An $Ω( (\log n / \log \log n)^2 )$ Cell-Probe Lower Bound for Dynamic Boolean Data Structures

2026-03-26

AI Summary

This work addresses the long-standing open problem of proving query time lower bounds for the Multiphase problem in dynamic Boolean data structures. We introduce a novel 2.5-round communication game that augments standard one-way communication with an additional verification round to enforce consistency in memory simulation. Departing from traditional approaches reliant on intricate tools such as the Peak-to-Average Lemma, our method integrates communication complexity, discrepancy-based lower bounds, and the Fredman–Saks chronogram technique. This yields the first unconditional $\Omega((\log n / \log \log n)^2)$ lower bound on query time in the cell-probe model. Our result not only matches the classic barrier established by Larsen for weighted problems but also extends to a broad class of hard Boolean functions, achieving a tight lower bound in the Boolean setting.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesSearch and Optimization: Learning to Search

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSecurity and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Models for Web evolution

Abstract

We resolve the long-standing open problem of Boolean dynamic data structure hardness, proving an unconditional lower bound of $Ω((\log n / \log\log n)^2)$ for the Multiphase Problem of Patrascu [STOC 2010] (instantiated with Inner Product over $\mathbb{F}_2$). This matches the celebrated barrier for weighted problems established by Larsen [STOC 2012] and closes the gap left by the $Ω(\log^{1.5} n)$ Boolean bound of Larsen, Weinstein, and Yu [STOC 2018]. The previous barrier was methodological: all prior works relied on ``one-way'' communication games, where the inability to verify query simulations necessitated complex machinery (such as the Peak-to-Average Lemma) that hit a hard ceiling at $\log^{1.5} n$. Our key contribution is conceptual: We introduce a 2.5-round Multiphase Communication Game that augments the standard one-way model with a verification round, where Bob confirms the consistency of Alice's simulation against the actual memory. This simple, qualitative change allows us to bypass technical barriers and obtain the optimal bound directly. As a consequence, our analysis naturally extends to other hard Boolean functions, offering a general recipe for translating discrepancy lower bounds into $Ω((\log n / \log\log n)^2)$ dynamic Boolean data structure lower bounds. We also argue that this result likely represents the structural ceiling of the Chronogram framework initiated by Fredman and Saks [STOC 1989]: any $ω(\log^2 n)$ lower bound would require either fundamentally new techniques or major circuit complexity breakthroughs.

Problem

• dynamic data structures
• cell-probe lower bound
• Boolean functions
• Multiphase Problem
• communication complexity

Innovation

• dynamic data structures
• cell-probe lower bound
• communication complexity
• verification round
• boolean functions