The PGM-index: a multicriteria, compressed and learned approach to data indexing

The recent introduction of learned indexes has shaken the foundations of the decades-old field of indexing data structures. Combining, or even replacing, classic design elements such as B-tree nodes with machine learning models has proven to give outstanding improvements in the space footprint and time efficiency of data systems. However, these novel approaches are based on heuristics, thus they lack any guarantees both in their time and space requirements. We propose the Piecewise Geometric Model index (shortly, PGM-index), which achieves guaranteed I/O-optimality in query operations, learns an optimal number of linear models, and its peculiar recursive construction makes it a purely learned data structure, rather than a hybrid of traditional and learned indexes (such as RMI and FITing-tree). We show that the PGM-index improves the space of the FITing-tree by 63.3% and of the B-tree by more than four orders of magnitude, while achieving their same or even better query time efficiency. We complement this result by proposing three variants of the PGM-index. First, we design a compressed PGM-index that further reduces its space footprint by exploiting the repetitiveness at the level of the learned linear models it is composed of. Second, we design a PGM-index that adapts itself to the distribution of the queries, thus resulting in the first known distribution-aware learned index to date. Finally, given its flexibility in the offered space-time trade-offs, we propose the multicriteria PGM-index that efficiently auto-tune itself in a few seconds over hundreds of millions of keys to the possibly evolving space-time constraints imposed by the application of use. We remark to the reader that this paper is an extended and improved version of our previous paper titled "Superseding traditional indexes by orchestrating learning and geometry" (arXiv:1903.00507).Comment: We remark to the reader that this paper is an extended and improved version of our previous paper titled "Superseding traditional indexes by orchestrating learning and geometry" (arXiv:1903.00507

A "Learned" Approach to Quicken and Compress Rank/Select Dictionaries

We address the well-known problem of designing, implementing and experimenting compressed data structures for supporting rank and select queries over a dictionary of integers. This problem has been studied far and wide since the end of the ‘80s with tons of important theoretical and practical results. Following a recent line of research on the so-called learned data structures, we first show that this problem has a surprising connection with the geometry of a set of points in the Cartesian plane suitably derived from the input integers. We then build upon some classical results in computational geometry to introduce the first “learned” scheme for implementing a compressed rank/select dictionary. We prove theoretical bounds on its time and space performance both in the worst case and in the case of input distributions with finite mean and variance. We corroborate these theoretical results with a large set of experiments over datasets originating from a variety of sources and applications (Web, DNA sequencing, information retrieval and natural language processing), and we show that a carefully engineered version of our approach provides new interesting space-time trade-offs with respect to several well-established implementations of Elias-Fano, RRR-vector, and random-access vectors of Elias γ/δ-coded gaps

Repetition- and Linearity-Aware Rank/Select Dictionaries

We revisit the fundamental problem of compressing an integer dictionary that supports efficient rank and select operations by exploiting two kinds of regularities arising in real data: repetitiveness and approximate linearity. Our first contribution is a Lempel-Ziv parsing properly enriched to also capture approximate linearity in the data and still be compressed to the kth order entropy. Our second contribution is a variant of the block tree structure whose space complexity takes advantage of both repetitiveness and approximate linearity, and results highly competitive in time too. Our third and final contribution is an implementation and experimentation of this last data structure, which achieves new space-time trade-offs compared to known data structures that exploit only one of the two regularities

Why are learned indexes so effective?

A recent trend in algorithm design consists of augmenting classic data structures with machine learning models, which are better suited to reveal and exploit patterns and trends in the input data so to achieve outstanding practical improvements in space occupancy and time efficiency. This is especially known in the context of indexing data structures where, despite few attempts in evaluating their asymptotic efficiency, theoretical results are yet missing in showing that learned indexes are provably better than classic indexes, such as B+-trees and their variants. In this paper, we present the first mathematically-grounded answer to this open problem. We obtain this result by discovering and exploiting a link between the original problem and a mean exit time problem over a proper stochastic process which, we show, is related to the space and time occupancy of those learned indexes. Our general result is then specialised to five well-known distributions: Uniform, Lognormal, Pareto, Exponential, and Gamma; and it is corroborated in precision and robustness by a large set of experiments

Combined biomechanical and tomographic keratoconus staging: Adding a biomechanical parameter to the ABCD keratoconus staging system

