Finding dense spanning trees (DST) in unweighted graphs is a variation of the well studied minimum spanning tree problem (MST). We utilize established mathematical properties of extremal structures with the minimum sum of distances between vertices to formulate some general conditions on the sum of vertex degrees. We analyze the performance of various combinations of these degree sum conditions in finding dense spanning subtrees and apply our approach to practical examples. After briefly describing our algorithm we also show how it can be used on variations of DST, motivated by variations of MST. Our work provide some insights on the role of various degree sums in forming dense spanning trees and hopefully lay the foundation for finding fast algorithms or heuristics for related problems.
Performance effective task scheduling algorithms are essential for taking advantage of the heterogeneous multi-processor in heterogeneous computing environments. In this paper, we present a task scheduling algorithm named as Dependency-ratio Bundling Earliest Finish Time (DBEFT). DBEFT is a list based scheduling algorithm combined with task duplication, which can achieve high performance and low time complexity simultaneously. DBEFT selects the task from the perspective of extending parallelism between tasks instead of giving priorities to tasks on the critical path. Also, DBEFT reduces communication cost by adopting a bundling scheduling strategy. The experiments were conducted on both random graph set and real-world applications, and the results show that DBEFT obtained significant performance improvement, outperforming CEFT by 15%, PEFT by 30% and HEFT 33% in terms of SLR respectively. INDEX TERMS Heterogeneous system, task scheduling, task duplication, task dependency ratio, bundling scheduling.
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