The concept of maximal perfect haplotype blocks is introduced as a simple pattern allowing to identify genomic regions that show signatures of natural selection. The model is formally defined and a simple algorithm is presented to find all perfect haplotype blocks in a set of phased chromosome sequences. Application to three whole chromosomes from the 1000 genomes project phase 3 data set shows the potential of the concept as an effective approach for quick detection of selection in large sets of thousands of genomes.
A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges. The t-tessellability problem aims to decide whether there is a tessellation cover of the graph with t tessellations. This problem is motivated by its applications to quantum walk models, in especial, the evolution operator of the staggered model is obtained from a graph tessellation cover. We establish upper bounds on the tessellation cover number given by the minimum between the chromatic index of the graph and the chromatic number of its clique graph and we show graph classes for which these bounds are tight. We prove N P-completeness for t-tessellability if the instance is restricted to planar graphs, chordal (2, 1)-graphs, (1, 2)-graphs, diamond-free graphs with diameter five, or for any fixed t at least 3. On the other hand, we improve the complexity for 2-tessellability to a linear-time algorithm.
In this paper, we propose a new quantum algorithm for solving the minimum searching problem. This algorithm has the same order of time and space complexities as the algorithm proposed by Dürr and Høyer, but it provides a quadratic reduction in the number of measurements. In addition to the correctness and complexity analysis of the algorithm, we present simulation results considering an NP-hard problem.
<p class="Abstract">Statistical sampling and simulations produced by algorithms require fast random number generators; however, true random number generators are often too slow for the purpose, so pseudorandom number generators are usually more suitable. But choosing and using a pseudorandom number generator is no simple task; most pseudorandom number generators fail statistical tests. Default pseudorandom number generators offered by programming languages usually do not offer sufficient statistical properties. Testing random number generators so as to choose one for a project is essential to know its limitations and decide whether the choice fits the project’s objectives. However, this study presents a reproducible experiment that demonstrates that, despite all the contributions it made when it was first published, the popular NIST SP 800-22 statistical test suite as implemented in the software package is inadequate for testing generators.</p>
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