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Coflow is a network abstraction used to represent communication patterns in data centers. The coflow scheduling problem encountered in large data centers is a challenging NP -hard problem. Many previous studies on coflow scheduling mainly focus on the single-core model. However, with the growth of data centers, this single-core model is no longer sufficient. This paper addresses the coflow scheduling problem within heterogeneous parallel networks, which feature an architecture consisting of multiple network cores running in parallel. In this paper, two polynomial-time approximation algorithms are developed for the flow-level scheduling problem and the coflow-level scheduling problem in heterogeneous parallel networks, respectively. For the flow-level scheduling problem, the proposed algorithm achieves an approximation ratio of O (log ⁡ m / log ⁡ log ⁡ m) when all coflows are released at arbitrary times, where m represents the number of network cores. On the other hand, in the coflow-level scheduling problem, the proposed algorithm achieves an approximation ratio of O (m (log ⁡ m / log ⁡ log ⁡ m) 2) when all coflows are released at arbitrary times. Moreover, we propose a heuristic algorithm for the flow-level scheduling problem. Simulation results using synthetic traffic traces validate the performance of our algorithms and show improvements over the previous algorithm. • Algorithm achieves O (log ⁡ m / log ⁡ log ⁡ m) approximation ratio for flow-level scheduling, where m is network cores. • For coflow-level scheduling, algorithm achieves O (m (log ⁡ m / log ⁡ log ⁡ m) 2) approximation ratio. • Moreover, this paper proposes a heuristic algorithm for the flow-level scheduling problem. [ABSTRACT FROM AUTHOR]

Randomized algorithms play a central role in low rank approximations of large matrices. In this paper, the scheme of the randomized SVD is extended to a randomized LU algorithm. Several error bounds are introduced, that are based on recent results from random matrix theory related to subgaussian matrices. The bounds also improve the existing bounds of already known randomized SVD algorithm. The algorithm is fully parallelized and thus can utilize efficiently GPUs without any CPU–GPU data transfer. Numerical examples, which illustrate the performance of the algorithm and compare it to other decomposition methods, are presented. [ABSTRACT FROM AUTHOR]

Abstract: For more and more applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast. This paper gives an overview of several upper bound heuristics that have been proposed and tested for the problem of determining the treewidth of a graph and finding tree decompositions. Each of the heuristics produces tree decompositions whose width may be larger than the optimal width. However, experiments show that in many cases, the heuristics give tree decompositions whose width is close to the exact treewidth of the input graphs. [Copyright &y& Elsevier]