走向查询答案多样性的可处理性：Ultrametrics 来救援
Towards Tractability of the Diversity of Query Answers: Ultrametrics to the Rescue
AI总结：
本文研究隐式查询答案集合中多样化子集的计算难题，基于 ultrametric 与多样性函数给出多项式可处理条件及难处理下界，并应用于无环合取查询。
摘要：
查询的答案集合可能非常庞大，若将整个集合呈现给用户，可能会让用户应接不暇。在这种情况下，只向用户呈现答案的一个较小子集可能更为可取。对这个子集的一个自然要求是，它应尽可能多样化，以反映整个总体的多样性。为实现这一点，子集的多样性通过一个度量来衡量，该度量决定两个解之间有多大差异；同时通过一个多样性函数将该度量从成对对象扩展到集合。过去，若干研究已经表明，即使对于简单度量（如 Hamming 距离）和简单多样性函数（如对所有成对距离求和），从一个显式给定的集合中寻找多样化子集也是难以处理的。当试图从一个仅隐式给定的集合（例如查询和数据库的查询答案）中输出多样化子集时，这一复杂性障碍变得更具挑战性。迄今为止，可处理情形只在受限问题和特定多样性函数中被发现。为克服这些局限，我们关注 ultrametric（超度量）概念，该概念已在许多应用中得到广泛研究和使用。从任意 ultrametric d 以及扩展 d 的多样性函数 δ 出发，我们给出了关于 δ 的充分条件，使构造多样化答案存在多项式时间算法。据我们所知，文献中考虑的所有多样性函数都满足这些条件。此外，我们以下界对这些结果进行补充，展示了当这些条件不满足时，寻找多样化子集变得难以处理的具体情形。最后，我们将这些结果应用于合取查询求值，证明在使用属性顺序衡量多样性时，存在高效算法可为无环合取查询寻找多样化的解子集。
英文摘要：
The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this subset is that it should be as diverse as possible to reflect the variety of the entire population. To achieve this, the diversity of a subset is measured using a metric that determines how different two solutions are and a diversity function that extends this metric from pairs to sets. In the past, several studies have shown that finding a diverse subset from an explicitly given set is intractable even for simple metrics (like Hamming distance) and simple diversity functions (like summing all pairwise distances). This complexity barrier becomes even more challenging when trying to output a diverse subset from a set that is only implicitly given such as the query answers of a query and a database. Until now, tractable cases have been found only for restricted problems and particular diversity functions. To overcome these limitations, we focus on the notion of ultrametrics, which have been widely studied and used in many applications. Starting from any ultrametric $d$ and a diversity function $δ$ extending $d$, we provide sufficient conditions over $δ$ for having polynomial-time algorithms to construct diverse answers. To the best of our knowledge, these conditions are satisfied by all diversity functions considered in the literature. Moreover, we complement these results with lower bounds that show specific cases when these conditions are not satisfied and finding diverse subsets becomes intractable. We conclude by applying these results to the evaluation of conjunctive queries, demonstrating efficient algorithms for finding a diverse subset of solutions for acyclic conjunctive queries when the attribute order is used to measure diversity.
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