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In order to focus on the bottleneck expression in the running time analysis of the algorithm, we first summarize the complexity of its secondary computations in the following lemma: Lemma 2. It is possible to implement Algorithm 1 so that all operations, besides computation of solutions to the MCM problem, require O(n T n S ) running time.
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Such systems generally adopt rather simple mathematics, and, therefore, need extensive secondary index computation.
However, when the size of the graph becomes so high that is not possible to maintain it entirely in primary memory, either approaches based on parallel distributed computation [ 13- 15], or secondary memory-based computation [ 16- 18] can be considered.
A large number of programs have been developed over the last ten years [41], focusing on issues such as the scoring scheme, the use of templates for secondary structures and computation speed.
2. Secondary memory-based computation.
With secondary memory-based computation, the graph is stored on the disk of a single machine and only limited portions of the graph are loaded in primary memory for computation.
We applied our methods based on the local implementation of network-based algorithms and secondary memory-based computation to the multi-species protein function prediction in bacteria and eukarya.
It is worth noting that we do not propose novel algorithms, but simply their "local implementation", according to a vertex-centric programming model, necessary for secondary memory-based computation [ 14].
Our approach to big-graph analysis for AFP leverages on both a novel computational model for network analysis and on novel technologies for fast and efficient secondary memory-based computation.
In the next section we introduce our proposed approach based on the local implementation of network-based algorithms and secondary memory-based computation for the multi-species AFP problem.
MM analyzed the technological items behind secondary memory based computation and implemented the RW algorithm using Neo4j; MR curated the biological items of multi-species AFP and implemented the same algorithm with GraphChi; GV wrote the draft (with the contribution of the other authors) and implemented the main-memory version of the algorithms.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com