Direct Access for Answers to Conjunctive Queries with Aggregation
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We study the fine-grained complexity of conjunctive queries with grouping and aggregation. For some common aggregate functions (e.g., min, max, count, sum), such a query can be phrased as an ordinary conjunctive query over a database annotated with a suitable commutative semiring. Specifically, we investigate the ability to evaluate such queries by constructing in log-linear time a data structure that provides logarithmic-time direct access to the answers ordered by a given lexicographic order. This task is nontrivial since the number of answers might be larger than log-linear in the size of the input, and so, the data structure needs to provide a compact representation of the space of answers. In the absence of aggregation and annotation, past research provides a sufficient tractability condition on queries and orders. For queries without self-joins, this condition is not just sufficient, but also necessary (under conventional lower-bound assumptions in fine-grained complexity). We show that all past results continue to hold for annotated databases, assuming that the annotation itself is not part of the lexicographic order. On the other hand, we show infeasibility for the case of count-distinct that does not have any efficient representation as a commutative semiring. We then investigate the ability to include the aggregate and annotation outcome in the lexicographic order. Among the hardness results, standing out as tractable is the case of a semiring with an idempotent addition, such as those of min and max. Notably, this case captures also count-distinct over a logarithmic-size domain.
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