Tight Fine-Grained Bounds for Direct Access on Join Queries

Karl Bringmann; Nofar Carmeli; Stefan Mengel

Home/Archive/arXiv

arXiv · 2201.02401

Tight Fine-Grained Bounds for Direct Access on Join Queries

Karl Bringmann·Nofar Carmeli·Stefan Mengel

View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the task of lexicographic direct access to query answers. That is, we want to simulate an array containing the answers of a join query sorted in a lexicographic order chosen by the user. A recent dichotomy showed for which queries and orders this task can be done in polylogarithmic access time after quasilinear preprocessing, but this dichotomy does not tell us how much time is required in the cases classified as hard. We determine the preprocessing time needed to achieve polylogarithmic access time for all join queries and all lexicographical orders. To this end, we propose a decomposition-based general algorithm for direct access on join queries. We then explore its optimality by proving lower bounds for the preprocessing time based on the hardness of a certain online Set-Disjointness problem, which shows that our algorithm's bounds are tight for all lexicographic orders on join queries. Then, we prove the hardness of Set-Disjointness based on the Zero-Clique Conjecture which is an established conjecture from fine-grained complexity theory. Interestingly, while proving our lower bound, we show that self-joins do not affect the complexity of direct access (up to logarithmic factors). Our algorithm can also be used to solve queries with projections and relaxed order requirements, though in these cases, its running time not necessarily optimal. We also show that similar techniques to those used in our lower bounds can be used to prove that, for enumerating answers to Loomis-Whitney joins, it is not possible to significantly improve upon trivially computing all answers at preprocessing. This, in turn, gives further evidence (based on the Zero-Clique Conjecture) to the enumeration hardness of self-join free cyclic joins with respect to linear preprocessing and constant delay.

Explore related subjects

cs.DB cs.CC

Keep this discovery

BibTeX RIS

Karl Bringmann, Nofar Carmeli, Stefan Mengel. 2022-01-07. Tight Fine-Grained Bounds for Direct Access on Join Queries. https://arxiv.org/abs/2201.02401

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

OmniTable: A Unified Wide-Table System for Petabyte-Scale LLM Data Curation and Exploration

Data curation is a critical bottleneck in industrial-grade LLM development, where petabyte-scale unstructured corpora are scattered across hundreds of physical tables, feature engineering relies on manual, table-centric pipeline orchestration, and data lineage is largely absent. We present OmniTable as an architecture blueprint for a unified wide-table layer built on Logical Unification, Physical Separation, targeting petabyte-scale LLM data curation and exploration. OmniTable makes four contributions: (1) a unified wide-table abstraction that consolidates multi-source heterogeneous data and thousands of derived features under a single logical schema via logical-physical mapping; (2) declarative feature lifecycle management that automates dependency resolution, execution planning, operator fusion, and lineage tracking, replacing manual pipeline orchestration with a "declare-and-execute" paradigm; (3) an adaptive execution engine with autonomous governance that achieves stable PB-scale feature backfill through heterogeneous compute routing (CPU/GPU), adaptive tuning, UDF-level fault tolerance, and automated storage layout optimization; and (4) hybrid-accelerated data exploration combining a global ID index, transparent OLAP offloading, and background materialized views to deliver second-level point lookups and filtered exports exceeding 20 TB/hour. In production, OmniTable manages over 35 PB of training data across web, code, PDF, and SFT domains, reducing the human-in-the-loop curation cycle from approximately 14 days to approximately 2.5 days (5.6x over the pre-OmniTable production workflow), with consistent feature versioning, auditable lineage, and minimal manual intervention.

cs.DB↗

arXiv2026

When Does Low-Bit Quantization Preserve the Decisions of Vector Search?

Low-bit quantization can achieve high recall on some vector representations and fail sharply on others, while average distortion and global rank correlation do not explain the difference. We study quantized vector search at the level of the comparisons consumed by ranking and graph-pruning algorithms. Our first result is a distribution-free decomposition: the probability that a comparison flips is bounded by the probability mass of exact margins near zero plus the tail probability of the calibrated residual. We then account for dependence between residuals that share a query or graph node, and derive covariance-aware second-moment identities and tail bounds under a joint MGF proxy. For a frozen candidate permutation, we prove a deterministic coupling theorem for Vamana neighbour selection: the approximate replay returns the exact neighbour list exactly when all candidate-level pruning actions agree on the frozen exact states. We connect these results to representation geometry through an exact Gaussian oracle, establish a strict correlation gain from a deterministic magnitude bit in an aligned bilinear model, and give a rare-contamination construction showing why marginal Gaussian diagnostics do not imply the required residual tails. When analytical assumptions are unavailable, a held-out block certificate bounds the selective failure risk of a frozen quantized rule. Across learned, classical, and synthetic embeddings, standardized exact margins predict held-out ranking and pruning flip rates substantially better than global rank correlation. The framework applies to coordinate binary codes, RaBitQ, Lucene BBQ, and product quantizers through a common decision interface.

cs.DB↗

arXiv2026

Contextual Utility of Quantization Moves in Extreme Low-Bit LLMs

Post-training quantizers select finite code changes using reconstruction proxies or local loss approximations, but the utility of a quantization move depends on the state through which it is executed. We identify two sources of this contextual dependence. First, the displacement of the move matters: evaluating the gradient at the move midpoint captures curvature accumulated along the move that a current-state linearization omits. Across frozen two-bit moves from Llama-3.2 models, midpoint evaluation predicts the direction of exact endpoint loss changes substantially more accurately than current-state gradients. Second, moves interact: exhaustive lattices of legal quantized states are well approximated by quadratic pseudo-Boolean functions, yet their small pairwise components can determine Pareto fronts and cause different evaluation functionals to prefer opposite directions. These effects explain failures of reconstruction-optimal code re-selection and additive composition. Reading each move at its own midpoint repairs the local selection step and improves downstream accuracy and held-out perplexity, while larger supports require evaluating exact endpoints from the state actually reached. Exact-endpoint beam search finds sparse changes that dominate much larger one-shot updates, and repricing the same moves after intervening changes produces widespread sign reversals. These results show that quantization utility is contextual at the granularity of a few moves: reliable construction must evaluate finite changes along their own paths and compose them from the evolving quantized state.

cs.DB↗