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We design the first learned index that solves the dictionary problem withtime and space complexity provably better than classic data structures forhierarchical memories, such as B-trees, and modern learned indexes. We call oursolution the Piecewise Geometric Model index (PGM-index) because it turns theindexing of a sequence of keys into the coverage of a sequence of 2D-points vialinear models (i.e. segments) suitably learned to trade query time vs spaceefficiency. This idea comes from some known heuristic results which westrengthen by showing that the minimal number of such segments can be computedvia known and optimal streaming algorithms. Our index is then obtained byrecursively applying this geometric idea that guarantees a smoothed adaptationto the "geometric complexity" of the input data. Finally, we propose a variantof the index that adapts not only to the distribution of the dictionary keysbut also to their access frequencies, thus obtaining the firstdistribution-aware learned index.  The second main contribution of this paper is the proposal and study of theconcept of Multicriteria Data Structure, namely one that asks a data structureto adapt in an automatic way to the constraints imposed by the application ofuse. We show that our index is a multicriteria data structure because itssignificant flexibility in storage and query time can be exploited by aproperly designed optimisation algorithm that efficiently finds its best designsetting in order to match the input constraints.  A thorough experimental analysis shows that our index and its multicriteriavariant improve uniformly, over both time and space, classic and learnedindexes up to several orders of magnitude.

Superseding traditional indexes by orchestrating learning and geometry