Unifying the Landscape of Super-Logarithmic Dynamic Cell-Probe Lower Bounds

2025-10-20

AI Summary

This work addresses the challenge of transforming lower bounds from one-way communication complexity into lower bounds for the dynamic cell-probe model. We consider a broad class of dynamic data structure problems involving a public binary string and update/query operations. When the query function (f) exhibits high one-way communication complexity under a product distribution (e.g., uniform), we introduce the first generic reduction framework. It integrates cell-sampling, the chronogram method, and static lower-bound techniques, augmented by an entropy-based analysis centered on average min-entropy. Our framework systematically yields super-logarithmic time lower bounds: for problem size (m = Omega(n^{0.99})), it implies (max{t_u, t_q} geq ilde{Omega}(log^{3/2} n)). Notably, we establish the first super-logarithmic dynamic lower bound for the Multiphase problem (inner-product variant), unifying and extending prior isolated results from [LWY20] and [LY25].

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Stochastic OptimizationMachine Learning: Other Foundations of Machine Learning

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Large-scale security measurementsUser Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We prove a general translation theorem for converting one-way communication lower bounds over a product distribution to dynamic cell-probe lower bounds. Specifically, we consider a class of problems considered in [Pat10] where: 1. $S_1, ldots, S_m in {0, 1}^n$ are given and publicly known. 2. $T in {0, 1}^n$ is a sequence of updates, each taking $t_u$ time. 3. For a given $Q in [m]$, we must output $f(S_Q, T)$ in $t_q$ time. Our main result shows that for a "hard" function $f$, for which it is difficult to obtain a non-trivial advantage over random guessing with one-way communication under some product distribution over $S_Q$ and $T$ (for example, a uniform distribution), then the above explicit dynamic cell-probe problem must have $max { t_u, t_q } geq ildeΩ(log^{3/2}(n))$ if $m = Ω(n^{0.99})$. This result extends and unifies the super-logarithmic dynamic data structure lower bounds from [LWY20] and [LY25] into a more general framework. From a technical perspective, our approach merges the cell-sampling and chronogram techniques developed in [LWY20] and [LY25] with the new static data structure lower bound methods from [KW20] and [Ko25], thereby merging all known state-of-the-art cell-probe lower-bound techniques into one. As a direct consequence of our method, we establish a super-logarithmic lower bound against the Multiphase Problem [Pat10] for the case where the data structure outputs the Inner Product (mod 2) of $S_Q$ and $T$. We suspect further applications of this general method towards showing super-logarithmic dynamic cell-probe lower bounds. We list some example applications of our general method, including a novel technique for a one-way communication lower bound against small-advantage protocols for a product distribution using average min-entropy, which could be of independent interest.

Problem

• Converting communication lower bounds to dynamic cell-probe complexity bounds
• Establishing super-logarithmic lower bounds for dynamic data structures
• Providing unified framework for cell-probe lower bound techniques

Innovation

• Converts one-way communication bounds to cell-probe lower bounds
• Merges cell-sampling and chronogram techniques with static methods
• Uses average min-entropy for small-advantage protocol analysis