Purpose This retrospective cross-sectional study evaluated the potential of an additional biomechanical parameter ‘E’ as an addition to the tomographic ABCD ectasia/keratoconus (KC) staging. Methods The Corvis Biomechanical Factor (CBiF) represents the modified linear term of the Corvis Biomechanical Index (CBI) developed based on 448 KC corneas from the Homburg Keratoconus Center (HKC). The CBiF range was divided into five stages (E0 to E4) to create a grading system according to the ABCD stages. Stage E0 was characterized by values smaller than the 2.5 percentile. The thresholds were created by dividing the CBiF range between the 2.5 and 97.5 percentiles into four groups of equal values (E1–E4). The frequency distribution of ‘E’ was analysed and independently validated based on another 860 KC corneas dataset from Milano and Rio de Janeiro (MR). The relationship between ‘E’ and the ABCD staging was analysed by cross-tabulation. The specificity of ‘E’ was assessed based on healthy controls (112|851) from both datasets (HKC|MR). Results ‘E’ was normally distributed with E0 = 37|30, E1 = 86|200, E2 = 155|354, E3 = 101|206, E4 = 69|70 in the KC group and 96.4%|90.5% of the controls classified E0 in the HKC|MR dataset, respectively. Cross-tabulation revealed that ‘E’ was most comparable to posterior corneal curvature (‘B’) in both datasets, while showing a trend towards more advanced stages in comparison to anterior corneal curvature (‘A’) and thinnest corneal thickness (‘C’). Conclusion The novel Corvis-derived parameter ‘E’ provides a biomechanical staging for ectasia/KC potentially enhancing the ABCD staging and may detect abnormalities before tomographic changes, which requires further studies

Positions of Ocular Geometrical and Visual Axes in Brazilian, Chinese and Italian Populations

ABSTRACTPurpose: To identify the relative positions of geometrical and visual axes of the eye and present a method to locate the visual center when the geometrical axis is taken as a reference.Meth..

Learned Monotone Minimal Perfect Hashing

A Monotone Minimal Perfect Hash Function (MMPHF) constructed on a set S of keys is a function that maps each key in S to its rank. On keys not in S, the function returns an arbitrary value. Applications range from databases, search engines, data encryption, to pattern-matching algorithms. In this paper, we describe LeMonHash, a new technique for constructing MMPHFs for integers. The core idea of LeMonHash is surprisingly simple and effective: we learn a monotone mapping from keys to their rank via an error-bounded piecewise linear model (the PGM-index), and then we solve the collisions that might arise among keys mapping to the same rank estimate by associating small integers with them in a retrieval data structure (BuRR). On synthetic random datasets, LeMonHash needs 34% less space than the next larger competitor, while achieving about 16 times faster queries. On real-world datasets, the space usage is very close to or much better than the best competitors, while achieving up to 19 times faster queries than the next larger competitor. As far as the construction of LeMonHash is concerned, we get an improvement by a factor of up to 2, compared to the competitor with the next best space usage. We also investigate the case of keys being variable-length strings, introducing the so-called LeMonHash-VL: it needs space within 13% of the best competitors while achieving up to 3 times faster queries than the next larger competitor

Learned Monotone Minimal Perfect Hashing

A Monotone Minimal Perfect Hash Function (MMPHF) constructed on a set S of keys is a function that maps each key in S to its rank. On keys not in S, the function returns an arbitrary value. Applications range from databases, search engines, data encryption, to pattern-matching algorithms. In this paper, we describe LeMonHash, a new technique for constructing MMPHFs for integers. The core idea of LeMonHash is surprisingly simple and effective: we learn a monotone mapping from keys to their rank via an error-bounded piecewise linear model (the PGM-index), and then we solve the collisions that might arise among keys mapping to the same rank estimate by associating small integers with them in a retrieval data structure (BuRR). On synthetic random datasets, LeMonHash needs 34% less space than the next larger competitor, while achieving about 16 times faster queries. On real-world datasets, the space usage is very close to or much better than the best competitors, while achieving up to 19 times faster queries than the next larger competitor. As far as the construction of LeMonHash is concerned, we get an improvement by a factor of up to 2, compared to the competitor with the next best space usage. We also investigate the case of keys being variable-length strings, introducing the so-called LeMonHash-VL: it needs space within 13% of the best competitors while achieving up to 3 times faster queries than the next larger competitor

Learned Monotone Minimal Perfect Hashing

A Monotone Minimal Perfect Hash Function (MMPHF) constructed on a set S of keys is a function that maps each key in S to its rank. On keys not in S, the function returns an arbitrary value. Applications range from databases, search engines, data encryption, to pattern-matching algorithms. In this paper, we describe LeMonHash, a new technique for constructing MMPHFs for integers. The core idea of LeMonHash is surprisingly simple and effective: we learn a monotone mapping from keys to their rank via an error-bounded piecewise linear model (the PGM-index), and then we solve the collisions that might arise among keys mapping to the same rank estimate by associating small integers with them in a retrieval data structure (BuRR). On synthetic random datasets, LeMonHash needs 35% less space than the next best competitor, while achieving about 16 times faster queries. On real-world datasets, the space usage is very close to or much better than the best competitors, while achieving up to 19 times faster queries than the next larger competitor. As far as the construction of LeMonHash is concerned, we get an improvement by a factor of up to 2, compared to the competitor with the next best space usage. We also investigate the case of keys being variable-length strings, introducing the so-called LeMonHash-VL: it needs space within 10% of the best competitors while achieving up to 3 times faster queries