications in the area of probabilistic reasoning have received renewed attention in recent years. As a result, also the need for quickly solving counting-based problems with automated solvers is critical for certain areas. In this paper, we present experiments evaluating various techniques in order to improve the performance of parallel model counting on general purpose graphics processing units (GPGPUs). Thereby, we mainly consider engineering efficient algorithms for model counting on GPGPUs that utilize the treewidth of a propositional formula by means of dynamic programming. The combination of our techniques results in the solver GPUSAT3, which is based on the programming framework Cuda that -compared to other frameworks- shows superior extensibility and driver support. When combining all findings of this work, we show that GPUSAT3 not only solves more instances of the recent Model Counting Competition 2020 (MCC 2020) than existing GPGPU-based systems, but also solves those significantly faster. A portfolio with one of the best solvers of MCC 2020 and GPUSAT3 solves 19% more instances than the former alone in less than half of the runtime

On the van der Waerden numbers w(2;3,t)

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

Parallel SAT Solving using Bit-level Operations

We show how to exploit the 32/64 bit architecture of modern computers to accelerate some of the algorithms used in satisfiability solving by modifying assignments to variables in parallel on a single processor. Techniques such as random sampling demonstrate that while using bit vectors instead of Boolean values solutions to satisfiable formulae can be obtained faster. Here, we reveal that more complex algorithms, like unit propagation and detection of autarkies, can be parallelized efficiently, as well. We capitalize on the developed parallel algorithms by modifying the state-of-the-art local search Sat solver UnitWalk accordingly. Experiments show that the parallel version performs much faster than the original implementation.Software TechnologyElectrical Engineering, Mathematics and Computer Scienc

Some Results in Extremal Combinatorics

Extremal Combinatorics is one of the central and heavily contributed areas in discrete mathematics, and has seen an outstanding growth during the last few decades. In general, it deals with problems regarding determination and/or estimation of the maximum or the minimum size of a combinatorial structure that satisfies a certain combinatorial property. Problems in Extremal Combinatorics are often related to theoretical computer science, number theory, geometry, and information theory. In this thesis, we work on some well-known problems (and on their variants) in Extremal Combinatorics concerning the set of integers as the combinatorial structure. The van der Waerden number w(k;t_0,t_1,...,t_{k-1}) is the smallest positive integer n such that every k-colouring of 1, 2, . . . , n contains a monochromatic arithmetic progression of length t_j for some colour j in {0,1,...,k-1}. We have determined five new exact values with k=2 and conjectured several van der Waerden numbers of the form w(2;s,t), based on which we have formulated a polynomial upper-bound-conjecture of w(2; s, t) with fixed s. We have provided an efficient SAT encoding for van der Waerden numbers with k>=3 and computed three new van der Waerden numbers using that encoding. We have also devised an efficient problem-specific backtracking algorithm and computed twenty-five new van der Waerden numbers with k>=3 using that algorithm. We have proven some counting properties of arithmetic progressions and some unimodality properties of sequences regarding arithmetic progressions. We have generalized Szekeres’ conjecture on the size of the largest sub-sequence of 1, 2, . . . , n without an arithmetic progression of length k for specific k and n; and provided a construction for the lower bound corresponding to the generalized conjecture. A Strict Schur number S(h,k) is the smallest positive integer n such that every 2-colouring of 1,2,...,n has either a blue solution to x_1 +x_2 +···+x_{h-1} = x_h where x_1 < x_2 < ··· < x_h, or a red solution to x_1+x_2+···+x_{k-1} =x_k where x_1 <x_2 <···<x_k. We have proven the exact formula for S(3, k)

Efficient local search for Pseudo Boolean Optimization

Algorithms and the Foundations of Software technolog