Cell-Probe Lower Bounds via Semi-Random CSP Refutation: Simplified and the Odd-Locality Case

2025-07-29

AI Summary

This work establishes lower bounds for static data structures in the cell-probe model, resolving the long-standing open problem for odd-arity settings. Methodologically, it reduces lower-bound derivation to an XOR refutation problem, uncovering deep connections to range avoidance in NC⁰ circuits and refutation of semi-random constraint satisfaction problems (CSPs). It achieves the first fully derandomized analysis of semi-random k-XOR refutation, closing a major theoretical gap for odd k. The approach integrates techniques from constant-depth multi-output circuit analysis, input-expansion-based decision procedures, higher-order independence bias detection, and efficient certifiably-falsifiable mechanisms. The results yield significantly improved cell-probe lower bounds across several key parameter regimes; they unify and simplify prior frameworks, while also providing novel algorithmic insights and theoretical foundations for NC⁰ circuit construction and pseudorandomness design.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

Abstract

A recent work (Korten, Pitassi, and Impagliazzo, FOCS 2025) established an insightful connection between static data structure lower bounds, range avoidance of $ ext{NC}^0$ circuits, and the refutation of pseudorandom CSP instances, leading to improvements to some longstanding lower bounds in the cell-probe/bit-probe models. Here, we improve these lower bounds in certain cases via a more streamlined reduction to XOR refutation, coupled with handling the odd-arity case. Our result can be viewed as a complete derandomization of the state-of-the-art semi-random $k$-XOR refutation analysis (Guruswami, Kothari and Manohar, STOC 2022, Hsieh, Kothari and Mohanty, SODA 2023), which complements the derandomization of the even-arity case obtained by Korten et al. As our main technical statement, we show that for any multi-output constant-depth circuit that substantially stretches its input, its output is very likely far from strings sampled from distributions with sufficient independence, and further this can be efficiently certified. Via suitable shifts in perspectives, this gives applications to cell-probe lower bounds and range avoidance algorithms for $mathsf{NC}^0$ circuits.

Problem

• Improving cell-probe lower bounds via XOR refutation
• Derandomizing semi-random k-XOR refutation analysis
• Certifying output distance from high-independence distributions

Innovation

• Streamlined reduction to XOR refutation
• Handling odd-arity case in CSP refutation
• Derandomization of semi-random k-XOR analysis