Q-error -- the standard metric for quantifying the error of individual cardinality estimates -- has been widely adopted as a surrogate for query plan optimality in recent work on learning-based cardinality estimation. However, the only result connecting Q-error with plan optimality is an upper-bound on the cost of the worst possible query plan computed from a set of cardinality estimates---there is no connection between Q-error and the real plans generated by standard query optimizers. Therefore, in order to identify sub-optimal query plans, we propose a learning-based method having as its main feature a novel measure called L1-error. Similar to Q-error, L1-error requires complete knowledge of true cardinalities and estimates for all the sub-plans of a query plan. Unlike Q-error, which considers the estimates independently, L1-error is defined as a permutation distance between true cardinalities and estimates for all the sub-plans having the same number of joins. Moreover, L1-error takes into account errors relative to the magnitude of their cardinalities and gives larger weight to small multi-way joins. Our experimental results confirm that, when L1-error is integrated into a standard decision tree classifier, it leads to the accurate identification of sub-optimal plans across four different benchmarks. This accuracy can be further improved by combining L1-error with Q-error into a composite feature that can be computed without overhead from the same data.

Sketches are single-pass small-space data summaries that can quickly estimate the cardinality of join queries. However, sketches are not directly applicable to join queries with dynamic filter conditions --- where arbitrary selection predicate(s) are applied --- since a sketch is limited to a fixed selection. While multiple sketches for various selections can be used in combination, they each incur individual storage and maintenance costs. Alternatively, exact sketches can be built during runtime for every selection. To make this process scale, a high-degree of parallelism --- available in hardware accelerators such as GPUs --- is required. Therefore, sketch usage for cardinality estimation in query optimization is limited. Following recent work that applies transformers to cardinality estimation, we design a novel learning-based method to approximate the sketch of any arbitrary selection, enabling sketches for join queries with filter conditions. We train a transformer on each table to estimate the sketch of any subset of the table, i.e., any arbitrary selection. Transformers achieve this by learning the joint distribution amongst table attributes, which is equivalent to a multidimensional sketch. Subsequently, transformers can approximate any sketch, enabling sketches for join cardinality estimation. In turn, estimating joins via approximate sketches allows tables to be modeled individually and thus scales linearly with the number of tables. We evaluate the accuracy and efficacy of approximate sketches on queries with selection predicates consisting of conjunctions of point and range conditions. Approximate sketches achieve similar accuracy to exact sketches with at least one order of magnitude less overhead.

Cost-based query optimization remains a critical task in relational databases even after decades of research and industrial development. Query optimizers rely on a large range of statistical synopses for accurate cardinality estimation. As the complexity of selections and the number of join predicates increase, two problems arise. First, statistics cannot be incrementally composed to effectively estimate the cost of the sub-plans generated in plan enumeration. Second, small errors are propagated exponentially through joins, which can lead to severely sub-optimal plans. In this paper, we introduce COMPASS, a novel query optimization paradigm for in-memory databases based on a single type of statistics---Fast-AGMS sketches. In COMPASS, query optimization and execution are intertwined. Selection predicates and sketch updates are pushed-down and evaluated online during query optimization. This allows Fast-AGMS sketches to be computed only over the relevant tuples---which enhances cardinality estimation accuracy. Plan enumeration is performed over the query join graph by incrementally composing attribute-level sketches---not by building a separate sketch for every sub-plan. We prototype COMPASS in MapD -- an open-source parallel database -- and perform extensive experiments over the complete JOB benchmark. The results prove that COMPASS generates better execution plans -- both in terms of cardinality and runtime -- compared to four other database systems. Overall, COMPASS achieves a speedup ranging from 1.35X to 11.28X in cumulative query execution time over the considered competitors.