# Partitioning Strategies for Parallel Computation of Flexible Skylines
## 7 Conclusions  
In this paper, we tackled the problem of computing flexible skylines by exploiting common partitioning schemes that divide a dataset horizontally and send each partition to a different compute node for an initial round of computation of the result. The partial results, after this parallel phase, are collected and ultimately processed for a final, sequential round that eliminates all redundancies. While this consolidated scheme has been successfully used for computing the classical skyline operator, ours is the first attempt to apply it to the case of flexible skylines--an extension of skylines that accommodates user preferences expressed through constraints on weights and makes them closer to top-\(k\) queries.  
Our results show that the partitioning is generally beneficial, especially with schemes that divide the dataset into more balanced subsets, such as Angular and Sliced. In addition to optimization opportunities based on the notion of representative tuples, we also adopted another optimization, indicated as NoSeq, that removes the final sequential round completely, which is typically the most expensive part of the entire computation. Indeed, NoSeq proves to be the most robust of all tested algorithmic solutions.  
The computation of the two flexible skyline operators, ND and PO, differs in the ability of the partitioning strategy to effectively reduce the set of candidates before the final round. We noticed that, while Sliced does its job egregiously in the case of ND, the smaller sizes involved in the computation of potentially optimal tuples make Sliced a less suitable candidate than Angular for PO. In both cases, however, the NoSeq strategy proves to be the best option.  
Future research may follow several new lines of investigation. One regards the impact of the constraints on weights on the effectiveness of the partitioning strategy. Another concern of interest regards the potential improvements that might stem from adopting bounding box approximation for the dominance test, which currently involves a quadratic nested loop to iterate all vertices of a convex polytope. Another setting that might require a further study is that of domains with low cardinality, as studied in [47] for skylines, whose effect might dramatically affect the complexity of \(\mathcal{F}\)-dominance. Finally, we observe that addressing the skyline size estimation problem (a currently open problem that has been solved only in special cases [4; 48; 49; 50; 51]) would lead to significant advancements for characterizing the communication complexity of a distributed computation such as the one proposed here. However, our case would also require extending these estimates to the case of flexible skylines, which is much more challenging due to the presence of constraints on the sets of scoring functions, which heavily affect the result size. Future work may try to identify classes of constraints and data distributions for which such estimates are possible.  
Conceptualization, E.D.L. and D.M.; methodology, E.D.L. and D.M.; software, E.D.L.; formal analysis, E.D.L. and D.M.; writing--original draft preparation, E.D.L. and D.M.; writing--review and editing, D.M.; supervision, D.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The data presented in this study are available on request from the author. The authors declare no conflicts of interest.